--- title: "Just Do It: A Collection of Hartshorne Problems" author: - name: D. Zack Garza affiliation: University of Georgia email: dzackgarza@gmail.com date: "Fall 2021" author-note: "" book: true --- # I: Varieties ## I.1: Affine Varieties ### 1.1 #to_work (a) Let $Y$ be the plane curve $y=x^2$ (i.e., $Y$ is the zero set of the polynomial $f=$ $y-x^2$ ). Show that $A(Y)$ is isomorphic to a polynomial ring in one variable over $k$. (b) Let $Z$ be the plane curve $x y=1$. Show that $A(Z)$ is not isomorphic to a polynomial ring in one variable over $k$. (c) \* Let $f$ be any irreducible quadratic polynomial in $k[x, y]$, and let $W$ be the conic defined by $f$. Show that $A(W)$ is isomorphic to $A(Y)$ or $A(Z)$. Which one is it when? ::: {.solution .proofenv data-callout="solution" title="Solution (Part a)"} $A(Y) = k[x,y]/\left\langle{y-x^2}\right\rangle \cong k[t, t^2] \cong k[t]$. ::: ::: {.solution .proofenv data-callout="solution" title="Solution (Part b)"} $A(Z) = k[x, y]/\left\langle{xy-1}\right\rangle \cong k[x^{\pm 1}]$. This is not isomorphic to any polynomial ring, since it contains an invertible element $x^{-1}$ which is not in $k$. ::: ::: {.solution .proofenv data-callout="solution" title="Solution (Part c, incomplete) Idea: for F a conic (degree 2 polynomial), an affine change of coordinates yields either F(x, y) =xy-1 or F(x,y) = y-x^2. One can write F(x,y) ={\\left[ {x,y, 1} \\right]}^t\\left(\\begin{array}{ccc} A & B / 2 & D / 2 \\\\ B / 2 & C & E / 2 \\\\ D / 2 & E / 2 & F \\end{array}\\right){\\left[ {x,y, 1} \\right]} This is symmetric and can thus be diagonalized, correspond to an affine change of coordinates where F(x,y) = \\lambda_1 x^2 + \\lambda_2 y^2 + \\lambda_3 1."} ::: ### 1.2 The Twisted Cubic Curve #completed Let $Y \subseteq \mathbf{A}^3$ be the set $Y = \left\{{(t, t^2,t^3) {~\mathrel{\Big\vert}~}t\in k}\right\}$. - Show that $Y$ is an affine variety of dimension 1. - Find generators for the ideal $I\left(Y\right)$. - Show that $A(Y)$ is isomorphic to a polynomial ring in one variable over $k$. We say that $Y$ is given by the *parametric representation* $x=t . y=t^2, z=t^3$. > Useful facts: $\sqrt{I} = \sqrt{\prod p_i^{a_i}} = \prod p_i$ in a UFD when $I$ is a principal ideal factored into irreducibles. An ideal is also radical iff the quotient is reduced, and $\left\langle{f}\right\rangle$ is radical when $f$ is irreducible. ::: {.solution .proofenv data-callout="solution" title="Solution"} - For dimension in part 1, note ` \begin{align*}A(Y) = k[x,y]/\left\langle{y-x^2, z-x^3}\right\rangle \cong k[t, t^2,t^3] \cong k[t].\end{align*} `{=html}One can now use that $\operatorname{krulldim}k[t] = 1$ and Prop 1.7 ($\dim Y = \operatorname{krulldim}A(Y)$) to conclude $\dim Y = 1$. This also shows part (3) - To see that $Y$ is an affine variety, it STS it is an irreducible closed subset of ${\mathbf{A}}^3$. It is closed since $Y = V(y-x^2, z-x^3)$. It is irreducible iff $I(Y) = \left\langle{y-x^2, z-x^3}\right\rangle$ is prime by Cor 1.4. It is prime, since $k[x,y]/I(Y) \cong k[t]$ is a domain and $I(Y)$ is radical since it is generated by irreducibles. ::: ### 1.3 #completed Let $Y$ be the algebraic set in $\mathbf{A}^3$ defined by the two polynomials $x^2-yz$ and $x z-x$. Show that $Y$ is a union of three irreducible components. Describe them and find their prime ideals. ::: {.solution .proofenv data-callout="solution" title="Solution Y = V(x^2-yz)\\cap V(xz-x) = V(x^2-yz) \\cap(V(x) \\cup V(z-1)). Any point in Y is in either V(x) or V(z-1), so either x=0 or z=1. If a point is in V(x), then x=0 and 0^2=yz, so either - x=0 and y=0, and z is free, or - x=0 and z=0, and y is free. In either case, this is an intersection of two orthogonal hyperplanes in {\\mathbf{A}}^3 and thus a line. So we obtain the two lines V(x, y) \\cup V(x, z). If a point is in V(z-1) then z=1 then x^2 = y yielding a parabola V(x^2-y)."} So $Y =V(x, y) \cup V(x, z) \cup V(x^2-y)$ is a union of two lines and a parabola. ::: ### 1.4 #completed If we identify $\mathbf{A}^2$ with $\mathbf{A}^1 \times \mathbf{A}^1$ in the natural way, show that the Zariski topology on $\mathbf{A}^2$ is not the product topology of the Zariski topologies on the two copies of $\mathbf{A}^1$. ::: {.Hint .proofenv data-callout="hint" title="Hints Zariski-closed subsets of {\\mathbf{A}}^2 are finite unions of plane curves or the entire space. The Zariski topology on {\\mathbf{A}}^1 is the cofinite topology if k is infinite, and is not Hausdorff."} ::: ::: {.solution .proofenv data-callout="solution" title="Solution If k is infinite and algebraically closed, then {\\mathbf{A}}^1_{/ {k}} is not Hausdorff: the Zariski topology is the cofinite topology, so every open set has finite complement, and every two distinct open sets intersect. Now use that \\Delta_{{\\mathbf{A}}^1_{/ {k}} } is closed in the product topology on {\\mathbf{A}}^1\\times {\\mathbf{A}}^1 iff the base is Hausdorff, but \\Delta is closed in the Zariski topology since it can be written as V(x-y)."} Alternative: consider $V(xy-1) \subseteq {\mathbf{A}}^2$. If this had the product topology, the projection maps $\pi_x, \pi_y: {\mathbf{A}}^2\to {\mathbf{A}}^1$ would be closed, but $\pi_x(V(xy-1)) = {\mathbf{A}}^1\setminus\left\{{0}\right\}$ which is open and not closed since its complement $0 = V(x)$ is closed and not open. ::: ### 1.5 #to_work Show that a $k$-algebra $B$ is isomorphic to the affine coordinate ring of some algebraic set in $\mathbf{A}^n$. for some $n$, if and only if $B$ is a finitely generated $k$-algebra with no nilpotent elements. ### 1.6 #to_work Any nonempty open subset of an irreducible topological space is dense and irreducible. If $Y$ is a subset of a topological space $X$, which is irreducible in its induced topology, then the closure $\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu$ is also irreducible. ### 1.7 #to_work (a) Show that the following conditions are equivalent for a topological space $X$ : - $X$ is noetherian: - Every nonempty family of closed subsets has a minimal element: - $X$ satisfies the ascending chain condition for open subsets: - Every nonempty family of open subsets has a maximal element. (b) A noetherian topological space is quasi-compact, i.e., every open cover has a finite subcover. (c) Any subset of a noetherian topological space is noetherian in its induced topology. (d) A noetherian space which is also Hausdorff must be a finite set with the discrete topology. ### 1.8 #to_work Let $Y$ be an affine variety of dimension $r$ in $\mathbf{A}^n$. Let $H$ be a hypersurface in $\mathbf{A}^n$, and assume that $Y \nsubseteq H$. Then every irreducible component of $Y \cap H$ has dimension $r-1$.[^1] ### 1.9 #to_work Let $a \subseteq A=k\left[x_1, \ldots, x_n\right]$ be an ideal which can be generated by $r$ elements. Then every irreducible component of $Z(a)$ has dimension $\geqslant n-r$. ### 1.10 #to_work (a) If $Y$ is any subset of a topological space $X$, then $\operatorname{dim} Y \leqslant \operatorname{dim} X$. (b) If $X$ is a topological space which is covered by a family of open subsets $\left\{L_1 ;\right.$, then $\operatorname{dim} X=\sup \operatorname{dim} U_i$. (c) Give an example of a topological space $X$ and a dense open subset $U$ with $\operatorname{dim} L^{\prime}<\operatorname{dim} X$. (d) If $Y$ is a closed subset of an irreducible finite-dimensional topological space $X$, and if $\operatorname{dim} Y=\operatorname{dim} X$, then $Y=X$. (e) Give an example of a noetherian topological space of infinite dimension. ### 1.11 \* #to_work Let $Y \subseteq \mathbf{A}^3$ be the curve given parametrically by $x=t^3, y=t^4, z=t^5$. Show that $I(Y)$ is a prime ideal of height 2 in $k[x, y ;-]$ which cannot be generated by 2 elements. We say $Y$ is not a local complete intersection-cf. (Ex. 2.17). ### 1.12 #to_work Give an example of an irreducible polynomial $f \in \mathbf{R}[x, y]$. whose zero set $Z(f)$ in $\mathbf{A}_{\mathbf{R}}^2$ is not irreducible (cf. 1.4.2). ## I.2: Projective Varieties ### 2.1 #to_work Prove the "homogeneous Nullstellensatz," which says if $a \subseteq S$ is a homogeneous ideal, and if $f \in S$ is a homogeneous polynomial with deg $f>0$, such that $f(P)=0$ for all $P \in Z(a)$ in $\mathbf{P}^n$, then $f^u \in a$ for some $q>0$. [^2] ### 2.2 #to_work For a homogeneous ideal $a \subseteq S$, show that the following conditions are equivalent: (i) $Z(a) = \varnothing$ (the empty set); (ii) $\sqrt{a}=$ either $S$ or the ideal $S_{+}=\bigoplus_{d>0} S_d$; (iii) $a \supseteq S_d$ for some $d>0$. ### 2.3 #to_work (a) If $T_1 \subseteq T_2$ are subsets of $S^h$. then $Z\left(T_1\right) \supseteq Z\left(T_2\right)$. (b) If $Y_1 \subseteq Y_2$ are subsets of $\mathbf{P}^n$, then $I\left(Y_1\right) \supseteq I\left(Y_2\right)$. (c) For any two subsets $Y_1, Y_2$ of $\mathbf{P}^n, I\left(Y_1 \cup Y_2\right)=I\left(Y_1\right) \cap I\left(Y_2\right)$. (d) If $a \subseteq S$ is a homogeneous ideal with $Z(a) \neq \varnothing$. then $I(Z(a))=\sqrt a$. (e) For any subset $Y \subseteq \mathbf{P}^n, Z(I(Y))=\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu$. ### 2.4 #to_work (a) There is a 1-1 inclusion-reversing correspondence between algebraic sets in $\mathbf{P}^n$. and homogeneous radical ideals of $S$ not equal to $S_{+}$ given by $Y \mapsto I(Y)$ and $a \mapsto Z(a)$. [^3] (b) An algebraic set $Y \subseteq \mathbf{P}^n$ is irreducible if and only if $I\left(Y^{\prime}\right)$ is a prime ideal. (c) Show that $\mathbf{P}^n$ itself is irreducible. ### 2.5 #to_work (a) $\mathbf{P}^n$ is a noetherian topological space. (b) Every algebraic set in $\mathrm{P}^n$ can be written uniquely as a finite union of irreducible algebraic sets. no one containing another. These are called its irreducible components. ### 2.6 #to_work If $Y$ is a projective variety with homogeneous coordinate ring $S(Y)$, show that $\operatorname{dim} S(Y)=\operatorname{dim} Y+1$.[^4] ### 2.7 #to_work (a) $\operatorname{dim} \mathbf{P}^n=n$. (b) If $Y \subseteq \mathbf{P}^n$ is a quasi-projective variety, then $\operatorname{dim} Y=\operatorname{dim} \mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu$.[^5] ### 2.8 #to_work A projective variety $Y \subseteq \mathbf{P}^n$ has dimension $n-1$ if and only if it is the zero set of a single irreducible homogeneous polynomial $f$ of positive degree. $Y$ is called a hypersurface in $\mathbf{P}^n$. ### 2.9 Projective Closure of an Affine Variety #to_work If $Y \subseteq \mathbf{A}^n$ is an affine variety, we identify $\mathbf{A}^n$ with an open set $U_0 \subseteq \mathbf{P}^n$ by the homeomorphism $\varphi_0$. Then we can speak of $\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu$, the closure of $Y$ in $\mathbf{P}^n$, which is called the projective closure of $Y$. (a) Show that $I(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu)$ is the ideal generated by $\beta(I(Y))$, using the notation of the proof of $(2.2)$. (b) Let $Y \subseteq \mathbf{A}^3$ be the twisted cubic of (Ex. 1.2). Its projective closure $\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu \subseteq \mathbf{P}^3$ is called the twisted cubic curve in $\mathbf{P}^3$. Find generators for $I(Y)$ and $I(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu)$, and use this example to show that if $f_1, \ldots, f_r$ generate $I(Y)$, then $\beta\left(f_1\right), \ldots, \beta\left(f_r\right)$ do not necessarily generate $I(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu)$. ### 2.10 The Cone Over a Projective Variety (Fig. 1) #to_work Let $Y \subseteq \mathbf{P}^n$ be a nonempty algebraic set, and let $\theta: \mathbf{A}^{n+1}-\{(0, \ldots, 0)\} \rightarrow \mathbf{P}^n$ be the map which sends the point with affine coordinates $\left(a_0, \ldots, a_n\right)$ to the point with homogeneous coordinates $\left(a_0, \ldots, a_n\right)$. We define the affine cone over $Y$ to be ` \begin{align*} C(Y)=\theta^{-1}(Y) \cup\{(0, \ldots, 0)\} . \end{align*} `{=html} (a) Show that $C(Y)$ is an algebraic set in $\mathbf{A}^{n+1}$, whose ideal is equal to $I(Y)$, considered as an ordinary ideal in $k\left[x_0, \ldots, x_n\right]$. (b) $C(Y)$ is irreducible if and only if $Y$ is. (c) $\operatorname{dim} C(Y)=\operatorname{dim} Y+1$. Sometimes we consider the projective closure $\overline{C(Y)}$ of $C(Y)$ in $\mathbf{P}^{n+1}$. This is called the **projective cone** over $Y$. ![](attachments/2022-09-17_22-10-43.png) ### 2.11 Linear Varieties in $\mathbf{P}^n$ #to_work {#linear-varieties-in-mathbfpn-to_work} A hypersurface defined by a linear polynomial is called a hyperplane. (a) Show that the following two conditions are equivalent for a variety $Y$ in $\mathbf{P}^n$ : (i) $I(Y)$ can be generated by linear polynomials. (ii) $Y$ can be written as an intersection of hyperplanes. In this case we say that $Y$ is a **linear variety** in $\mathbf{P}^n$. (b) If $Y$ is a linear variety of dimension $r$ in $\mathbf{P}^n$, show that $I(Y)$ is minimally generated by $n-r$ linear polynomials. (c) Let $Y, Z$ be linear varieties in $\mathbf{P}^n$, with $\operatorname{dim} Y=i, \operatorname{dim} Z=$ s. If $r+s-n \geqslant 0$, then $Y \cap Z \neq \varnothing$. Furthermore, if $Y \cap Z \neq \varnothing$, then $Y \cap Z$ is a linear variety of dimension $\geqslant r+s-n$. [^6] ### 2.12 The $d$-uple Embedding #to_work {#the-d-uple-embedding-to_work} For given $n, d>0$, let $M_0, M_1, \ldots, M_N$ be all the monomials of degree $d$ in the $n+1$ variables $x_0, \ldots, x_n$, where $N={n+d\choose n}-1$. We define a mapping $\rho_d: \mathbf{P}^n \rightarrow \mathbf{P}^{N}$ by sending the point $P=\left(a_0, \ldots, a_n\right)$ to the point $\rho_d(P)=\left(M_0(a), \ldots, M_N(a)\right)$ obtained by substituting the $a_t$ in the monomials $M_J$. This is called the $d$-uple embedding of $\mathbf{P}^n$ in $\mathbf{P}^N$. For example, if $n=1, d=2$, then $N=2$, and the image $Y$ of the 2-uple embedding of $\mathbf{P}^1$ in $\mathbf{P}^2$ is a conic. (a) Let $\theta: k\left[y_0, \ldots, y_v\right] \rightarrow k\left[x_0, \ldots, x_n\right]$ be the homomorphism defined by sending $y_i$ to $M_i$, and let a be the kernel of $\theta$. Then $a$ is a homogeneous prime ideal, and so $Z$ (a) is a projective variety in $\mathbf{P}^{N}$. (b) Show that the image of $\rho_d$ is exactly $Z(a)$. [^7] (c) Now show that $\rho_d$ is a homeomorphism of $\mathbf{P}^n$ onto the projective variety $Z$ (a). (d) Show that the twisted cubic curve in $\mathbf{P}^3$ (Ex. 2.9) is equal to the 3-uple embedding of $\mathbf{P}^1$ in $\mathbf{P}^3$, for suitable choice of coordinates. ### 2.13 #to_work Let $Y$ be the image of the 2-uple embedding of $\mathbf{P}^2$ in $\mathbf{P}^5$. This is the Veronese surface. If $Z \subseteq Y$ is a closed curve (a **curve** is a variety of dimension 1), show that there exists a hypersurface $V \subseteq \mathbf{P}^5$ such that $V \cap Y = Z$. ### 2.14 The Segre Embedding #to_work Let $\psi: \mathbf{P}^r \times \mathbf{P}^{s} \rightarrow \mathbf{P}^{N}$ be the map defined by sending the ordered pair $\left(a_0, \ldots, a_r\right) \times\left(b_0, \ldots, b_s\right)$ to $\left(\ldots, a_i b_j, \ldots\right)$ in lexicographic order. where $N=r s+r+s$. Note that $\psi$ is well-defined and injective. It is called the Segre embedding. Show that the image of $\psi$ is a subvariety of $\mathbf{P}^N$. [^8] ### 2.15 The Quadric Surface in $\mathbf{P}^3$ (Fig. 2) #to_work {#the-quadric-surface-in-mathbfp3-fig.-2-to_work} Consider the surface $Q$ (a surface is a variety of dimension 2) in $\mathbf{P}^3$ defined by the equation $x y-zw =0$. (a) Show that $Q$ is equal to the Segre embedding of $\mathbf{P}^1 \times \mathbf{P}^1$ in $\mathbf{P}^3$. for suitable choice of coordinates. (b) Show that $Q$ contains two families of lines (a line is a linear variety of dimension 1) $\left\{L_t\right\},\left\{{M_t}\right\}$, each parametrized by $t \in \mathbf{P}^1$. with the properties that if $L_t \neq L_u$. then $L_t \cap L_u=\varnothing$ : if $M_t \neq M_u, M_t \cap M_u=\varnothing$, and for all $t,u$, $L_t \cap M_u=$ one point. (c) Show that $Q$ contains other curves besides these lines, and deduce that the Zariski topology on $Q$ is not homeomorphic via $\psi$ to the product topology on $\mathbf{P}^1 \times \mathbf{P}^1$ (where each $\mathbf{P}^1$ has its Zariski topology). ![](attachments/2022-09-17_22-24-16.png) ### 2.16 #to_work (a) The intersection of two varieties need not be a variety. For example, let $Q_1$ and $Q_2$ be the quadric surfaces in $\mathbf{P}^3$ given by the equations $x^2-y w=0$ and $x y-z w=0$, respectively. Show that $Q_1 \cap Q_2$ is the union of a twisted cubic curve and a line. (b) Even if the intersection of two varieties is a variety, the ideal of the intersection may not be the sum of the ideals. For example, let $C$ be the conic in $\mathbf{P}^2$ given by the equation $xy-zw=0$. Let $L$ be the line given by $y=0$. Show that $C \cap L$ consists of one point $P$, but that $I(C)+I(L) \neq I(P)$. ### 2.17 Complete intersections #to_work A variety $Y$ of dimension $r$ in $\mathbf{P}^n$ is a (strict) complete intersection if $I(Y)$ can be generated by $n-r$ elements. $Y$ is a set-theoretic complete intersection if $Y$ can be written as the intersection of $n-r$ hypersurfaces. (a) Let $Y$ be a variety in $\mathbf{P}^n$, let $Y=Z(a)$; and suppose that a can be generated by $q$ elements. Then show that $\operatorname{dim} Y \geqslant n-q$. (b) Show that a strict complete intersection is a set-theoretic complete intersection. (c) \* The converse of (b) is false. For example let $Y$ be the twisted cubic curve in $\mathbf{P}^3$ (Ex. 2.9). Show that $I(Y)$ cannot be generated by two elements. On the other hand, find hypersurfaces $\mathrm{H}_1, \mathrm{H}_2$ of degrees 2,3 respectively, such that $Y=H_1 \cap H_2$. (d) \*\* It is an unsolved problem whether every closed irreducible curve in $\mathbf{P}^3$ is a set-theoretic intersection of two surfaces. See Hartshorne $[1]$ and Hartshorne $[5.III, \text{section } 5]$ for commentary. ## I.3: Morphisms ### 3.1 #to_work 1. Show that any conic in $\mathbf{A}^2$ is isomorphic either to $\mathbf{A}^1$ or $\mathbf{A}^1\setminus\left\{{0}\right\}$ (cf. Ex.1.1). 2. Show that $\mathbf{A}^1$ is not isomorphic to any proper open subset of itself.[^9] 3. Any conic in ${\mathbb{P}}^2$ is isomorphic to ${\mathbb{P}}^1$. 4. We will see later (Ex. 4.8) that any two curves are homeomorphic. But show now that $\mathbf{A}^2$ is not even homeomorphic to ${\mathbb{P}}^2$. 5. If an affine variety is isomorphic to a projective variety, then it consists of only one point. ### 3.2 #to_work A morphism whose underlying map on the topological spaces is a homeomorphism need not be an isomorphism. 1. For example, let $\varphi: \mathbf{A}^1 \rightarrow \mathbf{A}^2$ be defined by $t \mapsto\left(t^2, t^3\right)$. Show that $\varphi$ defines a bijective bicontinuous morphism of $\mathbf{A}^1$ onto the curve $y^2=x^3$, but that $\varphi$ is not an isomorphism. 2. For another example. let the characteristic of the base field $k$ be $p>0$, and define a map $\rho: \mathbf{A}^1 \rightarrow \mathbf{A}^1$ by $t \mapsto t^p$. Show that $\varphi$ is bijective and bicontinuous but not an isomorphism. This is called the Frobenius morphism. ### 3.3 #to_work 1. Let $\varphi: X \rightarrow Y$ be a morphism. Then for each $P \in X, \varphi$ induces a homomorphism of local rings $\varphi_P^*: {\mathcal{O}}_{\phi(P), Y} \rightarrow {\mathcal{O}}_{P, Y}$. 2. Show that a morphism $\varphi$ is an isomorphism if and only if $\varphi$ is a homeomorphism, and the induced map $\varphi_P^*$ on local rings is an isomorphism, for all $P \in X$. 3. Show that if $\varphi(X)$ is dense in $Y$, then the map $\rho_P^*$ is injective for all $P \in X$. ### 3.4 #to_work Show that the $d{\hbox{-}}$uple embedding of ${\mathbb{P}}^n(\mathrm{Ex} .2 .12)$ is an isomorphism onto its image. ### 3.5 #to_work By abuse of language, we will say that a variety "is affine" if it is isomorphic to an affine variety. If $H \subseteq {\mathbb{P}}^n$ is any hypersurface. show that ${\mathbb{P}}^n-H$ is affine.[^10] ### 3.6 There are quasi-affine varieties which are not affine. #to_work For example, show that $\mathrm{I}=\mathbf{A}^2\setminus\left\{{ (0, 0) }\right\}$ is not affine.[^11] ### 3.7 #to_work 1. Show that any two curves in ${\mathbb{P}}^2$ have a nonempty intersection. 2. More generally, show that if $Y \subseteq {\mathbb{P}}^n$ is a projective variety of dimension $\geqslant 1$. and if $H$ is a hypersurface. then $Y \cap H \neq \varnothing$.[^12] ### 3.8 #to_work Let $H_1$ and $H$, be the hyperplanes in ${\mathbb{P}}^n$ defined by $x_1=0$ and $x_{,}=0$, with $i \neq j$. Show that any regular function on ${\mathbb{P}}^n-\left(H_1 \cap H_1\right)$ is constant.[^13] ### 3.9 #to_work The homogeneous coordinate ring of a projective variety is not invariant under isomorphism. For example, let $X={\mathbb{P}}^1$. and let $Y$ be the 2-uple embedding of ${\mathbb{P}}^1$ in ${\mathbb{P}}^2$. Then $X \cong Y($ Ex. 3.4). But show that $S(X) \equiv S(Y)$. ### 3.10 Subvarieties. #to_work A subset of a topological space is locally closed if it is an open subset of its closure. or. equivalently. if it is the intersection of an open set with a closed set. If $X$ is a quasi-affine or quasi-projective variety and $Y$ is an irreducible locally closed subset. then $I$ is also a quasi-affine (respectively, quasi-projective) variety by virtue of being a locally closed subset of the same affine or projective space. We call this the induced structure on Y. and we call $Y$ a subvariety of $X$. Now let $\varphi: X \rightarrow Y$ he a morphism. let $X^{\prime} \subseteq X$ and $Y^{\prime} \subseteq Y$ be irreducible locally closed subsets such that $\varphi\left(X^{\prime}\right) \subseteq Y^{\prime}$. Show that $\left.\varphi\right|_{X}: X^{\prime } \rightarrow Y^{\prime}$ is a morphism. ### 3.11 #to_work Let $X$ be any variety and let $P \in X$. Show there is a 1-1 correspondence between the prime ideals of the local ring ${\mathcal{O}}_P$ and the closed subvarieties of $X$ containing $P$. ### 3.12 #to_work If $P$ is a point on a variety $X$, then $\operatorname{dim} {\mathcal{O}}_P =\operatorname{dim} X$.[^14] ### 3.13 The Local Ring of a Subvariety #to_work Let $Y \subseteq X$ be a subvariety. Let ${\mathcal{O}}_{Y,X}$ be the set of equivalence classes $\langle L, f\rangle$ where $L \subseteq X$ is open. $L \cap Y \neq \varnothing$, and $f$ is a regular function on $L$. We say $\langle L , f\rangle$ is equivalent to $\left\langle{V, g}\right\rangle$ if $f=g$ on $U \cap V$. Show that ${\mathcal{O}}_{Y , X}$ is a local ring, with residue field $K(Y)$ and $\operatorname{dimension}=\operatorname{dim} \mathrm{X}-$ $\operatorname{dim} Y$. It is the local ring of $Y$ on $X$. Note if $Y=P$ is a point we get ${\mathcal{O}}_P$. and if $Y=X$ we get $K(X)$. Note also that if $Y$ is not a point, then $K(Y)$ is not algebraically closed, so in this way we get local rings whose residue fields are not algebraically closed. ### 3.14 Projection from a Point. #to_work Let ${\mathbb{P}}^n$ be a hyperplane in ${\mathbb{P}}^{n+1}$ and let $P \in {\mathbb{P}}^{n+1}-{\mathbb{P}}^n$. Define a mapping $\varphi: {\mathbb{P}}^{n+1}\setminus\left\{{ P }\right\}\to {\mathbb{P}}^n$ by $\varphi(Q)=$ the intersection of the unique line containing $P$ and $Q$ with ${\mathbb{P}}^n$. 1. Show that $\varphi$ is a morphism. 2. Let $Y \subseteq {\mathbb{P}}^3$ be the twisted cubic curve which is the image of the 3-uple embedding of ${\mathbb{P}}^1$ (Ex. 2.12). If $t,u$ are the homogeneous coordinates on ${\mathbb{P}}^1$. we say that $Y$ is the curve given parametrically by $(x, y, z, w)=\left(t^3, t^2 u, t u^2, u^3\right)$. Let $P=(0,0,1,0)$, and let ${\mathbb{P}}^2$ be the hyperplane $z=0$. Show that the projection of $Y$ from $P$ is a cuspidal cubic curve in the plane, and find its equation. ### 3.15 Products of Affine Varieties. #to_work Let $X \subseteq \mathbf{A}^n$ and $Y \subseteq \mathbf{A}^m$ be affine varieties. 1. Show that $X \times Y \subseteq \mathbf{A}^{n+m}$ with its induced topology is irreducible.[^15] The affine variety $X \times Y$ is called the product of $X$ and $Y$. Note that its topology is in general not equal to the product topology (Ex. 1.4). 2. Show that $A(X \times Y) \cong A(X) \otimes_k A(Y)$. 3. Show that $X \times Y$ is a product in the category of varieties, i.e., show - the projections $X \times Y \rightarrow X$ and $X \times Y \rightarrow Y$ are morphisms, and - given a variety $Z$, and the morphisms $Z \rightarrow X, Z \rightarrow Y$. there is a unique morphism $Z \rightarrow X \times Y$ making a commutative diagram ```{=tex} \begin{tikzcd} Z && {X\times Y} \\ \\ & X && Y \arrow[from=1-1, to=3-2] \arrow[from=1-1, to=3-4] \arrow[from=1-3, to=3-2] \arrow[from=1-3, to=3-4] \arrow[from=1-1, to=1-3] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJaIl0sWzIsMCwiWFxcdGltZXMgWSJdLFsxLDIsIlgiXSxbMywyLCJZIl0sWzAsMl0sWzAsM10sWzEsMl0sWzEsM10sWzAsMV1d) 4. Show that $\operatorname{dim} X \times Y=\operatorname{dim} X+\operatorname{dim} Y$. ### 3.16 Products of Quasi-Projective Varieties. #to_work Use the Segre embedding (Ex. 2.14) to identify ${\mathbb{P}}^n \times {\mathbb{P}}^m$ with its image and hence give it a structure of projective varieties. Now for any two quasi-projective varieties $X \subseteq {\mathbb{P}}^n$ and $Y \subseteq {\mathbb{P}}^m$, consider $X \times Y \subseteq {\mathbb{P}}^n \times {\mathbb{P}}^m$. 1. Show that $X \times Y$ is a quasi-projective variety. 2. If $X, Y$ are both projective, show that $X \times Y$ is projective. 3. Show that $X \times Y$ is a product in the category of varieties. ### 3.17 Normal Varieties. #to_work A variety $Y$ is normal at a point $P \in Y$ if ${\mathcal{O}}_P$ is an integrally closed ring. $Y$ is normal if it is normal at every point. 1. Show that every conic in ${\mathbb{P}}^2$ is normal. 2. Show that the quadric surfaces $Q_1, Q_2$ in $\mathrm{P}^3$ given by equations $Q_1: x y=zw$; $Q_2: xy=z^2$ are normal. (cf. (II. Ex. 6.4) for the latter.) 3. Show that the cuspidal cubic $y^2=x^3$ in $\mathbf{A}^2$ is not normal. 4. If $Y$ is affine, then $Y$ is normal $\Leftrightarrow A(Y)$ is integrally closed. 5. Let $Y$ be an affine variety. Show that there is a normal affine variety $\tilde{Y}$, and a morphism $\pi: \tilde{Y} \rightarrow Y$, with the property that whenever $Z$ is a normal variety, and $\varphi: Z \rightarrow Y$ is a **dominant** morphism (i.e., $\varphi(Z)$ is dense in $Y$), then there is a unique morphism $\theta: Z \rightarrow \tilde{Y}$ such that $\varphi=\pi \quad \theta$. $\tilde{Y}$ is called the **normalization** of $Y$. You will need $(3.9 \mathrm{~A})$ above. ### 3.18 Projectively Normal Varieties. #to_work A projective variety $Y \subseteq \mathrm{P}^n$ is projectively normal (with respect to the given embedding) if its homogeneous coordinate ring $S\left(Y \right)$ is integrally closed. 1. If $Y$ is projectively normal, then $Y$ is normal. 2. There are normal varieties in projective space which are not projectively normal. For example, let $Y$ be the twisted quartic curve in ${\mathbb{P}}^3$ given parametrically by $(x, y: z, w)=\left(t^4, t^3 u, t u^3, u^4\right)$. Then $Y$ is normal but not projectively normal. See (III, Ex. 5.6) for more examples. 3. Show that the twisted quartic curve $Y$ above is isomorphic to ${\mathbb{P}}^1$. which is projectively normal. Thus projective normality depends on the embedding. ### 3.19 Automorphisms of $\mathbf{A}^n$. #to_work {#automorphisms-of-mathbfan.-to_work} Let $\varphi: \mathbf{A}^n \rightarrow \mathbf{A}^n$ be a morphism of $\mathbf{A}^n$ to $\mathbf{A}^n$ given by $n$ polynomials $f_1 \ldots . f_n$ of $n$ variables $x_1, \ldots x_n$. Let $J=\operatorname{det}\left[ {\frac{\partial f_i}{\partial x_j}\,} \right]$ be the Jacobian polynomial of $\varphi$. 1. If $\varphi$ is an isomorphism (in which case we call $\varphi$ an automorphism of $\mathbf{A}^n$ ) show that $J$ is a nonzero constant polynomial. 2. \*\* The converse of 1.is an unsolved problem, even for $n=2$. See, for example, Vitushkin. ### 3.20 #to_work Let $Y$ be a variety of dimension $\geqslant 2$, and let $P \in Y$ be a normal point. Let $f$ be a regular function on $Y-P$. 1. Show that $f$ extends to a regular function on $Y$. 2. Show this would be false for $\operatorname{dim} Y=1$. See (III. Ex. 3.5) for generalization. ### 3.21. Group Varieties. #to_work A group variety consists of a variety Y together with a morphism $\mu: Y \times Y \rightarrow Y$. such that the set of points of $Y$ with the operation given by $\mu$ is a group. and such that the inverse map $y^{-y^{-1}}$ is also a morphism of $Y \rightarrow Y$. 1. The additive group $\mathbf{G}_a$ is given by the variety $\mathbf{A}^1$ and the morphism $\mu: \mathbf{A}^2 \rightarrow \mathbf{A}^1$ defined by $\mu(a, b) = a+b$. Show it is a group variety. 2. The multiplicative croup $\mathbf{G}_m$ is given by the variety $\mathbf{A}^1\setminus\left\{{0}\right\}$, and the morphism $\mu(a, b) = ab$. Show $|$ in a group variety. 3. If $G$ is a group variety, and $X$ is any variety. show that the set $\operatorname{Hom}(X, G)$ has a natural group structure. 4. For any variety $X$, show that $\operatorname{Hom}\left(X, \mathbf{G}_a\right)$ is isomorphic to (' (X) as a group under addition. 5. For any variety $X$, show that $\operatorname{Hom}\left(X, \mathbf{G}_m\right)$ is isomorphic to the group of units in ${\mathcal{O}}(X)$, under multiplication. ## I.4: Rational Maps ### 4.1. #to_work If $f$ and $g$ are regular functions on open subsets $U$ and $V$ of a variety $X$, and if $f=g$ on $U \cap V$. show that the function which is $f$ on $U$ and $g$ on $V$ is a regular function on $U \cup V$. Conclude that if $f$ is a rational function on $X$, then there is a largest open subset $U$ of $X$ on which $f$ is represented by a regular function. We say that $f$ is defined at the points of $U$. ### 4.2. Same problem for rational maps. #to_work If $\varphi$ is a rational map of $X$ to $Y$, show there is a largest open set on which $\varphi$ is represented by a morphism. We say the rational map is defined at the points of that open set. ### 4.3. #to_work 1. Let $f$ be the rational function on ${\mathbb{P}}^2$ given by $f=x_1 / x_0$. Find the set of points where $f$ is defined and describe the corresponding regular function. 2. Now think of this function as a rational map from ${\mathbb{P}}^2$ to $\mathbf{A}^1$. Embed $\mathbf{A}^1$ in ${\mathbb{P}}^1$, and let $\varphi: {\mathbb{P}}^2 \rightarrow {\mathbb{P}}^1$ be the resulting rational map. Find the set of points where $\varphi$ is defined, and describe the corresponding morphism. ### 4.4. #to_work A variety $Y$ is rational if it is birationally equivalent to ${\mathbb{P}}^n$ for some $n$ (or, equivalently by (4.5), if $K(Y)$ is a pure transcendental extension of $k$ ). 1. Any conic in ${\mathbb{P}}^2$ is a rational curve. 2. The cuspidal cubic $y^2=x^3$ is a rational curve. 3. Let $Y$ be the nodal cubic curve $y^2 z=x^2(x+z)$ in ${\mathbb{P}}^2$. Show that the projection $\varphi$ from the point $P=(0,0,1)$ to the line $z=0$ (Ex. 3.14) induces a birational map from $Y$ to ${\mathbb{P}}^1$. Thus $Y$ is a rational curve. ### 4.5. #to_work Show that the quadric surface $Q: x y=z w$ in ${\mathbb{P}}^3$ is birational to ${\mathbb{P}}^2$, but not isomorphic to ${\mathbb{P}}^2$ (cf. Ex. 2.15). ### 4.6. Plane Cremona Transformations. #to_work A birational map of ${\mathbb{P}}^2$ into itself is called a plane Cremona transformation. We give an example, called a quadratic transformation. It is the rational map $\varphi: {\mathbb{P}}^2 \rightarrow {\mathbb{P}}^2$ given by $\left(a_0, a_1, a_2\right) \rightarrow\left(a_1 a_2, a_0 a_2, a_0 a_1\right)$ when no two of $a_0, a_1, a_2$ are $0$. 1. Show that $\varphi$ is birational, and is its own inverse. 2. Find open sets $U, V \subseteq {\mathbb{P}}^2$ such that $\varphi: U \rightarrow V$ is an isomorphism. 3. Find the open sets where $\varphi$ and $\varphi^{-1}$ are defined. and describe the corresponding morphisms. See also (Chapter V, 4.2.3). ### 4.7. #to_work Let $X$ and $Y$ be two varieties. Suppose there are points $P \in X$ and $Q \in Y$ such that the local rings ${\mathcal{O}}_{P, X}$ and ${\mathcal{O}}_{Q, Y}$ are isomorphic as $k{\hbox{-}}$algebras. Then show that there are open sets $P \in U \subseteq X$ and $Q \in V \subseteq Y$ and an isomorphism of $U$ to $V$ which sends $P$ to $Q$. ### 4.8. #to_work 1. Show that any variety of positive dimension over $k$ has the same cardinality as $k$.[^16] 2. Deduce that any two curves over $k$ are homeomorphic (cf. Ex. 3.1). ### 4.9. #to_work Let $X$ be a projective variety of dimension $r$ in $P^n$. with $n \geqslant r+2$. Show that for suitable choice of $P \notin X$. and a linear ${\mathbb{P}}^{n-1} \subseteq {\mathbb{P}}^n$. the projection from $P$ to ${\mathbb{P}}^{n-1}$ (Ex. 3.14) induces a birational morphism of $X$ onto its image $X' \subseteq {\mathbb{P}}^{n-1}$. You will need to use (4.6A). (4.7A). and (4.8A). This shows in particular that the birational map of (4.9) can be obtained by a finite number of such projections. ### 4.10. #to_work Let $Y$ be the cuspidal cubic curve $y^2=x^{3}$ in $\mathbf{A}^2$. Blow up the point $O=(0.0)$. Let $E$ be the exceptional curve. and let $\tilde{Y}$ be the strict transform of $Y$. Show that $E$ meets $\tilde{Y}$ in one point. and that $\tilde{Y} \cong \mathbf{A}^1$. In this case the morphism $\rho: \tilde{Y} \rightarrow Y$ is bijective and bicontinuous. but it is not an isomorphism. ## I.5: Nonsingular Varieties ### 5.1. #to_work Locate the singular points and sketch the following curves in ${\mathbb{A}}^2$ (assume char $k \neq 2$ ). Which is which in Figure 4 ? 1. $x^2=x^4+y^4$ : 2. $x y=x^6+y^6$ : 3. $x^3=y^2+x^4+y^4$ : 4. $x^2 y+x y^2=x^4+y^4$. ![](attachments/2022-09-21_00-14-42.png) ### 5.2. #to_work Locate the singular points and describe the singularities of the following surfaces in ${\mathbb{A}}^3$ (assume char $k \neq 2$ ). Which is which in Figure 5? 1. $x y^2=z^2$ 2. $x^2+y^2=z^2$ 3. $xy + x^3 + y^3 = 0$. ![](attachments/2022-09-21_00-15-32.png) ### 5.3. Multiplicities. #to_work Let $Y \subseteq {\mathbb{A}}^2$ be a curve defined by the equation $f(x, y)=0$. Let $P=(a, b)$ be a point of ${\mathbb{A}}^2$. Make a linear change of coordinates so that $P$ becomes the point $(0,0)$. Then write $f$ as a sum $f=f_0+f_1+\ldots+f_d$, where $f_i$ is a homogeneous polynomial of degree $i$ in $x$ and $y$. Then we define the multiplicity of $P$ on $Y$, denoted $\mu_P(Y)$, to be the least $r$ such that $f_r \neq 0$. (Note that $P \in Y \Leftrightarrow \mu_P(Y)>0$.) The linear factors of $f_r$ are called the tangent directions at $P$. 1. Show that $\mu_P(Y) =1 \iff P$ is a nonsingular point of $Y$. 2. Find the multiplicity of each of the singular points in (Ex. 5.1) above. ### 5.4. Intersection Multiplicity. #to_work If $Y, Z \subseteq {\mathbb{A}}^2$ are two distinct curves, given by equations $f=0, g=0$, and if $P \in Y \cap Z$, we define the intersection multiplicity $(Y \cdot Z)_P$ of $Y$ and $Z$ at $P$ to be the length of the ${\mathcal{O}}_P$-module ${\mathcal{O}}_P /\left\langle{f, g}\right\rangle$. 1. Show that $(Y \cdot Z)_P$ is finite, and $(Y \cdot Z)_P \geqslant \mu_P(Y) \cdot \mu_P(Z)$. 2. If $P \in Y$, show that for almost all lines $L$ through $P$ (i.e., all but a finite number), $(L \cdot Y)_P=\mu_P(Y)$. 3. If $Y$ is a curve of degree $d$ in ${\mathbb{P}}^2$, and if $L$ is a line in ${\mathbb{P}}^2, L \neq Y$, show that $(L \cdot Y)=d$. Here we define $(L \cdot Y)=\sum(L \cdot Y)_P$ taken over all points $P \in$ $L \cap Y$, where $(L \cdot Y)_p$ is defined using a suitable affine cover of ${\mathbb{P}}^2$. ### 5.5. #to_work For every degree $d>0$, and every $p=0$ or a prime number, give the equation of a nonsingular curve of degree $d$ in ${\mathbb{P}}^2$ over a field $k$ of characteristic $p$. ### 5.6. Blowing Up Curve Singularities. #to_work 1. Let $Y$ be the cusp or node of (Ex. 5.1). Show that the curve $\tilde{Y}$ obtained by blowing up $Y$ at $O=(0,0)$ is nonsingular (cf. (4.9.1) and (Ex. 4.10)). 2. We define a node (also called ordinary double point) to be a double point (i.e., a point of multiplicity 2 ) of a plane curve with distinct tangent directions (Ex. 5.3). If $P$ is a node on a plane curve $Y$, show that $\varphi^{-1}(P)$ consists of two distinct nonsingular points on the blown-up curve $\tilde{Y}$. We say that "blowing up $P$ resolves the singularity at $P$". 3. Let $P \in Y$ be the tacnode of $($ Ex. 5.1). If $\varphi: \tilde{Y} \rightarrow Y$ is the blowing-up at $P$. show that $\rho^{-1}(P)$ is a node. Using 2. we see that the tacnode can be resolved by two successive blowings-up. 4. Let $Y$ be the plane curve $y^3=x^5$, which has a "higher order cusp" at $O$. Show that $O$ is a triple point: that blowing up $O$ gives rise to a double point (what kind?) and that one further blowing up resolves the singularity. Note: We will see later $(\mathrm{V}, 3.8)$ that any singular point of a plane curve can be resolved by a finite sequence of successive blowings-up. ### 5.7. #to_work Let $Y \subseteq {\mathbb{P}}^2$ be a nonsingular plane curve of degree $>1$, defined by the equation $f(x, y, z)=0$. Let $X \subseteq {\mathbb{A}}^3$ be the affine variety defined by $f$ (this is the cone over $Y$; see (Ex. 2.10) ). Let $P$ be the point $(0,0,0)$, which is the vertex of the cone. Let $\varphi: \tilde{X} \rightarrow X$ be the blowing-up of $X$ at $P$. 1. Show that $X$ has just one singular point, namely $P$. 2. Show that $\tilde{X}$ is nonsingular (cover it with open affines). 3. Show that $\varphi^{-1}(P)$ is isomorphic to $Y$. ### 5.8. #to_work Let $Y \subseteq {\mathbb{P}}^n$ be a projective variety of dimension $r$. Let $f_1, \ldots, f_t \in S=$ $k\left[x_0, \ldots, x_n\right]$ be homogeneous polynomials which generate the ideal of $Y$. Let $P \in Y$ be a point, with homogeneous coordinates $P=\left(a_0, \ldots, a_n\right)$. Show that $P$ is nonsingular on $Y$ if and only if the rank of the matrix $\left[ {\frac{\partial f_i}{\partial x_j}\,}(a_0,\cdots, a_n)\right]$ is $n-r$.[^17] ### 5.9. #to_work Let $f \in k[x, y ; z]$ be a homogeneous polynomial, let $Y=Z(f) \subseteq {\mathbb{P}}^2$ be the algebraic set defined by $f$, and suppose that for every $P \in Y$, at least one of ${\frac{\partial f}{\partial x}\,}(P), {\frac{\partial f}{\partial y}\,}(P), {\frac{\partial f}{\partial z}\,}(P)$ is nonzero. Show that $f$ is irreducible (and hence that $Y$ is a nonsingular variety). [^18] ### 5.10. #to_work For a point $P$ on a variety $X$. let ${\mathfrak{m}}$ be the maximal ideal of the local ring ${\mathcal{O}}_P$. We define the Zariski tangent space $T_P(X)$ of $X$ at $P$ to be the dual $k$-vector space of ${\mathfrak{m}}/{\mathfrak{m}}^2$. 1. For any point $P \in X$. $\operatorname{dim} T_P(X) \geqslant \operatorname{dim} X$. with equality if and only if $P$ is nonsingular. 2. For any morphism $\varphi: X \rightarrow Y$, there is a natural induced $k$-linear map $T_P(\varphi)$ : $T_P(X) \rightarrow T_{\varphi(P)}(Y)$ 3. If $\varphi$ is the vertical projection of the parabola $x=y^2$ onto the $x$-axis, show that the induced map $T_0(\varphi)$ of tangent spaces at the origin is the zero map. ### 5.11. The Elliptic Quartic Curve in ${\mathbb{P}}^3$. #to_work {#the-elliptic-quartic-curve-in-mathbbp3.-to_work} Let $Y$ be the algebraic set in ${\mathbb{P}}^3$ defined by the equations $x^2-x z- yw=0$ and $yz -xw - zw = 0$. Let $P$ be the point $(x, y, z, w)=(0,0,0,1)$. and let $\varphi$ denote the projection from $P$ to the plane $w=0$. Show that $\varphi$ induces an isomorphism of $Y-P$ with the plane cubic curve $y^2 z-x^3+x z^2=0$ minus the point $(1,0,-1)$. Then show that $Y$ is an irreducible nonsingular curve. It is called the elliptic quartic curve in ${\mathbb{P}}^3$. Since it is defined by two equations it is another example of a complete intersection (Ex. 2.17). ### 5.12. Quadric Hypersurfaces. #to_work Assume char $k \neq 2$. and let $f$ be a homogeneous polynomial of degree 2 in $x_0 \ldots \ldots x_n$. 1. Show that after a suitable linear change of variables, $f$ can be brought into the form $f=x_0^2+\ldots+x_r^2$ for some $0 \leqslant r \leqslant n$. 2. Show that $f$ is irreducible if and only if $r \geqslant 2$. 3. Assume $r \geqslant 2$, and let $Q$ be the quadric hypersurface in ${\mathbb{P}}^n$ defined by $f$. Show that the singular locus $Z=\operatorname{Sing} Q$ of $Q$ is a linear variety (Ex. 2.11) of dimension $n-r-1$. In particular, $Q$ is nonsingular if and only if $r=n$. 4. In case $r \begin{align*} \begin{align*} g=g_s+g_{s+1}+\ldots \\ h=h_t+h_{t+1}+\ldots \end{align*} \end{align*} `{=html} in $k{\left[\left[ x, y \right]\right] }$ such that $f=gh$. 3. Let $Y$ be defined by the equation $f(x, y)=0$ in ${\mathbb{A}}^2$, and let $P=(0,0)$ be a point of multiplicity $r$ on $Y$, so that when $f$ is expanded as a polynomial in $x$ and $y$, we have $f=f_r+$ higher terms. We say that $P$ is an ordinary $r{\hbox{-}}$fold point if $f_r$ is a product of $r$ distinct linear factors. Show that any two ordinary double points are analytically isomorphic. Ditto for ordinary triple points. But show that there is a one-parameter family of mutually nonisomorphic ordinary 4-fold points. 4. \* Assume char $k \neq 2$. Show that any double point of a plane curve is analytically isomorphic to the singularity at $(0,0)$ of the curve $y^{2}=x^r$, for a uniquely determined $r \geqslant 2$. If $r=2$ it is a node (Ex. 5.6). If $r=3$ we call it a cusp: if $r=4$ a tacnode. See $(\mathrm{V}, 3.9 .5)$ for further discussion. ### 5.15. Families of Plane Curves. #to_work A homogeneous polynomial $f$ of degree $d$ in three variables $x, y, z$ has ${d+2\choose 2}$ coefficients. Let these coefficients represent a point in ${\mathbb{P}}^N$. where $N= {d+2\choose 2} - 1 = {1\over 2}d(d+3)$. 1. Show that this gives a correspondence between points of ${\mathbb{P}}^{N}$ and algebraic sets in ${\mathbb{P}}^2$ which can be defined by an equation of degree $d$. The correspondence is 1-1 except in some cases where $f$ has a multiple factor. 2. Show under this correspondence that the (irreducible) nonsingular curves of degree $d$ correspond 1-1 to the points of a nonempty Zariski-open subset of ${\mathbb{P}}^N$. [^20] ## I.6: Nonsingular Curves ### 6.1. #to_work Recall that a curve is rational if it is birationally equivalent to $\mathbf{P}^1(\mathrm{Ex} .4 .4)$. Let $Y$ be a nonsingular rational curve which is not isomorphic to $\mathbf{P}^1$. (a) Show that $Y$ is isomorphic to an open subset of $\mathbf{A}^1$. (b) Show that $Y$ is affine. (c) Show that $A(Y)$ is a unique factorization domain. ### 6.2. An Elliptic Curve. #to_work Let $Y$ be the curve $y^2=x^3-x$ in $\mathbf{A}^2$, and assume that the characteristic of the base field $k$ is $\neq 2$. In this exercise we will show that $Y$ is not a rational curve, and hence $K(Y)$ is not a pure transcendental extension of $k$. (a) Show that $Y$ is nonsingular, and deduce that $A=A(Y) \simeq k[x, y] /\left(y^2-x^3+x\right)$ is an integrally closed domain. (b) Let $k[x]$ be the subring of $K=K(Y)$ generated by the image of $x$ in $A$. Show that $k[x]$ is a polynomial ring, and that $A$ is the integral closure of $k[x]$ in $K$. (c) Show that there is an automorphism $\sigma: A \rightarrow A$ which sends $y$ to $-y$ and leaves $x$ fixed. For any $a \in A$, define the norm of $a$ to be $N(a)=a \cdot \sigma(a)$. Show that $N(a) \in k[x], N(1)=1$, and $N(a b)=N(a) \cdot N(b)$ for any $a, b \in A$. (d) Using the norm, show that the units in $A$ are precisely the nonzero elements of $k$. Show that $x$ and $y$ are irreducible elements of $A$. Show that $A$ is not a unique factorization domain. (e) Prove that $Y$ is not a rational curve (Ex. 6.1). See (II, 8.20.3) and (III, Ex. 5.3) for other proofs of this important result. ### 6.3. #to_work Show by example that the result of $(6.8)$ is false if either (a) $\operatorname{dim} X \geqslant 2$, or (b) $Y$ is not projective. ### 6.4. #to_work Let $Y$ be a nonsingular projective curve. Show that every nonconstant rational function $f$ on $Y$ defines a surjective morphism $\varphi: Y \rightarrow \mathbf{P}^1$, and that for every $P \in \mathbf{P}^1$, $\varphi^{-1}(P)$ is a finite set of points. ### 6.5. #to_work Let $X$ be a nonsingular projective curve. Suppose that $X$ is a (locally closed) subvariety of a variety $Y$ (Ex. 3.10). Show that $X$ is in fact a closed subset of $Y$. See (II, Ex. 4.4) for generalization. ### 6.6. Automorphisms of $\mathbf{P}^1$. #to_work {#automorphisms-of-mathbfp1.-to_work} Think of $\mathbf{P}^1$ as $\mathbf{A}^1 \cup\left\{{\infty}\right\}$. Then we define a fractional linear transformation of $\mathbf{P}^1$ by sending $x \mapsto(a x+b)/(c x+d)$, for $a, b, c, d \in k$, and $ad-b c \neq 0$. (a) Show that a fractional linear transformation induces an automorphism of $\mathbf{P}^1$ (i.e., an isomorphism of $\mathbf{P}^1$ with itself). We denote the group of all these fractional linear transformations by $\operatorname{PGL}(1)$. (b) Let Aut $\mathbf{P}^1$ denote the group of all automorphisms of $\mathbf{P}^1$. Show that Aut $\mathbf{P}^1 \simeq$ Aut $k(x)$, the group of $k$-automorphisms of the field $k(x)$. (c) Now show that every automorphism of $k(x)$ is a fractional linear transformation, and deduce that $\mathrm{PGL}(1) \rightarrow$ Aut $\mathbf{P}^1$ is an isomorphism. Note: We will see later (II. 7.1.1) that a similar result holds for $\mathbf{P}^n$ : every automorphism is given by a linear transformation of the homogeneous coordinates. ### 6.7. #to_work Let $P_1, \ldots, P_r, Q_1, \ldots, Q_s$ be distinct points of $\mathbf{A}^1$. If $\mathbf{A}^1-\left\{P_1, \ldots, P_r\right\}$ is isomorphic to $\mathbf{A}^1-\left\{Q_1, \ldots, Q_{\diamond}\right\}$, show that $r=s$. Is the converse true? Cf. (Ex. 3.1). ## I.7: Intersections in Projective Space ### 7.1. #to_work 1. Find the degree of the $d$-uple embedding of $\mathbf{P}^n$ in $\mathbf{P}^N\left(\right.$ Ex. 2.12).[^21] 2. Find the degree of the Segre embedding of $\mathbf{P}^r \times \mathbf{P}^s$ in $\mathbf{P}^{\prime}($ Ex. 2.14).[^22] ### 7.2. #to_work Let $Y$ be a variety of dimension $r$ in $\mathbf{P}^n$, with Hilbert polynomial $P_Y$. We define the arithmetic genus of $Y$ to be $p_a(Y) = (-1)^r\left(P_Y(0)-1\right)$. This is an important invariant which (as we will see later in (III, Ex. 5.3)) is independent of the projective embedding of $Y$. 1. Show that $p_a\left(\mathbf{P}^n\right)=0$. 2. If $Y$ is a plane curve of degree $d$, show that $p_a(Y)=\frac{1}{2}(d-1)(d-2)$. 3. More generally, if $H$ is a hypersurface of degree $d$ in $\mathbf{P}^n$, then $p_a(H)={d-1\choose n}$. 4. If $Y$ is a complete intersection (Ex. 2.17) of surfaces of degrees $a, h$ in $\mathbf{P}^3$, then $p_a(Y)=\frac{1}{2} a b(a+b-4)+1$. ```{=html} ``` (e) Let $Y^r \subseteq \mathbf{P}^n, Z^{s} \subseteq \mathbf{P}^m$ be projective varieties, and embed $Y \times Z \subseteq \mathbf{P}^n \times$ $\mathbf{P}^m \rightarrow \mathbf{P}^N$ by the Segre embedding. Show that ` \begin{align*} p_a(Y \times Z)=p_a(Y) p_a(Z)+(-1)^{s} p_a(Y)+(-1)^r p_a(Z) . \end{align*} `{=html} ### 7.3. The Dual Curve. #to_work Let $Y \subseteq \mathbf{P}^2$ be a curve. We regard the set of lines in $\mathbf{P}^2$ as another projective space, $\left(\mathbf{P}^2\right)^*$. by taking $(a_0, a_1, a_2)$ as homogeneous coordinates of the line $L: a_0 x_0+a_1 x_1+a_2 x_2=0$. For each nonsingular point $P \in Y$, show that there is a unique line $T_P(Y)$ whose intersection multiplicity with $Y$ at $P$ is $>1$. This is the tangent line to $Y$ at $P$. Show that the mapping $P \mapsto T_P(Y)$ defines a morphism of Reg $Y$ (the set of nonsingular points of $Y)$ into $\left(\mathbf{P}^2\right)^*$. The closure of the image of this morphism is called the dual curve $Y^* \subseteq\left(\mathbf{P}^2\right)^*$ of $Y$. ### 7.4. #to_work Given a curve $Y$ of degree $d$ in $\mathbf{P}^2$, show that there is a nonempty open subset $U$ of $\left(\mathbf{P}^2\right)^*$ in its Zariski topology such that for each $L \in U, L$ meets $Y$ in exactly $d$ points. [^23] This result shows that we could have defined the degree of $Y$ to be the number $d$ such that almost all lines in $\mathbf{P}^2$ meet $Y$ in $d$ points, where "almost all" refers to a nonempty open set of the set of lines, when this set is identified with the dual projective space $\left(\mathbf{P}^2\right)^*$ ### 7.5. #to_work 1. Show that an irreducible curve $Y$ of degree $d>1$ in $\mathbf{P}^2$ cannot have a point of multiplicity $\geqslant d($ Ex. 5.3). 2. If $Y$ is an irreducible curve of degree $d>1$ having a point of multiplicity $d-1$. then $Y$ is a rational curve (Ex. 6.1). ### 7.6. Linear Varieties. #to_work Show that an algebraic set $Y$ of pure dimension $r$ (i.e., every irreducible component of $Y$ has dimension $r$ ) has degree 1 if and only if $Y$ is a linear variety (Ex. 2.11).[^24] ### 7.7. #to_work Let $Y$ be a variety of dimension $r$ and degree $d>1$ in $\mathbf{P}^n$. Let $P \in Y$ be a nonsingular point. Define $X$ to be the closure of the union of all lines $P Q$, where $Q \in Y, Q \neq P$. 1. Show that $X$ is a variety of dimension $r+1$. 2. Show that $\operatorname{deg} X \begin{align*}\ldots {\mathcal{F}}^{i-1} \xrightarrow{\phi^{i-1}} {\mathcal{F}}^{i} \xrightarrow{\phi^{i}} {\mathcal{F}}^{i+1} \rightarrow \ldots \end{align*} `{=html}of sheaves and morphisms is exact if and only if for each $P \in X$ the corresponding sequence of stalks is exact as a sequence of abelian groups. ### II.1.3. #to_work a. Let $\varphi: {\mathcal{F}}\rightarrow \mathscr{G}$ be a morphism of sheaves on $X$. Show that $\varphi$ is surjective if and only if the following condition holds: for every open set $U \subseteq X$, and for every $s \in \mathscr{G}\left(U \right)$, there is a covering $\left\{U_i\right\}$ of $U$, and there are elements $t_i \in {\mathcal{F}}\left(U_i\right)$, such that $\varphi\left(t_i\right)={ \left.{{s}} \right|_{{U_i}} }$ for all $i$. b. Give an example of a surjective morphism of sheaves $\varphi: {\mathcal{F}}\rightarrow \mathscr{G}$, and an open set $U$ such that $\varphi(U): {\mathcal{F}}(U) \rightarrow \mathscr{G}(U)$ is not surjective. ### II.1.4. #to_work a. Let $\varphi: {\mathcal{F}}\rightarrow \mathscr{G}$ be a morphism of presheaves such that $\varphi(U): {\mathcal{F}}(U) \rightarrow \mathscr{G}(U)$ is injective for each $U$. Show that the induced map $\varphi^{+}: {\mathcal{F}}^{+} \rightarrow \mathscr{G}^{+}$of associated sheaves is injective. b. Use part (a) to show that if $\varphi: {\mathcal{F}}\rightarrow \mathscr{G}$ is a morphism of sheaves, then $\operatorname{im}\varphi$ can be naturally identified with a subsheaf of $\mathscr{G}$. as mentioned in the text. ### II.1.5. #to_work Show that a morphism of sheaves is an isomorphism if and only if it is both injective and surjective. ### II.1.6. #to_work a. Let ${\mathcal{F}}^{\prime}$ be a subsheaf of a sheaf $\tilde{{\mathcal{F}}}$. Show that the natural map of $\tilde{{\mathcal{F}}}$ to the quotient sheaf ${\mathcal{F}}/{\mathcal{F}}'$ is surjective, and has kernel ${\mathcal{F}}^{\prime}$. Thus there is an exact sequence ` \begin{align*} 0 \rightarrow {\mathcal{F}}^{\prime} \rightarrow {\mathcal{F}}\rightarrow {\mathcal{F}}/ {\mathcal{F}}^{\prime} \rightarrow 0 .\end{align*} `{=html} b. Conversely, if ` \begin{align*} 0 \rightarrow {\mathcal{F}}' \rightarrow {\mathcal{F}}\rightarrow {\mathcal{F}}^{\prime \prime} \rightarrow 0 \end{align*} `{=html} is an exact sequence, show that ${\mathcal{F}}^{\prime}$ is isomorphic to a subsheaf of ${\mathcal{F}}$, and that ${\mathcal{F}}^{\prime \prime}$ is isomorphic to the quotient of ${\mathcal{F}}$ by this subsheaf. ### II.1.7. #to_work Let $\varphi: {\mathcal{F}}\rightarrow \mathscr{G}$ be a morphism of sheaves. a. Show that $\operatorname{im}\varphi \cong {\mathcal{F}}/ \operatorname{ker} \varphi$. b. Show that $\operatorname{coker} \varphi \cong \mathscr{G} / \operatorname{im} \varphi$. ### II.1.8. #to_work For any open subset $U \subseteq X$, show that the functor $\Gamma(U . \cdot)$ from sheaves on $X$ to abelian groups is a left exact functor, i.e.. if ` \begin{align*} 0 \rightarrow {\mathcal{F}}^{\prime} \rightarrow {\mathcal{F}}\rightarrow {\mathcal{F}}^{\prime \prime} \end{align*} `{=html} is an exact sequence of sheaves, then ` \begin{align*} 0 \rightarrow \Gamma\left(U, {\mathcal{F}}^{\prime}\right) \rightarrow \Gamma(U, {{\mathcal{F}}}) \rightarrow \Gamma\left(U, {\mathcal{F}}^{\prime \prime}\right) \end{align*} `{=html} is an exact sequence of groups. The functor $\Gamma(U, {-})$ need not be exact: see $($ Ex. 1.21) below. ### II.1.9. Direct Sum. #to_work Let ${\mathcal{F}}$ and $\mathscr{G}$ be sheaves on $X$. Show that the presheaf $U \mapsto {\mathcal{F}}(U) \oplus$ $\mathscr{G}(U)$ is a sheaf. It is called the **direct sum** of ${\mathcal{F}}$ and $\mathscr{G}$, and is denoted by ${\mathcal{F}}\oplus \mathscr{G}$. Show that it plays the role of direct sum and of direct product in the category of sheaves of abelian groups on $X$. ### II.1.10. Direct Limit. #to_work Let $\left\{{{\mathcal{F}}_i}\right\}$ be a direct system of sheaves and morphisms on $X$. We define the direct limit of the system $\left\{{{\mathcal{F}}_i}\right\}$, denoted $\lim {\mathcal{F}}_i$, to be the sheaf associated to the presheaf $U \mapsto \varinjlim {{\mathcal{F}}_i}(U)$. Show that this is a direct limit in the category of sheaves on $X$, i.e., that it has the following universal property: given a sheaf $\mathscr{G}$, and a collection of morphisms ${\mathcal{F}}_i \rightarrow \mathscr{G}$. compatible with the maps of the direct system, then there exists a unique map $\varinjlim {\mathcal{F}}_i \to \mathscr{G}$ such that for each $i$, the original map $\mathscr{F_i} \rightarrow \mathscr{G}$ is obtained by composing the maps ${\mathcal{F}}_i \rightarrow \varinjlim {\mathcal{F}}_i \rightarrow \mathscr{G}$. ### II.1.11. #to_work Let $\left\{{\mathcal{F}}_i\right\}$ be a direct system of sheaves on a noetherian topological space $X$. In this case show that the presheaf $U \mapsto \varinjlim {\mathcal{F}}_i(U)$ is already a sheaf. In particular, $\Gamma\left(X, \varinjlim {{\mathcal{F}}}_i\right) = \varinjlim \Gamma\left(X, {\mathcal{F}}_i\right)$ ### II.1.12. Inverse Limit. #to_work Let, $\left\{{{\mathcal{F}}_i}\right\}$ be an inverse system of sheaves on $X$. Show that the presheaf $U \mapsto \varprojlim {\mathcal{F}}_i\left(U \right)$ is a sheaf. It is called the **inverse limit** of the system $\left\{{\mathcal{F}}_i\right\}$, and is denoted by $\varprojlim {{\mathcal{F}}}_i$. Show that it has the universal property of an inverse limit in the category of sheaves. ### II.1.13. Espace Etale of a Presheaf. #to_work Given a presheaf ${\mathcal{F}}$ on $X$, we define a topological space $\operatorname{\operatorname{Spe}}({\mathcal{F}})$, called the **espace etale** of ${\mathcal{F}}$, as follows[^26]. As a set, $\operatorname{Spe} = \bigcup_{p\in X} {\mathcal{F}}_P$. We define a projection map $\pi: \operatorname{Spe} ({{\mathcal{F}}}) \rightarrow X$ by sending $s \in {\mathcal{F}}_p$ to $P$. For each open set $U \subseteq X$ and each section $s \in {\mathcal{F}}\left(U \right)$, we obtain a map $\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu: U \rightarrow \operatorname{Spe}(\left.{\mathcal{F}}\right)$ by sending $P \mapsto s_P$, its germ at $P$. This map has the property that $\pi \circ \mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu= \operatorname{id}$, in other words, it is a "section" of $\pi$ over $U$. We now make $\operatorname{Spe}({\mathcal{F}})$ into a topological space by giving it the strongest topology such that all the maps $\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu: U \rightarrow \operatorname{Spe}({\mathcal{F}})$ for all ${U}$. and all $s \in {\mathcal{F}}\left(U\right)$, are continuous. Now show that the sheaf ${\mathcal{F}}^{+}$associated to ${\mathcal{F}}$ can be described as follows: for any open set $U \subseteq X, {\mathcal{F}}^{+}\left(U \right)$ is the set of continuous sections of $\operatorname{Spe}({\mathcal{F}})$ over $U$. In particular, the original presheaf ${\mathcal{F}}$ was a sheaf if and only if for each $U, {\mathcal{F}}(U)$ is equal to the set of all continuous sections of $\operatorname{Spe}({\mathcal{F}})$ over $U$. ### II.1.14. Support. #to_work Let ${\mathcal{F}}$ be a sheaf on $X$, and let $s \in {\mathcal{F}}(U)$ be a section over an open set $U$. The **support of $s$**, denoted $\mathop{\mathrm{supp}}s$, is defined to be $\left\{P \in U \mathrel{\Big|}s_P \neq 0\right\}$, where $s_P$ denotes the germ of $s$ in the stalk ${\mathcal{F}}_P$. Show that Supp $s$ is a closed subset of $U$. We define the support of ${\mathcal{F}}$, $\mathop{\mathrm{supp}}{\mathcal{F}}$, to be $\left\{P \in X \mathrel{\Big|}\cdot {\mathcal{F}}_P \neq 0\right\}$. It need not be a closed subset. ### II.1.15. Sheaf $\mathop{\mathcal{H}\! \mathit{om}}$. #to_work {#ii.1.15.-sheaf-mathopmathcalh-mathitom.-to_work} Let ${\mathcal{F}}, \mathscr{G}$ be sheaves of abelian groups on $X$. For any open set $U \subseteq X$, show that the set $\mathop{\mathrm{Hom}}({ \left.{{{\mathcal{F}}}} \right|_{{U}} }, { \left.{{\mathscr G}} \right|_{{U}} })$ of morphisms of the restricted sheaves has a natural structure of abelian group. Show that the presheaf $U \mapsto \operatorname{Hom}\left(\left.{\mathcal{F}}\right|_U,\left.\mathscr{G}\right|_U\right)$ is a sheaf. It is called the **sheaf of local morphisms of ${\mathcal{F}}$ into $\mathscr{G}$, "sheaf hom" for short**, and is denoted $\mathop{\mathcal{H}\! \mathit{om}}({\mathcal{F}}, \mathscr{G})$. ### II.1.16. Flasque Sheaves. #to_work A sheaf ${\mathcal{F}}$ on a topological space $X$ is **flasque** if for every inclusion $V \subseteq U$ of open sets, the restriction map ${\mathcal{F}}(U) \rightarrow {\mathcal{F}}(V)$ is surjective. a. Show that a constant sheaf on an irreducible topological space is flasque. See (I, §1) for irreducible topological spaces. b. If ` \begin{align*} 0 \rightarrow {\mathcal{F}}^{\prime} \rightarrow {\mathcal{F}}\rightarrow {\mathcal{F}}^{\prime \prime} \rightarrow 0 \end{align*} `{=html} is an exact sequence of sheaves, and if ${\mathcal{F}}^{\prime}$ is flasque, then for any open set $L$, the sequence ` \begin{align*} 0 \rightarrow {\mathcal{F}}^{\prime}(U) \rightarrow {\mathcal{F}}(U) \rightarrow {\mathcal{F}}^{\prime \prime}\left(U\right) \rightarrow 0 \end{align*} `{=html} of abelian groups is also exact. c. If ` \begin{align*} 0 \rightarrow {\mathcal{F}}^{\prime} \rightarrow {\mathcal{F}}\rightarrow {\mathcal{F}}^{\prime \prime} \rightarrow 0 \end{align*} `{=html} is an exact sequence of sheaves, and if ${\mathcal{F}}^{\prime}$ and ${\mathcal{F}}$ are flasque, then ${\mathcal{F}}^{\prime \prime}$ is flasque. d. If $f: X \rightarrow Y$ is a continuous map, and if ${\mathcal{F}}$ is a flasque sheaf on $X$, then $f_* {\mathcal{F}}$ is a flasque sheaf on $Y$. e. Let ${\mathcal{F}}$ be any sheaf on $X$. We define a new sheaf $\mathscr{G}$, called the sheaf of **discontinuous sections** of ${\mathcal{F}}$ as follows. For each open set $U \subseteq X, \mathscr{G}(U)$ is the set of maps $s: U \rightarrow \bigcup_{P \in U} {\mathcal{F}}_P$ such that for each $P \in U, s(P) \in {\mathcal{F}}_P$. Show that $\mathscr{G}$ is a flasque sheaf, and that there is a natural injective morphism of ${\mathcal{F}}$ to $\mathscr{G}$. ### II.1.17. Skyscraper Sheaves. #to_work Let $X$ be a topological space, let $P$ be a point, and let $A$ be an abelian group. Define a sheaf $i_P(A)$ on $X$ as follows: $i_P(A)\left(U\right)=A$ if $P \in U^{\prime}, 0$ otherwise. Verify that the stalk of $i_P(A)$ is $A$ at every point $Q \in\{P\}^{-}$, and 0 elsewhere, where $\{P\}^{-}$denotes the closure of the set consisting of the point $P$. Hence the name "skyscraper sheaf." Show that this sheaf could also be described as $i_*(A)$, where $A$ denotes the constant sheaf $A$ on the closed subspace $\{P\}^{-}$, and $i:\{P\}^{-} \rightarrow X$ is the inclusion. ### II.1.18. Adjoint Property of $f^{-1}$. #to_work {#ii.1.18.-adjoint-property-of-f-1.-to_work} Let $f: X \rightarrow Y$ be a continuous map of topological spaces. Show that for any sheaf ${\mathcal{F}}$ on $X$ there is a natural map $f^{-1} f_* {\mathcal{F}}\rightarrow {\mathcal{F}}$, and for any sheaf $\mathscr{G}$ on $Y$ there is a natural map $\mathscr{G} \rightarrow f_* f^{-1} \mathscr{G}$. Use these maps to show that there is a natural bijection of sets, for any sheaves ${\mathcal{F}}$ on $X$ and $\mathscr{G}$ on $Y$, ` \begin{align*} \operatorname{Hom}_X\left(f^{-1} \mathscr{G}, {\mathcal{F}}\right)=\operatorname{Hom}_Y\left(\mathscr{G}, f_* {\mathcal{F}}\right) . \end{align*} `{=html} Hence we say that $f^{-1}$ is a **left adjoint** of $f_*$, and that $f_*$ is a **right adjoint** of $f^{-1}$. ### II.1.19. Extending a Sheaf by Zero. #to_work Let $X$ be a topological space, let $Z$ be a closed subset, let $i: Z \rightarrow X$ be the inclusion, let $U=X-Z$ be the complementary open subset, and let $j: U \rightarrow X$ be its inclusion. a. Let ${\mathcal{F}}$ be a sheaf on $Z$. Show that the stalk $\left(i_* {\mathcal{F}}\right)_P$ of the direct image sheaf on $X$ is ${\mathcal{F}}_P$ if $P \in Z, 0$ if $P \notin Z$. Hence we call $i_* \cdot {\mathcal{F}}$ the sheaf obtained by **extending ${\mathcal{F}}$ by zero outside $Z$.** By abuse of notation we will sometimes write ${\mathcal{F}}$ instead of $i_* {\mathcal{F}}$, and say "consider ${\mathcal{F}}$ as a sheaf on $X$," when we mean \"consider $i_* {\mathcal{F}}$. b. Now let ${\mathcal{F}}$ be a sheaf on $U$. Let $j_!({{\mathcal{F}}})$ be the sheaf on $X$ associated to the presheaf $V \mapsto {\mathcal{F}}(V)$ if $V \subseteq U, V \mapsto 0$ otherwise. Show that the stalk $(j_!({\mathcal{F}}))_P$ is equal to ${\mathcal{F}}_P$ if $P \in U, 0$ if $P \notin U$, and show that $j_! {\mathcal{F}}$ is the only sheaf on $X$ which has this property, and whose restriction to $U$ is ${\mathcal{F}}$. We call $j_! {\mathcal{F}}$ the sheaf obtained by **extending ${\mathcal{F}}$ by zero outside $U$.** c. Now let ${\mathcal{F}}$ be a sheaf on $X$. Show that there is an exact sequence of sheaves on $X$, ` \begin{align*} 0 \to j_!({ \left.{{{\mathcal{F}}}} \right|_{{U}} }) \to {\mathcal{F}}\to i_*({ \left.{{{\mathcal{F}}}} \right|_{{Z}} })\to 0 \end{align*} `{=html} ### II.1.20. Subsheaf with Supports. #to_work Let $Z$ be a closed subset of $X$, and let ${\mathcal{F}}$ be a sheaf on $X$. We define $\Gamma_Z(X, {\mathcal{F}})$ to be the subgroup of $\Gamma(X, {\mathcal{F}})$ consisting of all sections whose support (Ex. 1.14) is contained in $Z$. a. Show that the presheaf $V \mapsto \Gamma_{\mathrm{Z} \cap V}\left(V,\left.{\mathcal{F}}\right|_V\right)$ is a sheaf. It is called the **subsheaf of ${\mathcal{F}}$ with supports in $Z$**, and is denoted by $\mathscr{H}_Z^0({\mathcal{F}})$. b. Let $U=X-Z$, and let $j: U \rightarrow X$ be the inclusion. Show there is an exact sequence of sheaves on $X$ ` \begin{align*} 0 \rightarrow \mathscr{H}_Z^0(\tilde{{\mathcal{F}}}) \rightarrow {\mathcal{F}}\rightarrow i_*({ \left.{{{\mathcal{F}}}} \right|_{{U}} }) \end{align*} `{=html} Furthermore, if ${\mathcal{F}}$ is flasque, the map ${\mathcal{F}}\rightarrow i_*({ \left.{{{\mathcal{F}}}} \right|_{{U}} })$ is surjective. ### II.1.21. Some Examples of Sheaves on Varieties. #to_work Let $X$ be a variety over an algebraically closed field $k$, as in Ch. I. Let ${\mathcal{O}}_X$ be the sheaf of regular functions on $X$ (See (1.0 .1)). a. Let $Y$ be a closed subset of $X$. For each open set $U \subseteq X$, let $\mathscr{I}_Y(U)$ be the ideal in the ring ${\mathcal{O}}_X(U)$ consisting of those regular functions which vanish at all points of $Y \cap U$. Show that the presheaf $U \mapsto \mathscr{I}_Y(U)$ is a sheaf. It is called the sheaf of ideals ${\mathcal{I}}_Y$ of $Y$, and it is a subsheaf of the sheaf of rings ${\mathcal{O}}_X$. b. If $Y$ is a subvariety, then the quotient sheaf $C_1 \mathcal{T}_{,}$is isomorphic to $i_*\left(C_1\right)$, where $i: Y \rightarrow X$ is the inclusion, and ${\mathcal{O}}_Y$ is the sheaf of regular functions on $Y$. c. Now let $X=\mathbf{P}^1$, and let $Y$ be the union of two distinct points $P, Q \in X$. Then there is an exact sequence of sheaves on $X$ where ${\mathcal{F}}= i_* {\mathcal{O}}_P \oplus i_* {\mathcal{O}}_Q$: ` \begin{align*} 0 \to {\mathcal{I}}_Y \to {\mathcal{O}}_X \to {\mathcal{F}}\to 0 \end{align*} `{=html} Show however that the induced map on global sections $\Gamma(X; {\mathcal{O}}_X) \to \Gamma(X; {\mathcal{F}})$ is not surjective. This shows that the global section functor $\Gamma(X, \cdot)$ is not exact (cf. (Ex. 1.8) which shows that it is left exact). d. Again let $X=\mathbf{P}^1$, and let ${\mathcal{O}}$ be the sheaf of regular functions. Let $\mathscr{K}$ be the constant sheaf on $X$ associated to the function field $K$ of $X$. Show that there is a natural injection ${\mathcal{O}}\rightarrow \mathscr{K}$. Show that the quotient sheaf $\mathscr K / {\mathcal{O}}$ is isomorphic to the direct sum of sheaves $\sum_{P \in X} i_P\left(I_P\right)$, where $I_P$ is the group $K / {\mathcal{O}}_P$, and $i_P\left(I_P\right)$ denotes the skyscraper sheaf (Ex. 1.17) given by $I_P$ at the point $P$. e. Finally show that in the case of (d) the sequence ` \begin{align*} 0 \rightarrow \Gamma(X, \mathcal{O}) \rightarrow \Gamma(X, \mathscr{K}) \rightarrow \Gamma(X, \mathscr K/{\mathcal{O}}) \rightarrow 0 \end{align*} `{=html} is exact.[^27] ### II.1.22. Glueing Sheaves. #to_work Let $X$ be a topological space, let $\mathfrak{U} = \left\{{U_i}\right\}$ be an open cover of $X$, and suppose we are given for each $i$ a sheaf ${{\mathcal{F}}}_i$ on $U_i$, and for each $i, j$ an isomorphism ` \begin{align*} \phi_{ij}: { \left.{{{\mathcal{F}}_i}} \right|_{{U_i \cap U_j}} } \to { \left.{{{\mathcal{F}}_j}} \right|_{{U_i \cap U_j}} } \end{align*} `{=html} such that 1. for each $i, \varphi_{ii}=\mathrm{id}$, and 2. for each $i, j, k$, $\phi_{ik} = \phi_{jk} \circ \phi_{ij}$ on $U_i \cap U_j \cap U_k$. Then there exists a unique sheaf ${\mathcal{F}}$ on $X$, together with isomorphisms $\psi_i:\left.{\mathcal{F}}\right|_{U_i} { \, \xrightarrow{\sim}\, }{\mathcal{F}}_i$, such that for each $i, j, \psi_j= \varphi_{i j} \circ \psi_i$ on $U_i \cap U_j$. We say loosely that ${\mathcal{F}}$ is obtained by **glueing** the sheaves ${\mathcal{F}}_i$ via the isomorphisms $\varphi_i$. ## II.2: Schemes ### II.2.1 #to_work Let $A$ be a ring, let $X=\operatorname{Spec} A$, let $f \in A$ and let $D(f) \subseteq X$ be the open complement of $V(f)$. Show that the locally ringed space $\left(D(f),{ \left.{{{\mathcal{O}}_X}} \right|_{{D(f)}} } \right)$ is isomorphic to $\operatorname{spec} A_f$. ### II.2.2 #to_work Let $\left(X, \mathcal{O}_X\right)$ be a scheme, and let $U \subseteq X$ be any open subset. Show that $\left(U,\left.\mathcal{O}_X\right|_U\right)$ is a scheme. We call this the **induced scheme structure** on the open set $U$, and we refer to $\left(U,\left.\mathcal{O}_X\right|_U\right)$ as an **open subscheme** of $X$. ### II.2.3 Reduced Schemes #to_work A scheme $\left(X, \mathcal{O}_X\right)$ is **reduced** if for every open set $U \subseteq X$, the ring $\mathcal{O}_X(U)$ has no nilpotent elements. a. Show that $\left(X, \mathcal{O}_X\right)$ is reduced if and only if for every $P \in X$, the local ring $\mathcal{O}_{X, P}$ has no nilpotent elements. b. Let $\left(X, \mathcal{O}_X\right)$ be a scheme. Let $\left(\mathcal{O}_X\right)_{\text {red }}$ be the sheaf associated to the presheaf $U \mapsto \mathcal{O}_X(U)_{\text {red }}$, where for any ring $A$, we denote by $A_{\text {red }}$ the quotient of $A$ by its ideal of nilpotent elements. Show that $\left(X,\left({\mathcal{O}}_X\right)_{\text {red }}\right)$ is a scheme. We call it the **reduced scheme** associated to $X$, and denote it by $X_{\text {red }}$. Show that there is a morphism of schemes $X_{\text {red }} \rightarrow X$, which is a homeomorphism on the underlying topological spaces. c. Let $f: X \rightarrow Y$ be a morphism of schemes, and assume that $X$ is reduced. Show that there is a unique morphism $g: X \rightarrow Y_{\mathrm{red}}$ such that $f$ is obtained by composing $g$ with the natural map $Y_{\text {red }} \rightarrow Y$. ### II.2.4 #to_work Let $A$ be a ring and let $\left(X, \mathcal{O}_X\right)$ be a scheme. Given a morphism $f: X \rightarrow \operatorname{Spec} A$, we have an associated map on sheaves $f^{\sharp}: \mathcal{U}_{\operatorname{Spec}(A)} \rightarrow f_* \mathcal{O}_X$. Taking global sections we obtain a homomorphism $A \rightarrow \Gamma\left(X, {\mathcal{O}}_X\right)$. Thus there is a natural map ` \begin{align*} \alpha: \operatorname{Hom}_{\mathsf{Sch}}(X, \operatorname{Spec} A) \rightarrow \operatorname{Hom}_{\mathsf{Ring}} \left(A, \Gamma\left(X,\left.{\mathcal{O}}_X\right)\right) .\right. \end{align*} `{=html} Show that $\alpha$ is bijective (cf. (I, 3.5) for an analogous statement about varieties). ### II.2.5 #to_work Describe Spec $\mathbf{Z}$, and show that it is a final object for the category of schemes, i.e., each scheme $X$ admits a unique morphism to Spec $\mathbf{Z}$. ### II.2.6 #to_work Describe the spectrum of the zero ring, and show that it is an initial object for the category of schemes. (According to our conventions, all ring homomorphisms must take 1 to 1 . Since $0=1$ in the zero ring, we see that each ring $R$ admits a unique homomorphism to the zero ring, but that there is no homomorphism from the zero ring to $R$ unless $0=1$ in $R$.) ### II.2.7 #to_work Let $X$ be a scheme. For any $x \in X$, let $\mathcal{O}_x$ be the local ring at $x$, and $\mathfrak{m}_x$ its maximal ideal. We define the residue field of $x$ on $X$ to be the field $k(x)=\mathcal{O}_x / \mathfrak{m}_x$. Now let $K$ be any field. Show that to give a morphism of $\operatorname{Spec} K$ to $X$ it is equivalent to give a point $x \in X$ and an inclusion map $k(x) \rightarrow K$. ### II.2.8 #to_work Let $X$ be a scheme. For any point $x \in X$, we define the **Zariski tangent space** $T_x$ to $X$ at $x$ to be the dual of the $k(x)$-vector space ${\mathfrak{m}}_x / {\mathfrak{m}}_x^2$. Now assume that $X$ is a scheme over a field $k$, and let $k[\varepsilon] / \varepsilon^2$ be the ring of dual numbers over $k$. Show that to give a $k$-morphism of $\operatorname{Spec} k[\varepsilon] / \varepsilon^2$ to $X$ is equivalent to giving a point $x \in X$, rational over $k$ (i.e., such that $k(x)=k$ ), and an element of $T_x$. ### II.2.9 #to_work If $X$ is a topological space, and $Z$ an irreducible closed subset of $X$, a **generic point** for $Z$ is a point $\zeta$ such that $Z=\{\zeta\}^{-}$. If $X$ is a scheme, show that every (nonempty) irreducible closed subset has a unique generic point. ### II.2.10 #to_work Describe $\operatorname{Spec} \mathbf{R}[x]$. How does its topological space compare to the set $\mathbf{R}$ ? To $\mathbf{C}$ ? ### II.2.11 #to_work Let $k=\mathbf{F}_p$ be the finite field with $p$ elements. Describe $\operatorname{Spec} k[x]$. What are the residue fields of its points? How many points are there with a given residue field? ### II.2.12 Glueing Lemma #to_work Generalize the glueing procedure described in the text (2.3.5) as follows. Let $\left\{X_i\right\}$ be a family of schemes (possible infinite). For each $i \neq j$, suppose given an open subset $U_{i j} \subseteq X_i$, and let it have the induced scheme structure (Ex. 2.2). Suppose also given for each $i \neq j$ an isomorphism of schemes $\varphi_{i j}: U_{i j} \rightarrow U_{j i}$ such that (1) for each $i, j, \varphi_{j i}=\varphi_{i j}^{-1}$, and (2) for each $i, j, k$, $\varphi_{i j}\left(U_{i j} \cap U_{i k}\right)=U_{j i} \cap U_{j k}$, and $\varphi_{i k}=\varphi_{j k} \circ \varphi_{i j}$ on $U_{i j} \cap U_{i k}$. Then show that there is a scheme $X$, together with morphisms $\psi_i: X_i \rightarrow X$ for each $i$, such that (1) $\psi_i$ is an isomorphism of $X_i$ onto an open subscheme of $X$ (2) the $\psi_i\left(X_i\right)$ cover $X$, (3) $\psi_i\left(U_{i j}\right)=\psi_i\left(X_i\right) \cap \psi_j\left(X_j\right)$ and (4) $\psi_i=\psi_j \circ \varphi_{i j}$ on $U_{i j}$. We say that $X$ is obtained by **glueing** the schemes $X_i$ along the isomorphisms $\varphi_{i j}$. An interesting special case is when the family $X_i$ is arbitrary, but the $U_{i j}$ and $\varphi_{i j}$ are all empty. Then the scheme $X$ is called the **disjoint union** of the $X_i$, and is denoted $\coprod X_i$. ### II.2.13 #to_work A topological space is **quasi-compact** if every open cover has a finite subcover. a. Show that a topological space is noetherian (I.1) if and only if every open subset is quasi-compact. b. If $X$ is an affine scheme, show that $\operatorname{sp}(X)$ is quasi-compact, but not in general noetherian. We say a scheme $X$ is quasi-compact if $\operatorname{sp}(X)$ is. c. If $A$ is a noetherian ring, show that $\operatorname{sp}(\operatorname{Spec} A)$ is a noetherian topological space. d. Give an example to show that $\operatorname{sp}(\operatorname{Spec} A)$ can be noetherian even when $A$ is not. ### II.2.14 #to_work a. Let $S$ be a graded ring. Show that $\mathop{\mathrm{Proj}}S=\varnothing$ if and only if every element of $S_{+}$is nilpotent. b. Let $\varphi: S \rightarrow T$ be a graded homomorphism of graded rings (preserving degrees). Let $U=\left\{\mathfrak{p} \in\right.\left.T \mathrel{\Big|}\mathfrak{p} \nsupseteq \varphi\left(S_{+}\right)\right\}$. Show that $U$ is an open subset of $\mathop{\mathrm{Proj}}T$, and show that $\varphi$ determines a natural morphism $f: U \rightarrow \operatorname{Proj} S$. c. The morphism $f$ can be an isomorphism even when $\varphi$ is not. For example, suppose that $\varphi_d: S_d \rightarrow T_d$ is an isomorphism for all $d \geqslant d_0$, where $d_0$ is an integer. Then show that $U=\operatorname{Proj} T$ and the morphism $f: \operatorname{Proj} T \rightarrow \operatorname{Proj} S$ is an isomorphism. d. Let $V$ be a projective variety with homogeneous coordinate ring $S$ (See I.2). Show that $t(V) \cong \operatorname{Proj} S$. ### II.2.15 #to_work a. Let $V$ be a variety over the algebraically closed field $k$. Show that a point $P \in t(V)$ is a closed point if and only if its residue field is $k$. b. If $f: X \rightarrow Y$ is a morphism of schemes over $k$, and if $P \in X$ is a point with residue field $k$, then $f(P) \in Y$ also has residue field $k$. c. Now show that if $V, W$ are any two varieties over $k$, then the natural map is bijective. (Injectivity is easy. The hard part is to show it is surjective.) ### II.2.16 #to_work Let $X$ be a scheme, let $f \in \Gamma\left(X, \mathcal{O}_X\right)$, and define $X_f$ to be the subset of points $x \in X$ such that the stalk $f_x$ of $f$ at $x$ is not contained in the maximal ideal $\mathfrak{m}_x$ of the local ring $\mathcal{O}_x$. a. If $U=\operatorname{Spec} B$ is an open affine subscheme of $X$, and if $\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu \in B=\Gamma\left(U,\left.\mathcal{O}_X\right|_U\right)$ is the restriction of $f$, show that $U \cap X_f=D(\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu)$. Conclude that $X_f$ is an open subset of $X$. b. Assume that $X$ is quasi-compact. Let $A=\Gamma\left(X, \mathcal{O}_X\right)$, and let $a \in A$ be an element whose restriction to $X_f$ is 0 . Show that for some $n>0, f^n a=0$. > Hint: Use an open affine cover of $X$. c. Now assume that $X$ has a finite cover by open affines $U_i$ such that each intersection $U_i \cap U_j$ is quasi-compact. (This hypothesis is satisfied, for example, if $\operatorname{sp}(X)$ is noetherian.) Let $b \in \Gamma\left(X_f, \mathcal{O}_{X_f}\right)$. Show that for some $n>0, f^n b$ is the restriction of an element of $A$. d. With the hypothesis of (c), conclude that $\Gamma\left(X_f, \mathcal{O}_{X_f}\right) \cong A_f$. ### II.2.17 A Criterion for Affineness #to_work a. Let $f: X \rightarrow Y$ be a morphism of schemes, and suppose that $Y$ can be covered by open subsets $U_i$, such that for each $i$, the induced map $f^{-1}\left(U_i\right) \rightarrow U_i$ is an isomorphism. Then $f$ is an isomorphism. b. A scheme $X$ is affine if and only if there is a finite set of elements $f_1, \ldots, f_r \in$ $A=\Gamma\left(X, \mathcal{O}_X\right)$, such that the open subsets $X_{f_i}$ are affine, and $f_1, \ldots, f_r$ generate the unit ideal in A.[^28] ### II.2.18 #to_work In this exercise, we compare some properties of a ring homomorphism to the induced morphism of the spectra of the rings. a. Let $A$ be a ring, $X=\operatorname{Spec} A$, and $f \in A$. Show that $f$ is nilpotent if and only if $D(f)$ is empty. b. Let $\varphi: A \rightarrow B$ be a homomorphism of rings, and let $f: Y=\operatorname{Spec} B \rightarrow X = \operatorname{Spec}A$ be the induced morphism of affine schemes. Show that $\varphi$ is injective if and only if the map of sheaves $f^{\sharp}: \mathcal{O}_X \rightarrow f_* \mathcal{O}_Y$ is injective. Show furthermore in that case $f$ is dominant, i.e., $f(Y)$ is dense in $X$. c. With the same notation, show that if $\varphi$ is surjective, then $f$ is a homeomorphism of $Y$ onto a closed subset of $X$, and $f^{\sharp}: \mathcal{O}_X \rightarrow f_* \mathcal{O}_Y$ is surjective. d. Prove the converse to (c), namely, if $f: Y \rightarrow X$ is a homeomorphism onto a closed subset, and $f^{\sharp}: \mathcal{O}_X \rightarrow f_* \mathcal{O}_Y$ is surjective, then $\varphi$ is surjective.[^29] ### II.2.19 #to_work Let $A$ be a ring. Show that the following conditions are equivalent: (i) $\operatorname{Spec} A$ is disconnected; (ii) there exist nonzero elements $e_1, e_2 \in A$ such that $e_1 e_2=0, e_1^2=e_1, e_2^2=e_2$, $e_1+e_2=1$ (these elements are called **orthogonal idempotents**); (iii) $A$ is isomorphic to a direct product $A_1 \times A_2$ of two nonzero rings. ## II.3: First Properties of Schemes ### II.3.1 Show that a morphism $f: X \rightarrow Y$ is locally of finite type if and only if for every open affine subset $V=\operatorname{Spec} B$ of $Y, f^{-1}(V)$ can be covered by open affine subsets $U_j=\operatorname{Spec} A_j$, where each $A_j$ is a finitely generated $B$-algebra. ### II.3.2 A morphism $f: X \rightarrow Y$ of schemes is **quasi-compact** if there is a cover of $Y$ by open affines $V_i$ such that $f^{-1}\left(V_i\right)$ is quasi-compact for each $i$. Show that $f$ is quasicompact if and only if for every open affine subset $V \subseteq Y, f^{-1}(V)$ is quasi-compact. ### II.3.3 a. Show that a morphism $f: X \rightarrow Y$ is of finite type if and only if it is locally of finite type and quasi-compact. b. Conclude from this that $f$ is of finite type if and only if for every open affine subset $V=\operatorname{Spec} B$ of $Y, f^{-1}(V)$ can be covered by a finite number of open affines $U_j=\operatorname{Spec} A_j$, where each $A_j$ is a finitely generated $B$-algebra. c. Show also if $f$ is of finite type, then for every open affine subset $V=\operatorname{Spec} B \subseteq$ $Y$, and for every open affine subset $U=\operatorname{Spec} A \subseteq f^{-1}(V), A$ is a finitely generated $B$-algebra. ### II.3.4 Show that a morphism $f: X \rightarrow Y$ is finite if and only if for every open affine subset $V=\operatorname{Spec} B$ of $Y, f^{-1}(V)$ is affine, equal to $\operatorname{Spec} A$, where $A$ is a finite $B$-module. ### II.3.5 A morphism $f: X \rightarrow Y$ is **quasi-finite** if for every point $y \in Y, f^{-1}(y)$ is a finite set. a. Show that a finite morphism is quasi-finite. b. Show that a finite morphism is closed, i.e., the image of any closed subset is closed. c. Show by example that a surjective, finite-type, quasi-finite morphism need not be finite. ### II.3.6 Let $X$ be an integral scheme. Show that the local ring $\mathcal{O}_{\xi}$ of the generic point $\xi$ of $X$ is a field. It is called the **function field** of $X$, and is denoted by $K(X)$. Show also that if $U=\operatorname{Spec} A$ is any open affine subset of $X$, then $K(X)$ is isomorphic to the quotient field of $A$. ### II.3.7 A morphism $f: X \rightarrow Y$, with $Y$ irreducible, is **generically finite** if $f^{-1}(\eta)$ is a finite set, where $\eta$ is the generic point of $Y$. A morphism $f: X \rightarrow Y$ is **dominant** if $f(X)$ is dense in $Y$. Now let $f: X \rightarrow Y$ be a dominant, generically finite morphism of finite type of integral schemes. Show that there is an open dense subset $U \subseteq Y$ such that the induced morphism $f^{-1}(U) \rightarrow U$ is finite.[^30] ### II.3.8. Normalization. A scheme is **normal** if all of its local rings are integrally closed domains. Let $X$ be an integral scheme. For each open affine subset $U=\operatorname{Spec} A$ of $X$, let $\tilde{A}$ be the integral closure of $A$ in its quotient field, and let $\tilde{U}=\operatorname{Spec} \tilde{A}$. Show that one can glue the schemes $\tilde{U}$ to obtain a normal integral scheme $\tilde{X}$, called the **normalization** of $X$. Show also that there is a morphism $\tilde{X} \rightarrow X$, having the following universal property: for every normal integral scheme $Z$, and for every dominant morphism $f: Z \rightarrow X, f$ factors uniquely through $\tilde{X}$. If $X$ is of finite type over a field $k$, then the morphism $\tilde{X} \rightarrow X$ is a finite morphism. This generalizes (I, Ex. 3.17). ### II.3.9. The Topological Space of a Product. Recall that in the category of varieties, the Zariski topology on the product of two varieties is not equal to the product topology (I, Ex. 1.4). Now we see that in the category of schemes, the underlying point set of a product of schemes is not even the product set. a. Let $k$ be a field, and let $\mathbf{A}_k^1=\operatorname{Spec} k[x]$ be the affine line over $k$. Show that $\mathbf{A}_k^1 \underset{\scriptscriptstyle {\operatorname{Spec}k} }{\times} \mathbf{A}_k^1 \cong \mathbf{A}_k^2$, and show that the underlying point set of the product is not the product of the underlying point sets of the factors (even if $k$ is algebraically closed). b. Let $k$ be a field, let $s$ and $t$ be indeterminates over $k$. Then $\operatorname{Spec}k(s)$, $\operatorname{Spec}k(t)$, and $\operatorname{Spec}k$ are all one-point spaces. Describe the product scheme Spec $k(s) \underset{\scriptscriptstyle {\operatorname{Spec}k} }{\times} \operatorname{Spec} k(t)$. ### II.3.10. Fibres of a Morphism. a. If $f: X \rightarrow Y$ is a morphism, and $y \in Y$ a point, show that $\operatorname{sp}\left(X_y\right)$ is homeomorphic to $f^{-1}(y)$ with the induced topology. b. Let $X=\operatorname{Spec} k[s, t] /\left(s-t^2\right)$, let $Y=\operatorname{Spec} k[s]$, and let $f: X \rightarrow Y$ be the morphism defined by sending $s \rightarrow s$. - If $y \in Y$ is the point $a \in k$ with $a \neq 0$, show that the fibre $X_y$ consists of two points, with residue field $k$. - If $y \in Y$ corresponds to $0 \in k$, show that the fibre $X_y$ is a nonreduced one-point scheme. - If $\eta$ is the generic point of $Y$, show that $X_\eta$ is a one-point scheme, whose residue field is an extension of degree two of the residue field of $\eta$. (Assume $k$ algebraically closed.) ### II.3.11. Closed Subschemes. a. Closed immersions are stable under base extension: if $f: Y \rightarrow X$ is a closed immersion, and if $X^{\prime} \rightarrow X$ is any morphism, then $f^{\prime}: Y \times_X X^{\prime} \rightarrow X^{\prime}$ is also a closed immersion. ```{=html} ``` (b) \* If $Y$ is a closed subscheme of an affine scheme $X=\operatorname{Spec} A$, then $Y$ is also affine, and in fact $Y$ is the closed subscheme determined by a suitable ideal $\mathfrak{a} \subseteq A$ as the image of the closed immersion $\operatorname{Spec} A / \mathfrak{a} \rightarrow \operatorname{Spec} A$.[^31][^32] ```{=html} ``` c. Let $Y$ be a closed subset of a scheme $X$, and give $Y$ the reduced induced subscheme structure. If $Y^{\prime}$ is any other closed subscheme of $X$ with the same underlying topological space, show that the closed immersion $Y \rightarrow X$ factors through $Y^{\prime}$. We express this property by saying that **the reduced induced structure is the smallest subscheme structure on a closed subset**. d. Let $f: Z \rightarrow X$ be a morphism. Then there is a unique closed subscheme $Y$ of $X$ with the following property: the morphism $f$ factors through $Y$, and if $Y^{\prime}$ is any other closed subscheme of $X$ through which $f$ factors, then $Y \rightarrow X$ factors through $Y^{\prime}$ also. We call $Y$ the **scheme-theoretic image** of $f$. If $Z$ is a reduced scheme, then $Y$ is just the reduced induced structure on the closure of the image $f(Z)$. ### II.3.12. Closed Subschemes of Proj $S$. {#ii.3.12.-closed-subschemes-of-proj-s.} a. Let $\varphi: S \rightarrow T$ be a surjective homomorphism of graded rings, preserving degrees. Show that the open set $U$ of (Ex. 2.14) is equal to $\mathop{\mathrm{Proj}}T$, and the morphism $f: \mathop{\mathrm{Proj}}T\to \mathop{\mathrm{Proj}}S$ is a closed immersion. b. If $I \subseteq S$ is a homogeneous ideal, take $T=S / I$ and let $Y$ be the closed subscheme of $X=\operatorname{Proj} S$ defined as image of the closed immersion $\operatorname{Proj} S / I \rightarrow X$. Show that different homogeneous ideals can give rise to the same closed subscheme. For example, let $d_0$ be an integer, and let $I^{\prime}=\bigoplus_{d \geqslant d_0} I_d$. Show that $I$ and $I^{\prime}$ determine the same closed subscheme.[^33] ### II.3.13. Properties of Morphisms of Finite Type. a. A closed immersion is a morphism of finite type. b. A quasi-compact open immersion (Ex. 3.2) is of finite type. c. A composition of two morphisms of finite type is of finite type. d. Morphisms of finite type are stable under base extension. e. If $X$ and $Y$ are schemes of finite type over $S$, then $X \times{ }_S Y$ is of finite type over $S$. f. If $X \stackrel{f}{\rightarrow} Y \stackrel{g}{\rightarrow} Z$ are two morphisms, and if $f$ is quasi-compact, and $g \circ f$ is of finite type, then $f$ is of finite type. g. If $f: X \rightarrow Y$ is a morphism of finite type, and if $Y$ is noetherian, then $X$ is noetherian. ### II.3.14 If $X$ is a scheme of finite type over a field, show that the closed points of $X$ are dense. Give an example to show that this is not true for arbitrary schemes. ### II.3.15 Let $X$ be a scheme of finite type over a field $k$ (not necessarily algebraically closed). a. Show that the following three conditions are equivalent (in which case we say that $X$ is **geometrically irreducible**): - i: $X \underset{\scriptscriptstyle {k} }{\times} \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu$ is irreducible, where $\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu$ denotes the algebraic closure of $k$.[^34] - ii: $X \underset{\scriptscriptstyle {k} }{\times} k_s$ is irreducible, where $k_s$ denotes the separable closure of $k$. - iii: $X \underset{\scriptscriptstyle {k} }{\times} K$ is irreducible for every extension field $K$ of $k$. b. Show that the following three conditions are equivalent (in which case we say $X$ is **geometrically reduced**): - i: $X \underset{\scriptscriptstyle {k} }{\times} \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu$ is reduced. - ii: $X \underset{\scriptscriptstyle {k} }{\times} k_p$ is reduced, where $k_p$ denotes the perfect closure of $k$. - iii: $X \underset{\scriptscriptstyle {k} }{\times} K$ is reduced for all extension fields $K$ of $k$. c. We say that $X$ is **geometrically integral** if $X \underset{\scriptscriptstyle {k} }{\times} \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu$ is integral. Give examples of integral schemes which are neither geometrically irreducible nor geometrically reduced. ### II.3.16. Noetherian Induction. Let $X$ be a noetherian topological space, and let $\mathscr{P}$ be a property of closed subsets of $X$. Assume that for any closed subset $Y$ of $X$, if $\mathscr{P}$ holds for every proper closed subset of $Y$, then $\mathscr{P}$ holds for $Y$. (In particular, $\mathscr{P}$ must hold for the empty set.) Then $\mathscr{P}$ holds for $X$. ### II.3.17. Zariski Spaces. A topological space $X$ is a **Zariski space** if it is noetherian and every (nonempty) closed irreducible subset has a unique generic point (Ex. 2.9). For example, let $R$ be a discrete valuation ring, and let $T=\operatorname{sp}(\operatorname{Spec} R)$. Then $T$ consists of two points $t_0=$ the maximal ideal, $t_1=$ the zero ideal. The open subsets are $\varnothing,\left\{t_1\right\}$, and $T$. This is an irreducible Zariski space with generic point $t_1$. a. Show that if $X$ is a noetherian scheme, then $\operatorname{sp}(X)$ is a Zariski space. b. Show that any minimal nonempty closed subset of a Zariski space consists of one point. We call these closed points. c. Show that a Zariski space $X$ satisfies the axiom $T_0$ :given any two distinct points of $X$, there is an open set containing one but not the other. d. If $X$ is an irreducible Zariski space, then its generic point is contained in every nonempty open subset of $X$ of $x_1$, or that $x_1$ is a generization of $x_0$. Now let $X$ be a Zariski space. - Show that the minimal points, for the partial ordering determined by $x_1>x_0$ if $x_1 \leftrightarrow$ $x_0$, are the closed points, and the maximal points are the generic points of the irreducible components of $X$. - Show also that a closed subset contains every specialization of any of its points. (We say closed subsets are **stable under specialization**.) Similarly, open subsets are stable under generization. e. Let $t$ be the functor on topological spaces introduced in the proof of (2.6). If $X$ is a noetherian topological space, show that $t(X)$ is a Zariski space. Furthermore $X$ itself is a Zariski space if and only if the map $\alpha: X \rightarrow t(X)$ is a homeomorphism. ### II.3.18 Constructible Sets. Let $X$ be a Zariski topological space. A constructible subset of $X$ is a subset which belongs to the smallest family $\mathscr{F}$ of subsets such that (1) every open subset is in $\mathscr{F},(2)$ a finite intersection of elements of $\mathscr{F}$ is in $\mathscr{F}$, and (3) the complement of an element of $\mathfrak{F}$ is in $\mathfrak{F}$. a. A subset of $X$ is locally closed if it is the intersection of an open subset with a closed subset. Show that a subset of $X$ is constructible if and only if it can be written as a finite disjoint union of locally closed subsets. b. Show that a constructible subset of an irreducible Zariski space $X$ is dense if and only if it contains the generic point. Furthermore, in that case it contains a nonempty open subset. c. A subset $S$ of $X$ is closed if and only if it is constructible and stable under specialization. Similarly, a subset $T$ of $X$ is open if and only if it is constructible and stable under generization. d. If $f: X \rightarrow Y$ is a continuous map of Zariski spaces, then the inverse image of any constructible subset of $Y$ is a constructible subset of $X$. ### II.3.19 Let $f: X \rightarrow Y$ be a morphism of finite type of noetherian schemes. Then the image of any constructible subset of $X$ is a constructible subset of $Y$. In particular, $f(X)$, which need not be either open or closed, is a constructible subset of $Y$.[^35] Prove this theorem in the following steps. a. Reduce to showing that $f(X)$ itself is constructible, in the case where $X$ and $Y$ are affine, integral noetherian schemes, and $f$ is a dominant morphism. b. \* In that case, show that $f(X)$ contains a nonempty open subset of $Y$ by using the following result from commutative algebra: let $A \subseteq B$ be an inclusion of noetherian integral domains, such that $B$ is a finitely generated $A$-algebra. Then given a nonzero element $b \in B$, there is a nonzero element $a \in A$ with the following property: if $\varphi: A \rightarrow K$ is any homomorphism of $A$ to an algebraically closed field $K$, such that $\varphi(a) \neq 0$, then $\varphi$ extends to a homomorphism $\varphi^{\prime}$ of $B$ into $K$, such that $\varphi^{\prime}(b) \neq 0$.[^36] c. Now use noetherian induction on $Y$ to complete the proof. d. Give some examples of morphisms $f: X \rightarrow Y$ of varieties over an algebraically closed field $k$, to show that $f(X)$ need not be either open or closed. ### II.3.20. Dimension. Let $X$ be an integral scheme of finite type over a field $k$ (not necessarily algebraically closed). Use appropriate results from I.1 to prove the following. a. For any closed point $P \in X, \operatorname{dim} X=\operatorname{dim} \mathcal{O}_P$, where for rings, we always mean the Krull dimension. b. Let $K(X)$ be the function field of $X$ (Ex. 3.6). Then ` \begin{align*}\operatorname{dim} X=\operatorname{trdeg}K(X)/k.\end{align*} `{=html} c. If $Y$ is a closed subset of $X$, then ` \begin{align*}\operatorname{codim}(Y, X)=\inf \left\{\operatorname{dim} \mathcal{O}_{P, X} \mathrel{\Big|}P \in Y\right\}.\end{align*} `{=html} d. If $Y$ is a closed subset of $X$, then ` \begin{align*}\operatorname{dim} Y+\operatorname{codim}(Y, X)=\operatorname{dim} X.\end{align*} `{=html} e. If $U$ is a nonempty open subset of $X$, then $\operatorname{dim} U=\operatorname{dim} X$. f. If $k \subseteq k^{\prime}$ is a field extension, then every irreducible component of $X^{\prime}=X \underset{\scriptscriptstyle {k} }{\times} k^{\prime}$ has dimension $=\operatorname{dim} X$. ### II.3.21 Let $R$ be a discrete valuation ring containing its residue field $k$. Let $X=$ Spec $R[t]$ be the affine line over Spec $R$. Show that statements (a), (d), (e) of (Ex. 3.20) are false for $X$. ### 3.22. \* Dimension of the Fibres of a Morphism. Let $f: X \rightarrow Y$ be a dominant morphism of integral schemes of finite type over a field $k$. a. Let $Y^{\prime}$ be a closed irreducible subset of $Y$, whose generic point $\eta^{\prime}$ is contained in $f(X)$. Let $Z$ be any irreducible component of $f^{-1}\left(Y^{\prime}\right)$, such that $\eta^{\prime} \in f(Z)$, and show that $\operatorname{codim}(Z, X) \leqslant \operatorname{codim}\left(Y^{\prime}, Y\right)$. b. Let $e=\operatorname{dim} X-\operatorname{dim} Y$ be the relative dimension of $X$ over $Y$. For any point $y \in f(X)$, show that every irreducible component of the fibre $X_y$ has dimension $\geqslant e$.[^37] c. Show that there is a dense open subset $U \subseteq X$, such that for any $y \in f(U)$, $\operatorname{dim} U_y=e$.[^38] d. Going back to our original morphism $f: X \rightarrow Y$, for any integer $h$, let $E_h$ be the set of points $x \in X$ such that, letting $y=f(x)$, there is an irreducible component $Z$ of the fibre $X_y$, containing $x$, and having $\operatorname{dim} Z \geqslant h$. Show that - 1) $E_e=X$[^39]; - 2) if $h>e$, then $E_h$ is not dense in $X$[^40]; and - 3) $E_h$ is closed, for all $h$[^41]. e. Prove the following theorem of Chevalley-see Cartan and Chevalley (1, exposé 8): For each integer $h$, let $C_h$ be the set of points $y \in Y$ such that dim $X_y=h$. Then the subsets $C_h$ are constructible, and $C_e$ contains an open dense subset of $Y$. ### II.3.23 If $V, W$ are two varieties over an algebraically closed field $k$, and if $V \times W$ is their product, as defined in (I, Ex. 3.15, 3.16), and if $t$ is the functor of $(2.6)$, then $t(V \times W)=t(V) \underset{\scriptscriptstyle {\operatorname{Spec}k} }{\times} t(W)$. ## II.4: Separated and Proper Morphisms ### II.4.1 #to_work Show that a finite morphism is proper. ### II.4.2 #to_work Let $S$ be a scheme, let $X$ be a reduced scheme over $S$, and let $Y$ be a separated scheme over $S$. Let $f$ and $g$ be two $S$-morphisms of $X$ to $Y$ which agree on an open dense subset of $X$. Show that $f=g$. Give examples to show that this result fails if either a. $X$ is nonreduced, or b. $Y$ is nonseparated.[^42] ### II.4.3 #to_work Let $X$ be a separated scheme over an affine scheme $S$. Let $U$ and $V$ be open affine subsets of $X$. Then $U \cap V$ is also affine. Give an example to show that this fails if $X$ is not separated. ### II.4.4. The image of a proper scheme is proper. #to_work Let $f: X \rightarrow Y$ be a morphism of separated schemes of finite type over a noetherian scheme $S$. Let $Z$ be a closed subscheme of $X$ which is proper over $S$. Show that $f(Z)$ is closed in $Y$, and that $f(Z)$ with its image subscheme structure (Ex. 3.11d) is proper over $S$.[^43] ### II.4.5 #to_work Let $X$ be an integral scheme of finite type over a field $k$, having function field $K$. We say that a valuation of $K / k$ (see $\mathrm{I}, \S 6$ ) has **center** $x$ on $X$ if its valuation ring $R$ dominates the local ring $\mathcal{O}_{x, X}$. a. If $X$ is separated over $k$, then the center of any valuation of $K / k$ on $X$ (if it exists) is unique. b. If $X$ is proper over $k$, then every valuation of $K / k$ has a unique center on $X$.[^44] c. \* Prove the converses of (a) and (b).[^45] d. If $X$ is proper over $k$, and if $k$ is algebraically closed, show that $\Gamma\left(X, \mathcal{O}_X\right)=k$. This result generalizes (I, 3.4a).[^46] ### II.4.6 #to_work Let $f: X \rightarrow Y$ be a proper morphism of affine varieties over $k$. Then $f$ is a finite morphism.[^47] ### II.4.7. Schemes Over $\mathbf{R}$. #to_work {#ii.4.7.-schemes-over-mathbfr.-to_work} For any scheme $X_0$ over $\mathbf{R}$, let $X=X_0 \times_{\mathbf{R}} \mathbf{C}$. Let $\alpha: \mathbf{C} \rightarrow \mathbf{C}$ be complex conjugation, and let $\sigma: X \rightarrow X$ be the automorphism obtained by keeping $X_0$ fixed and applying $\alpha$ to $\mathbf{C}$. Then $X$ is a scheme over $\mathbf{C}$, and $\sigma$ is a semi-linear automorphism, in the sense that we have a commutative diagram: ```{=tex} \begin{tikzcd} X && X \\ \\ {\operatorname{Spec}{\mathbf{C}}} && {\operatorname{Spec}{\mathbf{C}}} \arrow["\sigma", from=1-1, to=1-3] \arrow["\alpha", from=3-1, to=3-3] \arrow[from=1-1, to=3-1] \arrow[from=1-3, to=3-3] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJYIl0sWzIsMCwiWCJdLFswLDIsIlxcc3BlYyBcXENDIl0sWzIsMiwiXFxzcGVjIFxcQ0MiXSxbMCwxLCJcXHNpZ21hIl0sWzIsMywiXFxhbHBoYSJdLFswLDJdLFsxLDNdXQ==) Since $\sigma^2=\mathrm{id}$, we call $\sigma$ an involution. a. Now let $X$ be a separated scheme of finite type over $\mathbf{C}$, let $\sigma$ be a semilinear involution on $X$, and assume that for any two points $x_1, x_2 \in X$, there is an open affine subset containing both of them. (This last condition is satisfied for example if $X$ is quasi-projective.) Show that there is a unique separated scheme $X_0$ of finite type over $\mathbf{R}$, such that $X_0 \times_{\mathbf{R}} \mathbf{C} \cong X$, and such that this isomorphism identifies the given involution of $X$ with the one on $X_0 \times_{\mathbf{R}} \mathbf{C}$ described above. For the following statements, $X_0$ will denote a separated scheme of finite type over $\mathbf{R}$, and $X, \sigma$ will denote the corresponding scheme with involution over $\mathbf{C}$. b. Show that $X_0$ is affine if and only if $X$ is. c. If $X_0, Y_0$ are two such schemes over $\mathbf{R}$, then to give a morphism $f_0: X_0 \rightarrow Y_0$ is equivalent to giving a morphism $f: X \rightarrow Y$ which commutes with the involutions, i.e., $f \circ \sigma_X=\sigma_Y \circ f$. d. If $X \cong \mathbf{A}_{\mathbf{C}}^1$, then $X_0 \cong \mathbf{A}_{\mathbf{R}}^1$. e. If $X \cong \mathbf{P}_{\mathbf{C}}^1$, then either $X_0 \cong \mathbf{P}_{\mathbf{R}}^1$, or $X_0$ is isomorphic to the conic in $\mathbf{P}_{\mathbf{R}}^2$ given by the homogeneous equation $x_0^2+x_1^2+x_2^2=0$. ### II.4.8 #to_work Let $\mathscr{P}$ be a property of morphisms of schemes such that: a. a closed immersion has $\mathscr{P}$; b. a composition of two morphisms having $\mathscr{P}$ has $\mathscr{P}$; c. $\mathscr{P}$ is stable under base extension. Then show that: d. a product of morphisms having $\mathscr{P}$ has $\mathscr{P}$; e. if $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are two morphisms, and if $g \circ f$ has $\mathscr{P}$ and $g$ is separated, then $f$ has $\mathscr{P}$;[^48] f. If $f: X \rightarrow Y$ has $\mathscr{P}$, then $f_{\text {red }}: X_{\text {red }} \rightarrow Y_{\text {red }}$ has $\mathscr{P}$. ### II.4.9 #to_work Show that a composition of projective morphisms is projective.[^49] Conclude that projective morphisms have properties (a)-(f) of (Ex. 4.8) above. ### II.4.10 \* Chow's Lemma. #to_work This result says that proper morphisms are fairly close to projective morphisms. Let $X$ be proper over a noetherian scheme $S$. Then there is a scheme $X^{\prime}$ and a morphism $g: X^{\prime} \rightarrow X$ such that $X^{\prime}$ is projective over $S$, and there is an open dense subset $U \subseteq X$ such that $g$ induces an isomorphism of $g^{-1}(U)$ to $U$. Prove this result in the following steps. a. Reduce to the case $X$ irreducible. b. Show that $X$ can be covered by a finite number of open subsets $U_i, i=1, \ldots, n$, each of which is quasi-projective over $S$. Let $U_i \rightarrow P_i$ be an open immersion of $U_i$ into a scheme $P_i$ which is projective over $S$. c. Let $U=\bigcap U_i$, and consider the map ` \begin{align*} f: U \to X \underset{\scriptscriptstyle {S} }{\times} P_1 \underset{\scriptscriptstyle {S} }{\times} \cdots \underset{\scriptscriptstyle {S} }{\times} P_n \end{align*} `{=html} deduced from the given maps $U \rightarrow X$ and $U \rightarrow P_i$. Let $X^{\prime}$ be the closed image subscheme structure[^50] $f(U)^{-}$. Let $g: X^{\prime} \rightarrow X$ be the projection onto the first factor, and let ` \begin{align*} h: X' \to P = P_1 \underset{\scriptscriptstyle {k} }{\times} \cdots \underset{\scriptscriptstyle {k} }{\times} P_n \end{align*} `{=html} be the projection onto the product of the remaining factors. Show that $h$ is a closed immersion, hence $X^{\prime}$ is projective over $S$. d. Show that $g^{-1}(U) \rightarrow U$ is an isomorphism, thus completing the proof. ### II.4.11 #to_work If you are willing to do some harder commutative algebra, and stick to noetherian schemes, then we can express the valuative criteria of separatedness and properness using only **discrete** valuation rings. a. If $\mathcal{O}, {\mathfrak{m}}$ is a noetherian local domain with quotient field $K$, and if $L$ is a finitely generated field extension of $K$, then there exists a discrete valuation ring $R$ of $L$ dominating $\mathcal{O}$. Prove this in the following steps. - By taking a polynomial ring over $\mathcal{O}$, reduce to the case where $L$ is a finite extension field of $K$. - Then show that for a suitable choice of generators $x_1, \ldots, x_n$ of $m$, the ideal $\mathfrak{a}=\left(x_1\right)$ in $\mathcal{O}^{\prime}=\mathbb{C}\left[x_2 / x_1, \ldots, x_n / x_1\right]$ is not equal to the unit ideal. - Then let $\mathfrak{p}$ be a minimal prime ideal of $a$, and let $\mathcal{O}_p^{\prime}$ be the localization of $\mathcal{O}^{\prime}$ at $\mathfrak{p}$. This is a noetherian local domain of dimension 1 dominating $\mathcal{O}$. - Let $\tilde{\mathscr{O}}_p^{\prime}$ be the integral closure of $\mathcal{O}_p^{\prime}$ in $L$. Use the theorem of Krull-Akizuki[^51] to show that $\tilde{U}_{\mathrm{p}}^{\prime}$ is noetherian of dimension 1. - Finally, take $R$ to be a localization of $\tilde{{\mathcal{O}}}_p^{\prime}$ at one of its maximal ideals. b. Let $f: X \rightarrow Y$ be a morphism of finite type of noetherian schemes. Show that $f$ is separated (respectively, proper) if and only if the criterion of $(4.3)$ (respectively, (4.7)) holds for all discrete valuation rings. ### II.4.12 Examples of Valuation Rings. #to_work Let $k$ be an algebraically closed field. a. If $K$ is a function field of dimension 1 over $k$ then every valuation ring of $K / k$ (except for $K$ itself) is discrete. Thus the set of all of them is just the abstract nonsingular curve $C_K$ of $(I, \S 6)$. b. If $K / k$ is a function field of dimension two, there are several different kinds of valuations. Suppose that $X$ is a complete nonsingular surface with function field $K$. - If $Y$ is an irreducible curve on $X$, with generic point $x_1$, then the local ring $R=\mathcal{O}_{x_1, X}$ is a discrete valuation ring of $K / k$ with center at the (nonclosed) point $x_1$ on $X$. - If $f: X^{\prime} \rightarrow X$ is a birational morphism, and if $Y^{\prime}$ is an irreducible curve in $X^{\prime}$ whose image in $X$ is a single closed point $x_0$, then the local ring $R$ of the generic point of $Y^{\prime}$ on $X^{\prime}$ is a discrete valuation ring of $K / k$ with center at the closed point $x_0$ on $X$. - Let $x_0 \in X$ be a closed point. Let $f: X_1 \rightarrow X$ be the blowing-up of $x_0$ (I, §4) and let $E_1=f^{-1}\left(x_0\right)$ be the exceptional curve. Choose a closed point $x_1 \in E_1$, let $f_2: X_2 \rightarrow X_1$ be the blowing-up of $x_1$, and let $E_2=$ $f_2^{-1}\left(x_1\right)$ be the exceptional curve. Repeat. In this manner we obtain a sequence of varieties $X_i$ with closed points $x_i$ chosen on them, and for each $i$, the local ring $\mathcal{O}_{x_{i+1}, X_{i+1}}$ dominates $\mathcal{O}_{x_i, X_i}$. Let $R_0=\bigcup_{i=0}^{\infty} \mathcal{O}_{x_i, X_i}$. Then $R_0$ is a local ring, so it is dominated by some valuation ring $R$ of $K / k$ by (I, 6.1A). Show that $R$ is a valuation ring of $K / k$, and that it has center $x_0$ on $X$. When is $R$ a discrete valuation ring?[^52] ## II.5: Sheaves of Modules ### II.5.1. #to_work Let $\left(X, {\mathcal{O}}_X\right)$ be a ringed space, and let $\mathcal{E}$ be a locally free ${\mathcal{O}}_X$-module of finite rank. We define the dual of $\mathcal{E}$, denoted ${\mathcal{E}} {}^{ \vee }$, to be the sheaf $\mathop{\mathcal{H}\! \mathit{om}}_{{\mathcal{O}}_X}({\mathcal{E}}, {\mathcal{O}}_X)$. a. Show that $({\mathcal{E}} {}^{ \vee }) {}^{ \vee }\cong \mathcal{E}$. b. For any ${\mathcal{O}}_X$-module $\mathcal{F}$, ` \begin{align*} \mathop{\mathcal{H}\! \mathit{om}}_{{\mathcal{O}}_X}({\mathcal{E}}\otimes{\mathcal{F}}, {\mathcal{G}}) \cong \mathop{\mathcal{H}\! \mathit{om}}_{{\mathcal{O}}_X}({\mathcal{F}}, \mathop{\mathcal{H}\! \mathit{om}}_{{\mathcal{O}}_X}({\mathcal{E}},{\mathcal{G}}) ) .\end{align*} `{=html} c. (Projection Formula). If $f:\left(X, {\mathcal{O}}_X\right) \rightarrow\left(Y, {\mathcal{O}}_Y\right)$ is a morphism of ringed spaces, if $\mathcal{F}$ is an ${\mathcal{O}}_X$-module, and if $\mathcal{E}$ is a locally free ${\mathcal{O}}_Y$-module of finite rank, then there is a natural isomorphism ` \begin{align*} f_*\left(\mathcal{F} \otimes_{{\mathcal{O}}_X} f^* \mathcal{E}\right) \cong f_*(\mathcal{F}) \otimes_{{\mathcal{O}}_Y} \mathcal{E} .\end{align*} `{=html} ### II.5.2. #to_work Let $R$ be a discrete valuation ring with quotient field $K$, and let $X=\operatorname{Spec} R$. a. To give an ${\mathcal{O}}_X$-module is equivalent to giving an $R$-module $M$, a $K$-vector space $L$, and a homomorphism $\rho: M \otimes_R K \rightarrow L$. b. That ${\mathcal{O}}_X$-module is quasi-coherent if and only if $\rho$ is an isomorphism. ### II.5.3. #to_work Let $X=\operatorname{Spec} A$ be an affine scheme. Show that the functors $\sim$ and $\Gamma$ are adjoint, in the following sense: for any $A$-module $M$, and for any sheaf of ${\mathcal{O}}_X$-modules $\mathcal{F}$, there is a natural isomorphism ` \begin{align*} \operatorname{Hom}_A(M, \Gamma(X, \mathcal{F})) \cong \operatorname{Hom}_{{\mathcal{O}}_X}( \tilde{M}, \mathcal{F} ) \end{align*} `{=html} ### II.5.4. #to_work Show that a sheaf of ${\mathcal{O}}_X$-modules $\mathcal{F}$ on a scheme $X$ is quasi-coherent if and only if every point of $X$ has a neighborhood $U$, such that $\left.\mathcal{F}\right|_U$ is isomorphic to a cokernel of a morphism of free sheaves on $U$. If $X$ is noetherian, then $\mathcal{F}$ is coherent if and only if it is locally a cokernel of a morphism of free sheaves of finite rank. (These properties were originally the definition of quasi-coherent and coherent sheaves.) ### II.5.5. #to_work Let $f: X \rightarrow Y$ be a morphism of schemes. a. Show by example that if $\mathcal{F}$ is coherent on $X$, then $f_* \mathcal{F}$ need not be coherent on $Y$, even if $X$ and $Y$ are varieties over a field $k$. b. Show that a closed immersion is a finite morphism ($\S 3$). c. If $f$ is a finite morphism of noetherian schemes, and if $\mathcal{F}$ is coherent on $X$, then $f_* \mathcal{F}$ is coherent on $Y$. ### II.5.6. Support. #to_work Recall the notions of support of a section of a sheaf, support of a sheaf, and subsheaf with supports from (Ex. 1.14) and (Ex. 1.20). a. Let $A$ be a ring, let $M$ be an $A$-module, let $X=\operatorname{Spec} A$, and let $\mathcal{F}=\tilde{M}$. For any $m \in M=\Gamma(X, \mathcal{F})$, show that $\mathop{\mathrm{supp}}m = V(\operatorname{Ann}m )$, where $\operatorname{Ann}m$ is the annihilator of $m=\{a \in A \mathrel{\Big|}a m=0\}$. b. Now suppose that $A$ is noetherian, and $M$ finitely generated. Show that $\mathop{\mathrm{supp}}\mathcal{F}=V( \operatorname{Ann}M)$. c. The support of a coherent sheaf on a noetherian scheme is closed. d. For any ideal $\mathfrak{a} \subseteq A$, we define a submodule $\Gamma_{{\mathfrak{a}}}(M)$ of $M$ by ` \begin{align*} \Gamma_{{\mathfrak{a}}}(M)= \left\{m \in M \mathrel{\Big|}\mathfrak{a}^n m=0 \text{ for some } n > 0 \right\} \end{align*} `{=html} Assume that $A$ is noetherian, and $M$ any $A$-module. Show that $\Gamma_{\mathfrak{a}}(M)^{\sim} \cong \mathcal{H}_Z^0(\mathcal{F})$, where $Z=V(\mathfrak{a})$ and $\mathcal{F}=\tilde{M}$.[^53] ```{=html} ``` e. Let $X$ be a noetherian scheme, and let $Z$ be a closed subset. If $\mathcal{F}$ is a quasicoherent (respectively, coherent) ${\mathcal{O}}_X$-module, then $\mathcal{H}_Z^0(\mathcal{F})$ is also quasicoherent (respectively, coherent). ### II.5.7. #to_work Let $X$ be a noetherian scheme, and let $\mathcal{F}$ be a coherent sheaf. a. If the stalk $\mathcal{F}_x$ is a free ${\mathcal{O}}_x$-module for some point $x \in X$, then there is a neighborhood $U$ of $x$ such that $\left.\mathcal{F}\right|_U$ is free. b. $\mathcal{F}$ is locally free if and only if its stalks $\mathcal{F}_x$ are free ${\mathcal{O}}_x$-modules for all $x \in X$. c. $\mathcal{F}$ is invertible (i.e., locally free of rank 1) if and only if there is a coherent sheaf $\mathcal{G}$ such that $\mathcal{F} \otimes \mathcal{G} \cong {\mathcal{O}}_X$.[^54] ### II.5.8. #to_work Again let $X$ be a noetherian scheme, and $\mathcal{F}$ a coherent sheaf on $X$. We will consider the function ` \begin{align*} \varphi(x)=\operatorname{dim}_{k(x)} \mathcal{F}_x \otimes_{{\mathcal{O}}_x} k(x), \end{align*} `{=html} where $k(x)={\mathcal{O}}_x / m_x$ is the residue field at the point $x$. Use Nakayama's lemma to prove the following results. a. The function $\varphi$ is upper semi-continuous, i.e., for any $n \in \mathbf{Z}$, the set $\{x \in X \mathrel{\Big|}\varphi(x) \geqslant n\}$ is closed. b. If $\mathcal{F}$ is locally free, and $X$ is connected, then $\varphi$ is a constant function. c. Conversely, if $X$ is reduced, and $\varphi$ is constant, then $\mathcal{F}$ is locally free. ### II.5.9. #to_work Let $S$ be a graded ring, generated by $S_1$ as an $S_0$-algebra, let $M$ be a graded $S$ module, and let $X=$ Proj S. a. Show that there is a natural homomorphism $\alpha: M \rightarrow \Gamma_*(\tilde{M})$. b. Assume now that $S_0=A$ is a finitely generated $k$-algebra for some field $k$, that $S_1$ is a finitely generated $A$-module, and that $M$ is a finitely generated $S$-module. Show that the map $\alpha$ is an isomorphism in all large enough degrees, i.e., there is a $d_0 \in \mathbf{Z}$ such that for all $d \geqslant d_0, \alpha_d: M_d \rightarrow \Gamma(X, \tilde{M}(d))$ is an isomorphism.[^55] c. With the same hypotheses, we define an equivalence relation $\approx$ on graded $S$-modules by saying $M \approx M^{\prime}$ if there is an integer $d$ such that $M_{\geqslant d} \cong M_{\geqslant d}^{\prime}$. Here $M_{\geqslant d}=\bigoplus_{n \geqslant d} M_n$. We will say that a graded $S$-module $M$ is **quasifinitely generated** if it is equivalent to a finitely generated module. Now show that the functors $\sim$ and $\Gamma_*$ induce an equivalence of categories between the category of quasi-finitely generated graded $S$-modules modulo the equivalence relation $\approx$, and the category of coherent ${\mathcal{O}}_X$-modules. ### II.5.10. #to_work Let $A$ be a ring, let $S=A\left[x_0, \ldots, x_r\right]$ and let $X=\operatorname{Proj} S$. We have seen that a homogeneous ideal $I$ in $S$ defines a closed subscheme of $X$ (Ex. 3.12), and that conversely every closed subscheme of $X$ arises in this way (5.16). a. For any homogeneous ideal $I \subseteq S$, we define the saturation $\mkern 1.5mu\overline{\mkern-1.5muI\mkern-1.5mu}\mkern 1.5mu$ of $I$ to be $\left\{s \in S \mathrel{\Big|}\right.$ for each $i=0, \ldots, r$, there is an $n$ such that $\left.x_i^n s \in I\right\}$. We say that $I$ is saturated if $I=\mkern 1.5mu\overline{\mkern-1.5muI\mkern-1.5mu}\mkern 1.5mu$. Show that $\mkern 1.5mu\overline{\mkern-1.5muI\mkern-1.5mu}\mkern 1.5mu$ is a homogeneous ideal of $S$. b. Two homogeneous ideals $I_1$ and $I_2$ of $S$ define the same closed subscheme of $X$ if and only if they have the same saturation. c. If $Y$ is any closed subscheme of $X$, then the ideal $\Gamma_*\left(\mathcal{I}_Y\right)$ is saturated. Hence it is the largest homogeneous ideal defining the subscheme $Y$. d. There is a 1-1 correspondence between saturated ideals of $S$ and closed subschemes of $X$. ### II.5.11. #to_work Let $S$ and $T$ be two graded rings with $S_0=T_0=A$. We define the **Cartesian product** $S \underset{\scriptscriptstyle {A} }{\times} T$ to be the graded ring $\bigoplus_{d \geqslant 0} S_d \otimes_A T_d$. If $X=\operatorname{Proj} S$ and $Y=\operatorname{Proj} T$, show that $\operatorname{Proj}\left(S \times{ }_A T\right) \cong X \times{ }_A Y$, and show that the sheaf ${\mathcal{O}}(1)$ on $\operatorname{Proj}\left(S \times{ }_A T\right)$ is isomorphic to the sheaf $p_1^*\left({\mathcal{O}}_X(1)\right) \otimes p_2^*\left({\mathcal{O}}_Y(1)\right)$ on $X \times Y$. The Cartesian product of rings is related to the **Segre embedding** of projective spaces (I, Ex. 2.14) in the following way. If $x_0, \ldots, x_r$ is a set of generators for $S_1$ over $A$, corresponding to a projective embedding $X \hookrightarrow \mathbf{P}_A^r$, and if $y_0, \ldots, y_s$ is a set of generators for $T_1$, corresponding to a projective embedding $Y \hookrightarrow P_A^s$, then $\left\{x_i \otimes y_j\right\}$ is a set of generators for $\left(S \times{ }_A T\right)_1$, and hence defines a projective embedding $\mathop{\mathrm{Proj}}(S \underset{\scriptscriptstyle {A} }{\times} T) \hookrightarrow{\mathbf{P}}^N_A$, with $N=r s+r+s$. This is just the image of $X \times Y \subseteq \mathbf{P}^r \times \mathbf{P}^s$ in its Segre embedding. ### II.5.12. #to_work a. Let $X$ be a scheme over a scheme $Y$, and let $\mathcal{L}, \mathcal{M}$ be two very ample invertible sheaves on $X$. Show that $\mathcal{L} \otimes \mathcal{M}$ is also very ample.[^56] b. Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be two morphisms of schemes. Let $\mathcal{L}$ be a very ample invertible sheaf on $X$ relative to $Y$, and let $\mathcal{M}$ be a very ample invertible sheaf on $Y$ relative to $Z$. Show that $\mathcal{L} \otimes f^* \mathcal{M}$ is a very ample invertible sheaf on $X$ relative to $Z$. ### II.5.13. #to_work Let $S$ be a graded ring, generated by $S_1$ as an $S_0$-algebra. For any integer $d>0$, let $S^{(d)}$ be the graded ring $\bigoplus_{n \geqslant 0} S_n^{(d)}$ where $S_n^{(d)}=S_{n d}$. Let $X=$ Proj $S$. Show that Proj $S^{(d)} \cong X$, and that the sheaf ${\mathcal{O}}(1)$ on Proj $S^{(d)}$ corresponds via this isomorphism to ${\mathcal{O}}_X(d)$. This construction is related to the $d$-uple embedding (I, Ex. 2.12) in the following way. If $x_0, \ldots, x_r$ is a set of generators for $S_1$, corresponding to an embedding $X \hookrightarrow \mathbf{P}_A^r$, then the set of monomials of degree $d$ in the $x_i$ is a set of generators for $S_1^{(d)}=S_d$. These define a projective embedding of Proj $S^{(d)}$ which is none other than the image of $X$ under the $d$-uple embedding of $\mathbf{P}_A^r$. ### II.5.14. #to_work Let $A$ be a ring, and let $X$ be a closed subscheme of ${\mathbf{P}}_A^r$. We define the **homogeneous coordinate ring** $S(X)$ of $X$ for the given embedding to be $A\left[x_0, \ldots, x_r\right] / I$, where $I$ is the ideal $\Gamma_*\left(\mathcal{I}_X\right)$ constructed in the proof of $(5.16)$. Of course if $A$ is a field and $X$ a variety, this coincides with the definition given in (I, §2)! Recall that a scheme $X$ is **normal** if its local rings are integrally closed domains. A closed subscheme $X \subseteq \mathbf{P}_A^r$ is **projectively normal** for the given embedding, if its homogeneous coordinate ring $S(X)$ is an integrally closed domain (cf. (I, Ex. 3.18)). Now assume that $k$ is an algebraically closed field, and that $X$ is a connected, normal closed subscheme of $\mathbf{P}_k^r$. Show that for some $d>0$, the $d$-uple embedding of $X$ is projectively normal, as follows. a. Let $S$ be the homogeneous coordinate ring of $X$, and let $S^{\prime}=\bigoplus_{n \geqslant 0} \Gamma\left(X, {\mathcal{O}}_X(n)\right)$. Show that $S$ is a domain, and that $S^{\prime}$ is its integral closure.[^57] b. Use (Ex. 5.9) to show that $S_d=S_d^{\prime}$ for all sufficiently large $d$. c. Show that $S^{(d)}$ is integrally closed for sufficiently large $d$, and hence conclude that the $d$-uple embedding of $X$ is projectively normal. d. As a corollary of (a), show that a closed subscheme $X \subseteq \mathbf{P}_A^r$ is projectively normal if and only if it is normal, and for every $n \geqslant 0$ the natural map $\Gamma\left(\mathbf{P}^r, {\mathcal{O}}_{\mathbf{P}^r}(n)\right) \rightarrow \Gamma\left(X, {\mathcal{O}}_X(n)\right)$ is surjective. ### II.5.15. Extension of Coherent Sheaves. #to_work We will prove the following theorem in several steps: Let $X$ be a noetherian scheme, let $U$ be an open subset, and let $\mathcal{F}$ be a coherent sheaf on $U$. Then there is a coherent sheaf $\mathcal{F}^{\prime}$ on $X$ such that $\left.\mathcal{F}^{\prime}\right|_U \cong \mathcal{F}$. a. On a noetherian affine scheme, every quasi-coherent sheaf is the union of its coherent subsheaves. We say a sheaf $\mathcal{F}$ is the union of its subsheaves $\mathcal{F}$ if for every open set $U$, the group $\mathcal{F}(U)$ is the union of the subgroups $\mathcal{F}_\alpha(U)$. b. Let $X$ be an affine noetherian scheme, $U$ an open subset, and $\mathcal{F}$ coherent on $U$. Then there exists a coherent sheaf $\mathcal{F}^{\prime}$ on $X$ with $\left.\mathcal{F}^{\prime}\right|_U \cong \mathcal{F}$.[^58] c. With $X, U, \mathcal{F}$ as in (b), suppose furthermore we are given a quasi-coherent sheaf $\mathcal{G}$ on $X$ such that $\left.\mathcal{F} \subseteq \mathcal{G}\right|_U$. Show that we can find $\mathcal{F}^{\prime}$ a coherent subsheaf of $\mathcal{G}$, with $\left.\mathcal{F}^{\prime}\right|_U \cong \mathcal{F}$.[^59] d. Now let $X$ be any noetherian scheme, $U$ an open subset, $\mathcal{F}$ a coherent sheaf on $U$, and $\mathcal{G}$ a quasi-coherent sheaf on $X$ such that $\left.\mathcal{F} \subseteq \mathcal{G}\right|_U$. Show that there is a coherent subsheaf $\mathcal{F}^{\prime} \subseteq \mathcal{G}$ on $X$ with $\left.\mathcal{F}^{\prime}\right|_U \cong \mathcal{F}$. Taking $\mathcal{G}=i_* \mathcal{F}$ proves the result announced at the beginning.[^60] e. As an extra corollary, show that on a noetherian scheme, any quasi-coherent sheaf $\mathcal{F}$ is the union of its coherent subsheaves.[^61] ### II.5.16. Tensor Operations on Sheaves. #to_work First we recall the definitions of various tensor operations on a module. Let $A$ be a ring, and let $M$ be an $A$-module. - Let $T^n(M)$ be the tensor product $M \otimes \ldots \otimes M$ of $M$ with itself $n$ times, for $n \geqslant 1$. For $n=0$ we put $T^0(M)=A$. Then $T(M)=\bigoplus_{n \geqslant 0} T^n(M)$ is a (noncommutative) $A$-algebra, which we call the **tensor algebra** of $M$. - We define the **symmetric algebra** $S(M)=\bigoplus_{n \geqslant 0} S^n(M)$ of $M$ to be the quotient of $T(M)$ by the two-sided ideal generated by all expressions $x \otimes y-y \otimes x$, for all $x, y \in M$. Then $S(M)$ is a commutative $A$-algebra. Its component $S^n(M)$ in degree $n$ is called the $n$th **symmetric product** of $M$. We denote the image of $x \otimes y$ in $S(M)$ by $x y$, for any $x, y \in M$. As an example, note that if $M$ is a free $A$-module of rank $r$, then $S(M) \cong$ $A\left[x_1, \ldots, x_r\right]$ - We define the **exterior algebra** $\bigwedge(M)=\bigoplus_{n \geqslant 0} \bigwedge^n(M)$ of $M$ to be the quotient of $T(M)$ by the two-sided ideal generated by all expressions $x \otimes x$ for $x \in M$. Note that this ideal contains all expressions of the form $x \otimes y+y \otimes x$, so that $\bigwedge(M)$ is a skew commutative graded $A$-algebra. This means that if $u \in$ $\bigwedge^r(M)$ and $v \in \bigwedge^s(M)$, then $u \wedge v=(-1)^{r s} v \wedge u$ (here we denote by $\wedge$ the multiplication in this algebra; so the image of $x \otimes y$ in $\bigwedge^2(M)$ is denoted by $x \wedge y$ ). The $n$th component $\bigwedge^n(M)$ is called the $n$th **exterior power** of $M$. Now let $\left(X, {\mathcal{O}}_X\right)$ be a ringed space, and let $\mathcal{F}$ be a sheaf of ${\mathcal{O}}_X$-modules. We define the tensor algebra, symmetric algebra, and exterior algebra of $\mathcal{F}$ by taking the sheaves associated to the presheaf, which to each open 'set $U$ assigns the corresponding tensor operation applied to $\mathcal{F}(U)$ as an ${\mathcal{O}}_X(U)$-module. The results are ${\mathcal{O}}_x$-algebras, and their components in each degree are ${\mathcal{O}}_x$-modules. a. Suppose that $\mathcal{F}$ is locally free of rank $n$. Then $T^r(\mathcal{F}), S^r(\mathcal{F})$, and $\bigwedge^r(\mathcal{F})$ are also locally free, of ranks $n^r,\left(\begin{array}{c}n+r-1 \\ n-1\end{array}\right)$, and $\left(\begin{array}{c}r \\ r\end{array}\right)$ respectively. b. Again let $\mathcal{F}$ be locally free of rank $n$. Then the multiplication map $\bigwedge^r \mathcal{F} \otimes$ $\bigwedge^{n-r} \mathcal{F} \rightarrow \bigwedge^n \cdot \mathcal{F}$ is a perfect pairing for any $r$, i.c., it induces an isomorphism of $\bigwedge^r \mathcal{F}$ with $\left(\bigwedge^{n-r} \mathcal{F}\right)^{\ulcorner} \otimes \bigwedge^n \mathcal{F}$. As a special case, note if $\mathcal{F}$ has rank 2 , then $\mathcal{F} \cong \mathcal{F}^{\top} \otimes \bigwedge^2 \mathcal{F}$. c. Let $0 \rightarrow \mathcal{F}^{\prime} \rightarrow \mathcal{F} \rightarrow \mathcal{F}^{\prime \prime} \rightarrow 0$ be an exact sequence of locally free sheaves. Then for any $r$ there is a finite filtration of $S^r(\mathcal{F})$, ` \begin{align*} S^r(\mathcal{F})=F^0 \supseteq F^1 \supseteq \ldots \supseteq F^r \supseteq F^{r+1}=0 \end{align*} `{=html} with quotients ` \begin{align*} F^p / F^{p+1} \cong S^p\left(\mathcal{F}^{\prime}\right) \otimes S^{r-p}\left(\mathcal{F}^{\prime \prime}\right) \end{align*} `{=html} for each $p$. d. Same statement as (c), with exterior powers instead of symmetric powers. In particular, if $\mathcal{F}^{\prime}, \mathcal{F}, \mathcal{F}^{\prime \prime}$ have ranks $n^{\prime}, n, n^{\prime \prime}$ respectively, there is an isomorphism e. Let $f: X \rightarrow Y$ be a morphism of ringed spaces, and let $\mathcal{F}$ be an ${\mathcal{O}}_Y$-module. Then $f^*$ commutes with all the tensor operations on $\mathcal{F}$, i.e., $f^*\left(S^n(\mathcal{F})\right)=$ $S^n\left(f^* \mathcal{F}\right)$ etc. ### II.5.17. Affine Morphisms. #to_work A morphism $f: X \rightarrow Y$ of schemes is **affine** if there is an open affine cover $\left\{V_i\right\}$ of $Y$ such that $f^{-1}\left(V_i\right)$ is affine for each $i$. a. Show that $f: X \rightarrow Y$ is an affine morphism if and only if for every open affine $V \subseteq Y, f^{-1}(V)$ is affine[^62] b. An affine morphism is quasi-compact and separated. Any finite morphism is affine. c. Let $Y$ be a scheme, and let $\mathcal{A}$ be a quasi-coherent sheaf of ${\mathcal{O}}_Y$-algebras (i.e., a sheaf of rings which is at the same time a quasi-coherent sheaf of ${\mathcal{O}}_Y$-modules). Show that there is a unique scheme $X$, and a morphism $f: X \rightarrow Y$, such that for every open affine $V \subseteq Y, f^{-1}(V) \cong \operatorname{Spec} \mathcal{A}(V)$, and for every inclusion $U \hookrightarrow V$ of open affines of $Y$, the morphism $f^{-1}(U) \hookrightarrow f^{-1}(V)$ corresponds to the restriction homomorphism $\mathcal{A}(V) \rightarrow \mathcal{A}(U)$. The scheme $X$ is called $\operatorname{Spec}\mathcal{A}$.[^63] d. If $\mathcal{A}$ is a quasi-coherent ${\mathcal{O}}_Y$-algebra, then $f: X=$ Spec $\mathcal{A} \rightarrow Y$ is an affine morphism, and $\mathcal{A} \cong f_* {\mathcal{O}}_X$. Conversely, if $f: X \rightarrow Y$ is an affine morphism, then $\mathcal{A}=f_* {\mathcal{O}}_X$ is a quasi-coherent sheaf of ${\mathcal{O}}_Y$-algebras, and $X \cong \operatorname{Spec} \mathcal{A}$. e. Let $f: X \rightarrow Y$ be an affine morphism, and let $\mathcal{A}=f_* {\mathcal{O}}_X$. Show that $f_*$ induces an equivalence of categories from the category of quasi-coherent ${\mathcal{O}}_X$-modules to the category of quasi-coherent $\mathcal{A}$-modules (i.e., quasi-coherent ${\mathcal{O}}_Y$-modules having a structure of $\mathcal{A}$-module).[^64] ### II.5.18. Vector Bundles. #to_work Let $Y$ be a scheme. A **(geometric) vector bundle** of rank $n$ over $Y$ is a scheme $X$ and a morphism $f: X \rightarrow Y$, together with additional data consisting of an open covering $\left\{U_i\right\}$ of $Y$, and isomorphisms $\psi_i: f^{-1}\left(U_i\right) \rightarrow \mathbf{A}_{U_i}^n$, such that for any $i, j$, and for any open affine subset $V=\operatorname{Spec} A \subseteq U_i \cap U_j$, the automorphism $\psi=\psi_j \circ \psi_i^{-1}$ of $\mathbf{A}_V^n=\operatorname{Spec} A\left[x_1, \ldots, x_n\right]$ is given by a *linear* automorphism $\theta$ of $A\left[x_1, \ldots, x_n\right]$, i.e., $\theta(a)=a$ for any $a \in A$, and $\theta\left(x_i\right)=$ $\sum a_{i j} x_j$ for suitable $a_{i j} \in A$. An **isomorphism** ` \begin{align*} g:\left(X, f,\left\{U_i\right\},\left\{\psi_i\right\}\right) \rightarrow\left(X^{\prime}, f^{\prime},\left\{U_i^{\prime}\right\},\left\{\psi_i^{\prime}\right\}\right) \end{align*} `{=html} of one vector bundle of rank $n$ to another one is an isomorphism $g: X \rightarrow X^{\prime}$ of the underlying schemes, such that $f=f^{\prime} \circ g$, and such that $X, f$, together with the covering of $Y$ consisting of all the $U_i$ and $U_i^{\prime}$, and the isomorphisms $\psi_i$ and $\psi_i^{\prime} \circ g$, is also a vector bundle structure on $X$. a. Let $\mathcal{E}$ be a locally free sheaf of rank $n$ on a scheme $Y$. Let $S(\mathcal{E})$ be the symmetric algebra on $\mathcal{E}$, and let $X=\operatorname{Spec} S(\mathcal{E})$, with projection morphism $f: X \rightarrow Y$. For each open affine subset $U \subseteq Y$ for which $\left.\mathcal{E}\right|_U$ is free, choose a basis of $\mathcal{E}$, and let $\psi: f^{-1}(U) \rightarrow \mathbf{A}_U^n$ be the isomorphism resulting from the identification of $S(\mathcal{E}(U))$ with ${\mathcal{O}}(U)\left[x_1, \ldots, x_n\right]$. Then $(X, f,\{U\},\{\psi\})$ is a vector bundle of rank $n$ over $Y$, which (up to isomorphism) does not depend on the bases of $\mathcal{E}_U$ chosen. We call it the geometric vector bundle associated to $\mathcal{E}$, and denote it by $\mathbf{V}(\mathcal{E})$. b. For any morphism $f: X \rightarrow Y$, a section of $f$ over an open set $U \subseteq Y$ is a morphism $s: U \rightarrow X$ such that $f \circ s=\mathrm{id}_U$. It is clear how to restrict sections to smaller open sets, or how to glue them together, so we see that the presheaf $U \mapsto\{$ set of sections of $f$ over $U\}$ is a sheaf of sets on $Y$, which we denote by $\mathcal{S}(X / Y)$. Show that if $f: X \rightarrow Y$ is a vector bundle of rank $n$, then the sheaf of sections $\mathcal{S}(X / Y)$ has a natural structure of ${\mathcal{O}}_Y$-module, which makes it a locally free ${\mathcal{O}}_Y$-module of rank $n$.[^65] c. Again let $\mathcal{E}$ be a locally free sheaf of rank $n$ on $Y$, let $X=\mathbf{V}(\mathcal{E})$, and let $\mathcal{S}=$ $\mathcal{S}(X / Y)$ be the sheaf of sections of $X$ over $Y$. Show that $\mathcal{S} \cong \mathcal{E}^2$, as follows. Given a section $s \in \Gamma\left(V, \mathcal{E}^{\curlyvee}\right)$ over any open set $V$, we think of $s$ as an element of $\operatorname{Hom}\left(\left.\mathcal{E}\right|_V, {\mathcal{O}}_V\right)$. So $s$ determines an ${\mathcal{O}}_V$-algebra homomorphism $S\left(\left.\mathcal{E}\right|_V\right) \rightarrow {\mathcal{O}}_V$. This determines a morphism of spectra $V=\operatorname{Spec} {\mathcal{O}}_V \rightarrow \operatorname{Spec} S\left(\left.\mathcal{E}\right|_V\right)=$ $f^{-1}(V)$, which is a section of $X / Y$. Show that this construction gives an isomorphism of $\mathcal{E}^2$ to $\mathcal{S}$. d. Summing up, show that we have established a one-to-one correspondence between isomorphism classes of locally free sheaves of rank $n$ on $Y$, and isomorphism classes of vector bundles of rank $n$ over $Y$. Because of this, we sometimes use the words "locally free sheaf" and "vector bundle" interchangeably, if no confusion seems likely to result. ## II.6: Divisors In this section we will consider schemes satisfying the following condition: $(*) X$ is a noetherian integral separated scheme which is regular in codimension one. ### II.6.1. #to_work Let $X$ be a scheme satisfying $(*)$. Then $X \times \mathbf{P}^n$ also satisfies $(*)$, and $\operatorname{Cl} \left(X \times \mathbf{P}^n\right) \cong( \operatorname{Cl} X) \times \mathbf{Z}$ ### II.6.2. \* Varieties in Projective Space. #to_work Let $k$ be an algebraically closed field, and let $X$ be a closed subvariety of $\mathbf{P}_k^n$ which is nonsingular in codimension one (hence satisfies $(*)$ ). For any divisor $D=\sum n_i Y_i$ on $X$, we define the **degree** of $D$ to be $\sum n_i \operatorname{deg} Y_i$, where $\operatorname{deg} Y_i$ is the degree of $Y_i$, considered as a projective variety itself (I, §7). a. Let $V$ be an irreducible hypersurface in $\mathbf{P}^n$ which does not contain $X$, and let $Y_i$ be the irreducible components of $V \cap X$. They all have codimension 1 by (I, Ex. 1.8). For each $i$, let $f_i$ be a local equation for $V$ on some open set $U_i$ of $\mathbf{P}^n$ for which $Y_i \cap U_i \neq \varnothing$, and let $n_i=v_{Y_i}(\mkern 1.5mu\overline{\mkern-1.5muf_i\mkern-1.5mu}\mkern 1.5mu)$, where $\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu_i$ is the restriction of $f_i$ to $U_i \cap X$. Then we define the **divisor** $V . X$ to be $\sum n_i Y_i$. Extend by linearity, and show that this gives a well-defined homomorphism from the subgroup of Div $\mathbf{P}^n$ consisting of divisors, none of whose components contain $X$, to $\operatorname{Div} X$. b. If $D$ is a principal divisor on $\mathbf{P}^n$, for which $D . X$ is defined as in (a), show that $D . X$ is principal on $X$. Thus we get a homomorphism $\operatorname{Cl} \mathbf{P}^n \rightarrow \operatorname{Cl} X$. c. Show that the integer $n_i$ defined in (a) is the same as the intersection multiplicity $i\left(X, V ; Y_i\right)$ defined in $(\mathrm{I}, \S 7)$. Then use the generalized Bézout theorem $(\mathrm{I}, 7.7)$ to show that for any divisor $D$ on $\mathbf{P}^n$, none of whose components contain $X$, ` \begin{align*} \operatorname{deg}(D \cdot X)=(\operatorname{deg} D) \cdot(\operatorname{deg} X) . \end{align*} `{=html} d. If $D$ is a principal divisor on $X$, show that there is a rational function $f$ on $\mathbf{P}^n$ such that $D=(f) . X$. Conclude that $\operatorname{deg} D=0$. Thus the degree function defines a homomorphism deg: $\operatorname{Cl} X \rightarrow \mathbf{Z}$.[^66] Finally, there is a commutative diagram ```{=tex} \begin{tikzcd} { \operatorname{Cl} {\mathbf{P}}^n} && { \operatorname{Cl} X} \\ \\ {\mathbf{Z}}&& {\mathbf{Z}} \arrow["{\cdot \deg(X)}", from=3-1, to=3-3] \arrow[from=1-1, to=1-3] \arrow["\deg", from=1-3, to=3-3] \arrow["{\deg, \cong}"', from=1-1, to=3-1] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXENsIFxcUFBebiJdLFsyLDAsIlxcQ2wgWCJdLFswLDIsIlxcWloiXSxbMiwyLCJcXFpaIl0sWzIsMywiXFxjZG90IFxcZGVnKFgpIl0sWzAsMV0sWzEsMywiXFxkZWciXSxbMCwyLCJcXGRlZywgXFxjb25nIiwyXV0=) and in particular, we see that the map $\operatorname{Cl} \mathbf{P}^n \rightarrow \operatorname{Cl} X$ is injective. ### II.6.3. \* Cones. #to_work In this exercise we compare the class group of a projective variety $V$ to the class group of its cone (I, Ex. 2.10). So let $V$ be a projective variety in $\mathbf{P}^n$, which is of dimension $\geqslant 1$ and nonsingular in codimension 1 . Let $X=C(V)$ be the affine cone over $V$ in $\mathbf{A}^{n+1}$, and let $\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu$ be its projective closure in $\mathbf{P}^{n+1}$. Let $P \in X$ be the vertex of the cone. a. Let $\pi: \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu-P \rightarrow V$ be the projection map. Show that $V$ can be covered by open subsets $U_i$ such that $\pi^{-1}\left(U_i\right) \cong U_i \times \mathbf{A}^1$ for each $i$, and then show as in (6.6) that $\pi^*: \operatorname{Cl} V \rightarrow \operatorname{Cl} (\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu-P)$ is an isomorphism. Since $\operatorname{Cl} \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu \cong$ $\operatorname{Cl} (\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu-P)$, we have also $\operatorname{Cl} V \cong \operatorname{Cl} \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu$. b. We have $V \subseteq \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu$ as the hyperplane section at infinity. Show that the class of the divisor $V$ in $\operatorname{Cl} \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu$ is equal to $\pi^*$ (class of $V . H$ ) where $H$ is any hyperplane of $\mathbf{P}^n$ not containing $V$. Thus conclude using $(6.5)$ that there is an exact sequence ` \begin{align*} 0 \rightarrow \mathbf{Z} \rightarrow \operatorname{Cl} V \rightarrow \operatorname{Cl} X \rightarrow 0, \end{align*} `{=html} where the first arrow sends $1 \mapsto V . H$, and the second is $\pi^*$ followed by the restriction to $X-P$ and inclusion in $X$. (The injectivity of the first arrow follows from the previous exercise.) c. Let $S(V)$ be the homogeneous coordinate ring of $V$ (which is also the affine coordinate ring of $X$ ). Show that $S(V)$ is a unique factorization domain if and only if - $V$ is projectively normal (Ex. 5.14), and - $\operatorname{Cl} V \cong \mathrm{Z}$ and is generated by the class of $V . H$. d. Let ${\mathcal{O}}_P$ be the local ring of $P$ on $X$. Show that the natural restriction map induces an isomorphism $\operatorname{Cl} X \rightarrow \operatorname{Cl} \left(\operatorname{Spec}{\mathcal{O}}_P\right)$. ### II.6.4. #to_work Let $k$ be a field of characteristic $\neq 2$. Let $f \in k\left[x_1, \ldots, x_n\right]$ be a square-free nonconstant polynomial, i.e., in the unique factorization of $f$ into irreducible polynomials, there are no repeated factors. Let $A=k\left[x_1, \ldots, x_n, z\right] /\left(z^2-f\right)$. Show that $A$ is an integrally closed ring.[^67] Conclude that $A$ is the integral closure of $k\left[x_1, \ldots, x_n\right]$ in $K$. ### II.6.5. \* Quadric Hypersurfaces. #to_work Let char $k \neq 2$, and let $X$ be the affine quadric hypersurface[^68] ` \begin{align*} \operatorname{Spec} k\left[x_0, \ldots, x_n\right] /\left(x_0^2+x_1^2+\ldots+x_r^2\right) .\end{align*} `{=html} a. Show that $X$ is normal if $r \geqslant 2$ (use (Ex. 6.4)). b. Show by a suitable linear change of coordinates that the equation of $X$ could be written as $x_0 x_1=x_2^2+\ldots+x_r^2$. Now imitate the method of $(6.5 .2)$ to show that: (1) If $r=2$, then $\operatorname{Cl} X \cong \mathbf{Z} / 2 \mathbf{Z}$; (2) If $r=3$, then $\operatorname{Cl} X \cong \mathbf{Z}$ (use (6.6.1) and (Ex. 6.3) above); (3) If $r \geqslant 4$ then $\operatorname{Cl} X=0$. c. Now let $Q$ be the projective quadric hypersurface in $\mathbf{P}^n$ defined by the same equation. Show that: (1) If $r=2, \operatorname{Cl} Q \cong \mathbf{Z}$, and the class of a hyperplane section $Q . H$ is twice the generator; (2) If $r=3, \operatorname{Cl} Q \cong \mathbf{Z} \oplus \mathbf{Z}$; (3) If $r \geqslant 4, \operatorname{Cl} Q \cong \mathbf{Z}$, generated by $Q . H$. d. Prove Klein's theorem, which says that if $r \geqslant 4$, and if $Y$ is an irreducible subvariety of codimension 1 on $Q$, then there is an irreducible hypersurface $V \subseteq \mathbf{P}^n$ such that $V \cap Q=Y$, with multiplicity one. In other words, $Y$ is a complete intersection. (First show that for $r \geqslant 4$, the homogeneous coordinate ring $S(Q)=k\left[x_0, \ldots, x_n\right] /\left(x_0^2+\ldots+x_r^2\right)$ is a UFD.) ### II.6.6. #to_work Let $X$ be the nonsingular plane cubic curve $y^2 z=x^3-x z^2$ of $(6.10 .2)$. a. Show that three points $P, Q, R$ of $X$ are collinear if and only if $P+Q+R=0$ in the group law on $X$. Note that the point $P_0=(0,1,0)$ is the zero element in the group structure on $X$. b. A point $P \in X$ has order 2 in the group law on $X$ if and only if the tangent line at $P$ passes through $P_0$. c. A point $P \in X$ has order 3 in the group law on $X$ if and only if $P$ is an inflection point.[^69] d. Let $k=\mathbf{C}$. Show that the points of $X$ with coordinates in $\mathbf{Q}$ form a subgroup of the group $X$. Can you determine the structure of this subgroup explicitly? ### II.6.7. \* #to_work Let $X$ be the nodal cubic curve $y^2 z=x^3+x^2 z$ in $\mathbf{P}^2$. Imitate (6.11.4) and show that the group of Cartier divisors of degree $0, \mathrm{CaCl}^{\circ} X$, is naturally isomorphic to the multiplicative group $\mathbf{G}_m$. ### II.6.8. #to_work a. Let $f: X \rightarrow Y$ be a morphism of schemes. Show that $\mathcal{L} \mapsto f^* \mathcal{L}$ induces a homomorphism of Picard groups, $f^*:\operatorname{Pic}Y \rightarrow \operatorname{Pic}X$. b. If $f$ is a finite morphism of nonsingular curves, show that this homomorphism corresponds to the homomorphism $f^*: \operatorname{Cl} Y \rightarrow \operatorname{Cl} X$ defined in the text, via the isomorphisms of (6.16). c. If $X$ is a locally factorial integral closed subscheme of $\mathbf{P}_k^n$, and if $f: X \rightarrow \mathbf{P}^n$ is the inclusion map, then $f^*$ on Pic agrees with the homomorphism on divisor class groups defined in (Ex. 6.2) via the isomorphisms of (6.16). ### II.6.9. \* Singular Curves. #to_work Here we give another method of calculating the Picard group of a singular curve. Let $X$ be a projective curve over $k$, let $\tilde{X}$ be its normalization, and let $\pi: \tilde{X} \rightarrow X$ be the projection map (Ex. 3.8). For each point $P \in X$, let ${\mathcal{O}}_P$ be its local ring, and let $\tilde{{\mathcal{O}}}_P$ be the integral closure of ${\mathcal{O}}_P$. We use a $*$ to denote the group of units in a ring. a. Show there is an exact sequence[^70] ` \begin{align*} 0 \rightarrow \bigoplus_{P \in X} \tilde{{\mathcal{O}}}_P^* / {\mathcal{O}}_P^* \rightarrow \operatorname{Pic}X \stackrel{\pi^*}{\rightarrow} \operatorname{Pic}\tilde{X} \rightarrow 0 . \end{align*} `{=html} b. Use (a) to give another proof of the fact that if $X$ is a plane cuspidal cubic curve, then there is an exact sequence ` \begin{align*} 0 \rightarrow \mathbf{G}_a \rightarrow \operatorname{Pic}X \rightarrow \mathbf{Z} \rightarrow 0, \end{align*} `{=html} and if $X$ is a plane nodal cubic curve, there is an exact sequence ` \begin{align*} 0 \rightarrow \mathbf{G}_m \rightarrow \operatorname{Pic}X \rightarrow \mathbf{Z} \rightarrow 0 \end{align*} `{=html} ### II.6.10. The Grothendieck Group $K(X)$. #to_work {#ii.6.10.-the-grothendieck-group-kx.-to_work} Let $X$ be a noetherian scheme. We define $K(X)$ to be the quotient of the free abelian group generated by all the coherent sheaves on $X$, by the subgroup generated by all expressions ${\mathcal{F}}-{\mathcal{F}}^{\prime}-{\mathcal{F}}^{\prime \prime}$, whenever there is an exact sequence $0 \rightarrow {\mathcal{F}}^{\prime} \rightarrow {\mathcal{F}}\rightarrow {\mathcal{F}}^{\prime \prime} \rightarrow 0$ of coherent sheaves on $X$. If ${\mathcal{F}}$ is a coherent sheaf, we denote by $\gamma({\mathcal{F}})$ its image in $K(X)$. a. If $X=\mathbf{A}_k^1$, then $K(X) \cong \mathbf{Z}$. b. If $X$ is any integral scheme, and ${\mathcal{F}}$ a coherent sheaf, we define the **rank** of ${\mathcal{F}}$ to be $\operatorname{dim}_K {\mathcal{F}}_{\xi}$, where $\xi$ is the generic point of $X$, and $K={\mathcal{O}}_{\xi}$ is the function field of $X$. Show that the rank function defines a surjective homomorphism rank: $K(X) \rightarrow \mathbf{Z}$. c. If $Y$ is a closed subscheme of $X$, there is an exact sequence ` \begin{align*} K(Y) \rightarrow K(X) \rightarrow K(X-Y) \rightarrow 0, \end{align*} `{=html} where the first map is extension by zero, and the second map is restriction.[^71] ### II.6.11. \*The Grothendieck Group of a Nonsingular Curve. #to_work Let $X$ be a nonsingular curve over an algebraically closed field $k$. We will show that $K(X) \cong \operatorname{Pic}X \oplus \mathbf{Z}$, in several steps. a. For any divisor $D=\sum n_i P_i$ on $X$, let $\psi(D)=\sum n_i \gamma\left(k\left(P_i\right)\right) \in K(X)$, where $k\left(P_i\right)$ is the skyscraper sheaf $k$ at $P_i$ and 0 elsewhere. If $D$ is an effective divisor, let ${\mathcal{O}}_D$ be the structure sheaf of the associated subscheme of codimension 1, and show that $\psi(D)=\gamma\left({\mathcal{O}}_D\right)$. Then use (6.18) to show that for any $D, \psi(D)$ depends only on the linear equivalence class of $D$, so $\psi$ defines a homomorphism $\psi: \operatorname{Cl} X \rightarrow K(X)$ b. For any coherent sheaf ${\mathcal{F}}$ on $X$, show that there exist locally free sheaves $\mathcal{E}_0$ and $\mathcal{E}_1$ and an exact sequence $0 \rightarrow \mathcal{E}_1 \rightarrow \mathcal{E}_0 \rightarrow {\mathcal{F}}\rightarrow 0$. Let $r_0=$ rank $\mathcal{E}_0$, $r_1=\operatorname{rank} \mathcal{E}_1$, and define ` \begin{align*} \operatorname{det}{\mathcal{F}}= \left(\bigwedge^{r_0} \mathcal{E}_0\right) \otimes\left(\bigwedge^{r_1} \mathcal{E}_1\right)^{-1} \in \operatorname{Pic}X .\end{align*} `{=html} Here $\bigwedge$ denotes the exterior power (Ex. 5.16). Show that $\operatorname{det}{\mathcal{F}}$ is independent of the resolution chosen, and that it gives a homomorphism det: $K(X) \rightarrow \operatorname{Pic}X$. Finally show that if $D$ is a divisor, then $\operatorname{det}(\psi(D))=\mathcal{L}(D)$. c. If ${\mathcal{F}}$ is any coherent sheaf of rank $r$, show that there is a divisor $D$ on $X$ and an exact sequence ` \begin{align*} 0 \rightarrow \mathcal{L}(D)^{\oplus r} \rightarrow {\mathcal{F}}\rightarrow \mathcal{T} \rightarrow 0 ,\end{align*} `{=html} where $\mathcal{T}$ is a torsion sheaf. Conclude that if ${\mathcal{F}}$ is a sheaf of rank $r$, then ` \begin{align*} \gamma({\mathcal{F}})-r \gamma\left({\mathcal{O}}_X\right) \in \operatorname{Im} \psi .\end{align*} `{=html} d. Using the maps $\psi, \operatorname{det}, \mathrm{rank}$, and $1 \mapsto \gamma\left({\mathcal{O}}_X\right)$ from $\mathbf{Z} \rightarrow K(X)$, show that $K(X) \cong \operatorname{Pic}X \oplus \mathbf{Z}$. ### II.6.12. Let $X$ be a complete nonsingular curve. Show that there is a unique way to define the degree of any coherent sheaf on $X$, deg ${\mathcal{F}}\in \mathbf{Z}$, such that: #to_work a. If $D$ is a divisor, $\operatorname{deg} \mathcal{L}(D)=\operatorname{deg} D$; b. If ${\mathcal{F}}$ is a torsion sheaf(meaning a sheaf whose stalk at the generic point is zero), then $\operatorname{deg} {\mathcal{F}}=\sum_{P \in X}$ length $\left({\mathcal{F}}_P\right)$; and c. If $0 \rightarrow {\mathcal{F}}^{\prime} \rightarrow \mathcal{\mathcal { F }} \rightarrow {\mathcal{F}}^{\prime \prime} \rightarrow 0$ is an exact sequence, then $\operatorname{deg} {\mathcal{F}}=\operatorname{deg} {\mathcal{F}}^{\prime}+$ $\operatorname{deg} {\mathcal{F}}^{\prime \prime}$. ## II.7: Projective Morphisms ### II.7.1. #to_work Let $\left(X, {\mathcal{O}}_X\right)$ be a locally ringed space, and let $f: \mathcal{L} \rightarrow \mathcal{M}$ be a surjective map of invertible sheaves on $X$. Show that $f$ is an isomorphism.[^72] ### II.7.2. #to_work Let $X$ be a scheme over a field $k$. Let $\mathcal{L}$ be an invertible sheaf on $X$, and let $\left\{s_0, \ldots, s_n\right\}$ and $\left\{t_0, \ldots, t_m\right\}$ be two sets of sections of $\mathcal{L}$, which generate the same subspace $V \subseteq \Gamma(X, \mathcal{L})$, and which generate the sheaf $\mathcal{L}$ at every point. Suppose $n \leqslant m$. Show that the corresponding morphisms $\varphi: X \rightarrow \mathbf{P}_k^n$ and $\psi: X \rightarrow$ $\mathbf{P}_k^m$ differ by a suitable linear projection $\mathbf{P}^m-L \rightarrow \mathbf{P}^n$ and an automorphism of $\mathbf{P}^n$, where $L$ is a linear subspace of $\mathbf{P}^m$ of dimension $m-n-1$. ### II.7.3. #to_work Let $\varphi: \mathbf{P}_k^n \rightarrow \mathbf{P}_k^m$ be a morphism. Then: a. Either $\varphi\left(\mathbf{P}^n\right) = {\operatorname{pt}}$ or $m \geqslant n$ and $\operatorname{dim} \varphi\left(\mathbf{P}^n\right)=n$; b. In the second case, $\varphi$ can be obtained as the composition of (1) a $d$-uple embedding $\mathbf{P}^n \rightarrow \mathbf{P}^N$ for a uniquely determined $d \geqslant 1,$ (2) a linear projection $\mathbf{P}^N-\mathbf{L} \rightarrow \mathbf{P}^m$, and (3) an automorphism of $\mathbf{P}^m$. Also, $\varphi$ has finite fibres. ### II.7.4. #to_work a. Use (7.6) to show that if $X$ is a scheme of finite type over a noetherian $\operatorname{ring} A$, and if $X$ admits an ample invertible sheaf, then $X$ is separated. b. Let $X$ be the affine line over a field $k$ with the origin doubled (4.0.1). Calculate $\operatorname{Pic}X$, determine which invertible sheaves are generated by global sections, and then show directly (without using (a)) that there is no ample invertible sheaf on $X$. ### II.7.5. #to_work Establish the following properties of ample and very ample invertible sheaves on a noetherian scheme $X$. $\mathcal{L}, \mathcal{M}$ will denote invertible sheaves, and for (d), (e) we assume furthermore that $X$ is of finite type over a noetherian ring $A$. a. If $\mathcal{L}$ is ample and $\mathcal{M}$ is generated by global sections, then $\mathcal{L} \otimes \mathcal{M}$ is ample. b. If $\mathcal{L}$ is ample and $\mathcal{M}$ is arbitrary, then $\mathcal{M} \otimes \mathcal{L}^n$ is ample for sufficiently large $n$. c. If $\mathcal{L}, \mathcal{M}$ are both ample, so is $\mathcal{L} \otimes \mathcal{M}$. d. If $\mathcal{L}$ is very ample and $\mathcal{M}$ is generated by global sections, then $\mathcal{L} \otimes \mathcal{M}$ is very ample. e. If $\mathcal{L}$ is ample, then there is an $n_0>0$ such that $\mathcal{L}^n$ is very ample for all $n \geqslant n_0$. ### II.7.6. The Riemann-Roch Problem. #to_work Let $X$ be a nonsingular projective variety over an algebraically closed field, and let $D$ be a divisor on $X$. For any $n>0$ we consider the complete linear system $|n D|$. Then the Riemann-Roch problem is to determine $\operatorname{dim}|n D|$ as a function of $n$, and, in particular, its behavior for large $n$. If $\mathcal{L}$ is the corresponding invertible sheaf, then $\operatorname{dim}|n D|=\operatorname{dim} \Gamma\left(X, \mathcal{L}^n\right)-1$, so an equivalent problem is to determine $\operatorname{dim} \Gamma\left(X, \mathcal{L}^n\right)$ as a function of $n$. a. Show that if $D$ is very ample, and if $X \hookrightarrow \mathbf{P}_k^n$ is the corresponding embedding in projective space, then for all $n$ sufficiently large, $\operatorname{dim}|n D|=P_X(n)-1$, where $P_X$ is the Hilbert polynomial of $X(\mathbf{I}, \S 7)$. Thus in this case $\operatorname{dim}|n D|$ is a polynomial function of $n$, for $n$ large. b. If $D$ corresponds to a torsion element of $\operatorname{Pic}X$, of order $r$, then $\operatorname{dim}|n D|=0$ if $r \mathrel{\Big|}n$ and $-1$ otherwise. In this case the function is periodic of period $r$. It follows from the general Riemann-Roch theorem that $\operatorname{dim}|n D|$ is a polynomial function for $n$ large, whenever $D$ is an ample divisor.[^73] In the case of algebraic surfaces, Zariski $[7]$ has shown for any effective divisor $D$, that there is a finite set of polynomials $P_1, \ldots, P_r$, such that for all $n$ sufficiently large, $\operatorname{dim}|n D|=P_{i(n)}(n)$, where $i(n) \in\{1,2, \ldots, r\}$ is a function of $n$. ### II.7.7. Some Rational Surfaces. #to_work Let $X=\mathbf{P}_k^2$, and let $|D|$ be the complete linear system of all divisors of degree 2 on $X$ (conics). $D$ corresponds to the invertible sheaf ${\mathcal{O}}(2)$, whose space of global sections has a basis $x^2, y^2, z^2, x y, x z, y z$, where $x, y, z$ are the homogeneous coordinates of $X$. a. The complete linear system $|D|$ gives an embedding of $\mathbf{P}^2$ in $\mathbf{P}^5$, whose image is the Veronese surface.[^74] b. Show that the subsystem defined by $x^2, y^2, z^2, y(x-z),(x-y) z$ gives a closed immersion of $X$ into $\mathbf{P}^4$. The image is called the Veronese surface in $\mathbf{P}^4$. Cf. (IV, Ex. 3.11). c. Let $\nu \subseteq|D|$ be the linear system of all conics passing through a fixed point $P$. Then $\nu$ gives an immersion of $U=X-P$ into $\mathbf{P}^4$. Furthermore, if we blow up $P$, to get a surface $\tilde{X}$, then this map extends to give a closed immersion of $\tilde{X}$ in $\mathbf{P}^4$. Show that $\tilde{X}$ is a surface of degree 3 in $\mathbf{P}^4$, and that the lines in $X$ through $P$ are transformed into straight lines in $\tilde{X}$ which do not meet. $\tilde{X}$ is the union of all these lines, so we say $\tilde{X}$ is a **ruled surface** $(\mathrm{V}, 2.19 .1)$. ### II.7.8. #to_work Let $X$ be a noetherian scheme, let $\mathcal{E}$ be a coherent locally free sheaf on $X$, and let $\pi: \mathbf{P}(\mathcal{E}) \rightarrow X$ be the corresponding projective space bundle. Show that there is a natural $1-1$ correspondence between sections of $\pi$ (i.e., morphisms $\sigma: X \rightarrow$ $\mathbf{P}(\mathcal{E})$ such that $\left.\pi \circ \sigma=\mathrm{id}_X\right)$ and quotient invertible sheaves $\mathcal{E} \rightarrow \mathcal{L} \rightarrow 0$ of $\mathcal{E}$. ### II.7.9. #to_work Let $X$ be a regular noetherian scheme, and $\mathcal{E}$ a locally free coherent sheaf of rank $\geqslant 2$ on $X$. a. Show that $\operatorname{Pic}\mathbf{P}(\mathcal{E}) \cong \operatorname{Pic}X \times \mathbf{Z}$. b. If $\mathcal{E}^{\prime}$ is another locally free coherent sheaf on $X$, show that $\mathbf{P}(\mathcal{E}) \cong \mathbf{P}\left(\mathcal{E}^{\prime}\right)$ (over $X$ ) if and only if there is an invertible sheaf $\mathcal{L}$ on $X$ such that $\mathcal{E}^{\prime} \cong \mathcal{E} \otimes \mathcal{L}$. ### II.7.10. $\mathbf{P}^n$-Bundles Over a Scheme. #to_work {#ii.7.10.-mathbfpn-bundles-over-a-scheme.-to_work} Let $X$ be a noetherian scheme. a. By analogy with the definition of a vector bundle (Ex. 5.18), define the notion of a projective $n$-space bundle over $X$, as a scheme $P$ with a morphism $\pi: P \rightarrow X$ such that $P$ is locally isomorphic to $U \times \mathbf{P}^n, U \subseteq X$ open, and the transition automorphisms on $\operatorname{Spec} A \times \mathbf{P}^n$ are given by $A$-linear automorphisms of the homogeneous coordinate ring $A\left[x_0, \ldots, x_n\right]$ E.g., $x_i^{\prime}=\sum a_{i j} x_j, a_{i j} \in A$. b. If $\mathcal{E}$ is a locally free sheaf of rankof rank $n+1$ on $X$ then $\mathbf{P}(\mathcal E)$ is a $\mathbf P^n{\hbox{-}}$bundle over $X$. c. \* Assume that $X$ is regular, and show that every $\mathbf{P}^n$-bundle $P$ over $X$ is isomorphic to $\mathbf{P}(\mathcal{E})$ for some locally free sheaf ${\mathcal{E}}$ on $X$.[^75] d. Conclude (in the case $X$ regular) that we have a 1-1 correspondence between $\mathbf{P}^n$-bundles over $X$, and equivalence classes of locally free sheaves $\mathcal{E}$ of rank $n+1$ under the equivalence relation ${\mathcal{E}}\sim {\mathcal{E}}'$ if and only if ${\mathcal{E}}' \cong {\mathcal{E}}\otimes{\mathcal{M}}$ for some invertible sheaf ${\mathcal{M}}$ on $X$. ### II.7.11. #to_work On a noetherian scheme $X$, different sheaves of ideals can give rise to isomorphic blown up schemes. a. If $\mathcal{I}$ is any coherent sheaf of ideals on $X$, show that blowing up $\mathcal{I}^d$ for any $d \geqslant 1$ gives a scheme isomorphic to the blowing up of $\mathcal{I}$ (cf. Ex. 5.13). b. If $\mathcal{I}$ is any coherent sheaf of ideals, and if $\mathcal{J}$ is an invertible sheaf of ideals, then $\mathcal{I}$ and $\mathcal{I} \cdot \mathcal{J}$ give isomorphic blowings-up. c. If $X$ is regular, show that (7.17) can be strengthened as follows. Let $U \subseteq X$ be the largest open set such that $f: f^{-1} U \rightarrow U$ is an isomorphism. Then $\mathcal{I}$ can be chosen such that the corresponding closed subscheme $Y$ has support equal to $X-U$ ### II.7.12. #to_work Let $X$ be a noetherian scheme, and let $Y, Z$ be two closed subschemes, neither one containing the other. Let $\tilde{X}$ be obtained by blowing up $Y \cap Z$ (defined by the ideal sheaf $\mathcal{I}_Y+\mathcal{I}_Z$ ). Show that the strict transforms $\tilde{Y}$ and $\tilde{Z}$ of $Y$ and $Z$ in $\tilde{X}$ do not meet. ### II.7.13. \* A Complete Nonprojective Variety. #to_work Let $k$ be an algebraically closed field of char $\neq 2$. Let $C \subseteq \mathbf{P}_k^2$ be the nodal cubic curve ` \begin{align*} y^2 z=x^3+x^2 z .\end{align*} `{=html} If $P_0=(0,0,1)$ is the singular point, then $C-P_0$ is isomorphic to the multiplicative group $\mathbf{G}_m=\operatorname{Spec} k\left[t, t^{-1}\right]$ (Ex. 6.7). For each $a \in k, a \neq 0$, consider the translation of $\mathbf{G}_m$ given by $t \mapsto a t$. This induces an automorphism of $C$ which we denote by $\varphi_a$. Now consider $C \times\left(\mathbf{P}^1-\{0\}\right)$ and $C \times\left(\mathbf{P}^1-\{\infty\}\right)$. We glue their open subsets $C \times\left(\mathbf{P}^1-\{0, \infty\}\right)$ by the isomorphism ` \begin{align*} \varphi:\langle P, u\rangle \mapsto\left\langle\varphi_u(P), u\right\rangle,\qquad P \in C, u \in \mathbf{G}_m=\mathbf{P}^1-\{0, \infty\} .\end{align*} `{=html} Thus we obtain a scheme $X$, which is our example. The projections to the second factor are compatible with $\varphi$, so there is a natural morphism $\pi: X \rightarrow \mathbf{P}^1$. a. Show that $\pi$ is a proper morphism, and hence that $X$ is a complete variety over $k$. b. Use the method of (Ex. 6.9) to show that[^76] ` \begin{align*} \operatorname{Pic}\left(C \times \mathbf{A}^1\right) \cong \mathbf{G}_m \times \mathbf{Z}\quad\text{and}\quad \operatorname{Pic}\left(C \times\left(\mathbf{A}^1-\{0\}\right)\right) \cong \mathbf{G}_m \times \mathbf{Z} \times \mathbf{Z} .\end{align*} `{=html} c. Now show that the restriction map ` \begin{align*} \operatorname{Pic}\left(C \times \mathbf{A}^1\right) \rightarrow \operatorname{Pic}\left(C \times\left(\mathbf{A}^1-\{0\}\right)\right) \end{align*} `{=html} is of the form $\langle t, n\rangle \mapsto\langle t, 0, n\rangle$, and that the automorphism $\varphi$ of $C \times\left(\mathbf{A}^1-\{0\}\right)$ induces a map of the form $\langle t, d, n\rangle \mapsto\langle t, d+n, n\rangle$ on its Picard group. d. Conclude that the image of the restriction map ` \begin{align*} \operatorname{Pic}X \rightarrow \operatorname{Pic}(C \times\{0\}) \end{align*} `{=html} consists entirely of divisors of degree 0 on $C$. Hence $X$ is not projective over $k$ and $\pi$ is not a projective morphism. ### II.7.14. #to_work a. Give an example of a noetherian scheme $X$ and a locally free coherent sheaf $\mathcal{E}$, such that the invertible sheaf ${\mathcal{O}}(1)$ on $\mathbf{P}(\mathcal{E})$ is not very ample relative to $X$. b. Let $f: X \rightarrow Y$ be a morphism of finite type, let $\mathcal{L}$ be an ample invertible sheaf on $X$, and let $\mathcal{S}$ be a sheaf of graded ${\mathcal{O}}_X$-algebras satisfying (†). Let $P=\operatorname{Proj} \mathcal{S}$, let $\pi: P \rightarrow X$ be the projection, and let ${\mathcal{O}}_P(1)$ be the associated invertible sheaf. Show that for all $n \gg 0$, the sheaf ${\mathcal{O}}_P(1) \otimes \pi^* \mathcal{L}^n$ is very ample on $P$ relative to $Y$.[^77] ## II.8: Differentials ### II.8.1 #to_work Here we will strengthen the results of the text to include information about the sheaf of differentials at a not necessarily closed point of a scheme $X$. a. Generalize (8.7) as follows. Let $B$ be a local ring containing a field $k$, and assume that the residue field $k(B)=B / \mathfrak{m}$ of $B$ is a separably generated extension of $k$. Then the exact sequence of (8.4A), ` \begin{align*} 0 \rightarrow \mathrm{m} / \mathrm{m}^2 \stackrel{\delta}{\rightarrow} \Omega_{B / k} \otimes k(B) \rightarrow \Omega_{k(B) / k} \rightarrow 0 \end{align*} `{=html} is exact on the left also.[^78] b. Generalize (8.8) as follows. With $B, k$ as above, assume furthermore that $k$ is perfect, and that $B$ is a localization of an algebra of finite type over $k$. Then show that $B$ is a regular local ring if and only if $\Omega_{B / k}$ is free of rank $=\operatorname{dim} B+$ tr.d. $k(B) / k$. c. Strengthen (8.15) as follows. Let $X$ be an irreducible scheme of finite type over a perfect field $k$, and let $\operatorname{dim} X=n$. For any point $x \in X$, not necessarily closed, show that the local ring $\mathcal{O}_{x, X}$ is a regular local ring if and only if the stalk $\left(\Omega_{X / k}\right)_x$ of the sheaf of differentials at $x$ is free of rank $n$. d. Strengthen (8.16) as follows. If $X$ is a variety over an algebraically closed field $k$, then $U=\left\{x \in X \mathrel{\Big|}\mathcal{O}_x\right.$ is a regular local ring $\}$ is an open dense subset of $X$. ### II.8.2. #to_work Let $X$ be a variety of dimension $n$ over $k$. Let $\mathcal{E}$ be a locally free sheaf of rank $>n$ on $X$, and let $V \subseteq \Gamma(X, \mathcal{E})$ be a vector space of global sections which generate $\mathcal{E}$. Then show that there is an element $s \in V$, such that for each $x \in X$, we have $s_x \notin \mathfrak{m}_x \mathcal{E}_x$. Conclude that there is a morphism $\mathcal{O}_X \rightarrow \mathcal{E}$ giving rise to an exact sequence ` \begin{align*} 0 \rightarrow \mathcal{O}_X \rightarrow \mathcal{E} \rightarrow \mathcal{E}^{\prime} \rightarrow 0 \end{align*} `{=html} where $\mathcal{E}'$ is also locally free.[^79] ### II.8.3. Product Schemes. #to_work a. Let $X$ and $Y$ be schemes over another scheme $S$. Use (8.10) and (8.11) to show that ` \begin{align*} \Omega_{X \underset{\scriptscriptstyle {S} }{\times} Y/S} \cong p_1^* \Omega_{X / S} \oplus p_2^* \Omega_{Y / S} .\end{align*} `{=html} b. If $X$ and $Y$ are nonsingular varieties over a field $k$, show that ` \begin{align*} \omega_{X \times Y} \cong p_1^* \omega_X \otimes p_2^* \omega_Y .\end{align*} `{=html} c. Let $Y$ be a nonsingular plane cubic curve, and let $X$ be the surface $Y \times Y$. Show that $p_g(X)=1$ but $p_a(X)=-1$ (I, Ex. 7.2). This shows that the arithmetic genus and the geometric genus of a nonsingular projective variety may be different. ### II.8.4. Complete Intersections in $\mathbf{P}^n$. #to_work {#ii.8.4.-complete-intersections-in-mathbfpn.-to_work} A closed subscheme $Y$ of $\mathbf{P}_k^n$ is called a **(strict, global) complete intersection** if the homogeneous ideal $I$ of $Y$ in $S=k\left[x_0, \ldots, x_n\right]$ can be generated by $r=\operatorname{codim}\left(Y, \mathbf{P}^n\right)$ elements (I, Ex. 2.17). a. Let $Y$ be a closed subscheme of codimension $r$ in $\mathbf{P}^n$. Then $Y$ is a complete intersection if and only if there are hypersurfaces (i.e., locally principal subschemes of codimension 1) $H_1, \ldots, H_r$, such that $Y=H_1 \cap \ldots \cap H_r$ as schemes, i.e., $\mathcal{I}_Y=\mathcal{I}_{H_1}+\ldots+\mathcal{I}_{H_r}$.[^80] b. If $Y$ is a complete intersection of dimension $\geqslant 1$ in $\mathbf{P}^n$, and if $Y$ is normal, then $Y$ is projectively normal (Ex. 5.14).[^81] c. With the same hypotheses as (b), conclude that for all $l \geqslant 0$, the natural map $\Gamma\left(\mathbf{P}^n, \mathcal{O}_{\mathbf{P}^n}(l)\right) \rightarrow \Gamma\left(Y, \mathcal{O}_Y(l)\right)$ is surjective. In particular, taking $l=0$, show that $Y$ is connected. d. Now suppose given integers $d_1, \ldots, d_r \geqslant 1$, with $r \begin{align*} \omega_Y \cong \mathcal{O}_Y\left(\sum d_i-n-1\right) .\end{align*} `{=html} f. If $Y$ is a nonsingular hypersurface of degree $d$ in $\mathbf{P}^n$, use (c) and (e) above to show that $p_g(Y)={d-1\choose n}$. Thus $p_g(Y)=p_a(Y)$ (I, Ex. 7.2). In particular, if $Y$ is a nonsingular plane curve of degree $d$, then ` \begin{align*} p_g(Y)=\frac{1}{2}(d-1)(d-2) .\end{align*} `{=html} g. If $Y$ is a nonsingular curve in $\mathbf{P}^3$, which is a complete intersection of nonsingular surfaces of degrees $d, e$, then ` \begin{align*} p_g(Y)=\frac{1}{2} d e(d+e-4)+1 .\end{align*} `{=html} Again the geometric genus is the same as the arithmetic genus (I, Ex. 7.2). ### II.8.5. Blowing up a Nonsingular Subvariety. #to_work As in (8.24), let $X$ be a nonsingular variety, let $Y$ be a nonsingular subvariety of codimension $r \geqslant 2$, let $\pi: \widetilde{X} \rightarrow X$ be the blowing-up of $X$ along $Y$, and let $Y^{\prime}=\pi^{-1}(Y)$. a. Show that the maps $\pi^*:\operatorname{Pic}X \rightarrow\operatorname{Pic}\tilde{X}$, and $\mathbf{Z} \rightarrow \operatorname{Pic}X$ defined by $n \mapsto$ class of $n Y^{\prime}$, give rise to an isomorphism $\operatorname{Pic}\tilde{X} \cong \operatorname{Pic} X \oplus \mathbf{Z}$. b. Show that[^82] ` \begin{align*} \omega_{\tilde{x}} \cong f^* \omega_X \otimes \mathcal{L}\left((r-1) Y^{\prime}\right) .\end{align*} `{=html} ### II.8.6. The Infinitesimal Lifting Property. #to_work The following result is very important in studying deformations of nonsingular varieties. Let $k$ be an algebraically closed field, let $A$ be a finitely generated $k$-algebra such that $\operatorname{Spec} A$ is a nonsingular variety over $k$. Let ` \begin{align*} 0 \rightarrow I \rightarrow B^{\prime} \rightarrow B \rightarrow 0 \end{align*} `{=html} be an exact sequence, where $B^{\prime}$ is a $k$-algebra, and $I$ is an ideal with $I^2=0$. Finally suppose given a $k$-algebra homomorphism $f: A \rightarrow B$. Then there exists a $k$-algebra homomorphism $g: A \rightarrow B^{\prime}$ making a commutative diagram ```{=tex} \begin{tikzcd} && {} \\ \\ && I \\ && {B'} \\ A && B \arrow["f", from=5-1, to=5-3] \arrow["g", dashed, from=5-1, to=4-3] \arrow[hook, from=3-3, to=4-3] \arrow[two heads, from=4-3, to=5-3] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCw0LCJBIl0sWzIsNCwiQiJdLFsyLDIsIkkiXSxbMiwwXSxbMiwzLCJCJyJdLFswLDEsImYiXSxbMCw0LCJnIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzIsNCwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbNCwxLCIiLDIseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XV0=) We call this result the **infinitesimal lifting property for $A$**. We prove this result in several steps. a. First suppose that $g: A \rightarrow B^{\prime}$ is a given homomorphism lifting $f$. If $g^{\prime}: A \rightarrow B^{\prime}$ is another such homomorphism, show that $\theta=g-g^{\prime}$ is a $k$-derivation of $A$ into $I$, which we can consider as an element of $\operatorname{Hom}_A\left(\Omega_{A / k}, I\right)$. Note that since $I^2=0, I$ has a natural structure of $B$-module and hence also of $A$-module. Conversely, for any $\theta \in \operatorname{Hom}_A\left(\Omega_{A / k}, I\right), g^{\prime}=g+\theta$ is another homomorphism lifting $f$. (For this step, you do not need the hypothesis about Spec $A$ being nonsingular.) b. Now let $P=k\left[x_1, \ldots, x_n\right]$ be a polynomial ring over $k$ of which $A$ is a quotient, and let $J$ be the kernel. Show that there does exist a homomorphism $h: P \rightarrow B^{\prime}$ making a commutative diagram, ```{=tex} \begin{tikzcd} J && I \\ P && B \\ A && {B'} \arrow["f"', from=3-1, to=3-3] \arrow["h"', from=2-1, to=2-3] \arrow[hook, from=1-1, to=2-1] \arrow[hook, from=1-3, to=2-3] \arrow[two heads, from=2-1, to=3-1] \arrow[two heads, from=2-3, to=3-3] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCJKIl0sWzAsMSwiUCJdLFswLDIsIkEiXSxbMiwwLCJJIl0sWzIsMSwiQiJdLFsyLDIsIkInIl0sWzIsNSwiZiIsMl0sWzEsNCwiaCIsMl0sWzAsMSwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMyw0LCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFsxLDIsIiIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dLFs0LDUsIiIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dXQ==) and show that $h$ induces an $A$-linear map $\mkern 1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern 1.5mu: J / J^2 \rightarrow I$. c. Now use the hypothesis Spec $A$ nonsingular and (8.17) to obtain an exact sequence ` \begin{align*} 0 \rightarrow J / J^2 \rightarrow \Omega_{P / k} \otimes A \rightarrow \Omega_{A / k} \rightarrow 0 . \end{align*} `{=html} Show furthermore that applying the functor $\operatorname{Hom}_A(\cdot, I)$ gives an exact sequence ` \begin{align*} 0 \rightarrow \operatorname{Hom}_A\left(\Omega_{A / k}, I\right) \rightarrow \operatorname{Hom}_P\left(\Omega_{P / k}, I\right) \rightarrow \operatorname{Hom}_A\left(J / J^2, I\right) \rightarrow 0 . \end{align*} `{=html} Let $\theta \in \operatorname{Hom}_P\left(\Omega_{P / k}, I\right)$ be an element whose image gives $\mkern 1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern 1.5mu\in \operatorname{Hom}_A\left(J / J^2, I\right)$. Consider $\theta$ as a derivation of $P$ to $B^{\prime}$. Then let $h^{\prime}=h-\theta$, and show that $h^{\prime}$ is a homomorphism of $P \rightarrow B^{\prime}$ such that $h^{\prime}(J)=0$. Thus $h^{\prime}$ induces the desired homomorphism $g: A \rightarrow B^{\prime}$. ### II.8.7. #to_work As an application of the infinitesimal lifting property, we consider the following general problem. Let $X$ be a scheme of finite type over $k$, and let $\mathcal{F}$ be a coherent sheaf on $X$. We seek to classify schemes $X^{\prime}$ over $k$, which have a sheaf of ideals $\mathcal{I}$ such that $\mathcal{I}^2=0$ and $\left(X^{\prime}, \mathcal{O}_X / \mathcal{I}\right) \cong\left(X, \mathcal{O}_X\right)$, and such that $\mathcal{I}$ with its resulting structure of $\mathcal{O}_X$-module is isomorphic to the given sheaf $\mathcal{F}$. Such a pair $X^{\prime}, \mathcal{I}$ we call an **infinitesimal extension of the scheme $X$ by the sheaf $\mathcal{F}$**. One such extension, the trivial one, is obtained as follows. Take $\mathcal{O}_{X^{\prime}}=\mathcal{O}_X \oplus \mathcal{F}$ as sheaves of abelian groups, and define multiplication by ` \begin{align*} (a \oplus f) \cdot\left(a^{\prime} \oplus f^{\prime}\right)=a a^{\prime} \oplus \left(a f^{\prime}+a^{\prime} f\right) .\end{align*} `{=html} Then the topological space $X$ with the sheaf of rings $\mathcal{O}_{X^{\prime}}$ is an infinitesimal extension of $X$ by $\mathcal{F}$. The general problem of classifying extensions of $X$ by $\mathcal{F}$ can be quite complicated. So for now, just prove the following special case: if $X$ is affine and nonsingular, then any extension of $X$ by a coherent sheaf $\mathcal{F}$ is isomorphic to the trivial one. See (III, Ex. 4.10) for another case. ### II.8.8. Plurigenus and (some) Hodge numbers are birational invariants. #to_work Let $X$ be a projective nonsingular variety over $k$. For any $n>0$ we define the **$n$th plurigenus of $X$** to be ` \begin{align*} P_n=\operatorname{dim}_k \Gamma\left(X, \omega_X^{\otimes n}\right) .\end{align*} `{=html} Thus in particular $P_1=p_g$. Also, for any $q, 0 \leqslant q \leqslant \operatorname{dim} X$ we define an integer ` \begin{align*} h^{q, 0}=\operatorname{dim}_k \Gamma\left(X, \Omega_{X / k}^q\right) \quad\text{where}\quad \Omega_{X / k}^q=\bigwedge^q \Omega_{X / k} .\end{align*} `{=html} is the sheaf of regular $q$-forms on $X$. In particular, for $q=\operatorname{dim} X$, we recover the geometric genus again. The integers $h^{q, 0}$ are called **Hodge numbers**. Using the method of $(8.19)$, show that $P_n$ and $h^{q, 0}$ are birational invariants of $X$, i.e., if $X$ and $X^{\prime}$ are birationally equivalent nonsingular projective varieties, then $P_n(X)=P_n\left(X^{\prime}\right)$ and $h^{q, 0}(X)=h^{q, 0}\left(X^{\prime}\right)$. ## II.9: Formal Schemes ### II.9.1. #to_work Let $X$ be a noetherian scheme, $Y$ a closed subscheme, and $\widehat{X}$ the completion of $X$ along $Y$. We call the ring $\Gamma\left(\widehat{X}, \mathcal{O}_{\widehat{X}}\right)$ the ring of **formal-regular** functions on $X$ along $Y$. In this exercise we show that if $Y$ is a connected, nonsingular, positive dimensional subvariety of $X=\mathbf{P}_k^n$ over an algebraically closed field $k$, then $\Gamma\left(\widehat{X}, \mathcal{O}_{\widehat{X}}\right)=k$ a. Let $\mathcal{I}$ be the ideal sheaf of $Y$. Use (8.13) and (8.17) to show that there is an inclusion of sheaves on $Y, \mathcal{I} / \mathcal{I}^2 \hookrightarrow \mathcal{O}_Y(-1)^{n+1}$. b. Show that for any $r \geqslant 1, \Gamma\left(Y, \mathcal{I}^r / \mathcal{I}^{r+1}\right)=0$. c. Use the exact sequences ` \begin{align*} 0 \rightarrow \mathcal{I}^r / \mathcal{I}^{r+1} \rightarrow \mathcal{O}_X / \mathcal{I}^{r+1} \rightarrow \mathcal{O}_X / \mathcal{I}^r \rightarrow 0 \end{align*} `{=html} and induction on $r$ to show that $\Gamma\left(Y, \mathcal{O}_X / \mathcal{I}^r\right)=k$ for all $r \geqslant 1$.[^83] ```{=html} ``` d. Conclude that $\Gamma\left(\widehat{X}, \mathcal{O}_{\widehat{X}}\right)=k$.[^84] ### II.9.2. #to_work Use the result of (Ex. 9.1) to prove the following geometric result. Let $Y \subseteq X=$ $\mathbf{P}_k^n$ be as above, and let $f: X \rightarrow Z$ be a morphism of $k$-varieties. Suppose that $f(Y)$ is a single closed point $P \in Z$. Then $f(X)=P$ also. ### II.9.3. #to_work Prove the analogue of $(5.6)$ for formal schemes, which says, if $\mathrm{X}$ is an affine formal scheme, and if ` \begin{align*} 0 \rightarrow \mathrm{F}^{\prime} \rightarrow \mathrm{F} \rightarrow \mathrm{F}^{\prime \prime} \rightarrow 0 \end{align*} `{=html} is an exact sequence of $\mathcal{O}_x$-modules, and if $\mathrm{F}^{\prime}$ is coherent, then the sequence of global sections ` \begin{align*} 0 \rightarrow \Gamma\left(\mathrm{X}, \mathrm{F}^{\prime}\right) \rightarrow \Gamma(\mathrm{X}, \mathrm{F}) \rightarrow \Gamma\left(\mathrm{X}, \mathrm{F}^{\prime \prime}\right) \rightarrow 0 \end{align*} `{=html} is exact. For the proof, proceed in the following steps. a. Let $\mathrm{I}$ be an ideal of definition for $\mathrm{X}$, and for each $n>0$ consider the exact sequence ` \begin{align*} 0 \rightarrow \mathrm{F}^{\prime} / \mathrm{J}^n \mathrm{F}^{\prime} \rightarrow \mathrm{F} / \mathrm{J}^n \mathrm{F}^{\prime} \rightarrow \mathrm{F}^{\prime \prime} \rightarrow 0 .\end{align*} `{=html} Use (5.6), slightly modified, to show that for every open affine subset $\mathrm{U} \subseteq \mathrm{X}$, the sequence ` \begin{align*} 0 \rightarrow \Gamma\left(\mathrm{U}, \mathrm{F}^{\prime} / \mathrm{I}^n \mathrm{F}^{\prime}\right) \rightarrow \Gamma\left(\mathrm{Y}, \mathrm{F} / \mathrm{J}^n \mathrm{F}^{\prime}\right) \rightarrow \Gamma\left(\mathrm{U}, \mathrm{F}^{\prime \prime}\right) \rightarrow 0 \end{align*} `{=html} is exact. b. Now pass to the limit, using (9.1), (9.2), and (9.6). Conclude that $\mathrm{F} \cong \lim \mathrm{F} / \mathrm{J}^n \mathrm{F}^{\prime}$ and that the sequence of global sections above is exact. ### II.9.4. #to_work Use (Ex. 9.3) to prove that if ` \begin{align*} 0 \rightarrow \mathrm{F}^{\prime} \rightarrow \mathrm{F} \rightarrow \mathrm{F}^{\prime \prime} \rightarrow 0 \end{align*} `{=html} is an exact sequence of $\mathcal{O}_x$-modules on a noetherian formal scheme $\mathrm{X}$, and if $\mathrm{F}^{\prime}, \mathrm{F}^{\prime \prime}$ are coherent, then $\mathrm{F}$ is coherent also. ### II.9.5. #to_work If $\mathrm{F}$ is a coherent sheaf on a noetherian formal scheme $\mathrm{X}$, which can be generated by global sections, show in fact that it can be generated by a finite number of its global sections. ### II.9.6. #to_work Let $\mathrm{X}$ be a noetherian formal scheme, let $\mathrm{I}$ be an ideal of definition, and for each $n$, let $Y_n$ be the scheme $\left(\mathrm{X}, \mathcal{O}_x / \mathrm{J}^n\right)$. Assume that the inverse system of groups $\left(\Gamma\left(Y_n, \mathcal{O}_{Y_n}\right)\right)$ satisfies the Mittag-Leffler condition. Then prove that $\operatorname{Pic}\mathrm{X}=\lim \operatorname{Pic}Y_n$. As in the case of a scheme, we define $\operatorname{Pic}\mathrm{X}$ to be the group of locally free $\mathcal{O}_x$-modules of rank 1 under the operation $\otimes$. Proceed in the following steps. a. Use the fact that $\operatorname{ker}\left(\Gamma\left(Y_{n+1}, \mathcal{O}_{Y_{n+1}}\right) \rightarrow \Gamma\left(Y_n, \mathcal{O}_{Y_n}\right)\right)$ is a nilpotent ideal to show that the inverse system $\left(\Gamma\left(Y_n, \mathcal{O}_{Y_n}^*\right)\right)$ of units in the respective rings also satisfies (ML). b. Let $\mathrm{F}$ be a coherent sheaf of $\mathcal{O}_x$-modules, and assume that for each $n$, there is some isomorphism $\varphi_n: \tilde{F} / \mathrm{J}^n \mathrm{F} \cong \mathcal{O}_{Y_n}$. Then show that there is an isomorphism $\tilde{F} \cong \mathcal{O}_x$.[^85] Conclude that the natural map Pic $\mathrm{X} \rightarrow \cocolim \operatorname{Pic}Y_n$ is injective. c. Given an invertible sheaf $\mathcal{L}_n$ on $Y_n$ for each $n$, and given isomorphisms $\mathcal{L}_{n+1} \otimes$ $\mathcal{O}_{Y_n} \cong \mathcal{L}_n$, construct maps $\mathcal{L}_{n^{\prime}} \rightarrow \mathcal{L}_n$ for each $n^{\prime} \geqslant n$ so as to make an inverse system, and show that $\mathrm{L}=\lim \mathcal{L}_n$ is a coherent sheaf on $\mathrm{X}$. Then show that $\mathrm{L}$ is locally free of rank 1, and thus conclude that the map $\operatorname{Pic}\mathrm{X} \rightarrow \lim \operatorname{Pic}Y_n$ is surjective.[^86] d. Show that the hypothesis "$\left(\Gamma\left(Y_n, \mathcal{O}_{Y_n}\right)\right)$ satisfies (ML)" is satisfied if either $\mathrm{X}$ is affine, or each $Y_n$ is projective over a field $k$.[^87] # III: Cohomology ## III.1: Derived Functors Amazing! No exercises in this section. ## III.2: Cohomology of Sheaves ### V.2.1. #to_work a. Let $X=\mathbf{A}_k^1$ be the affine line over an infinite field $k$. Let $P, Q$ be distinct closed points of $X$, and let $U=X-\{P, Q\}$. Show that $H^1\left(X, \mathbf{Z}_U\right) \neq 0$. b. \* More generally, let $Y \subseteq X=\mathbf{A}_k^n$ be the union of $n+1$ hyperplanes in suitably general position, and let $U=X-Y$. Show that $H^n\left(X, Z_U\right) \neq 0$. Thus the result of $(2.7)$ is the best possible. ### V.2.2. #to_work Let $X=\mathbf{P}_k^1$ be the projective line over an algebraically closed field $k$. Show that the exact sequence ` \begin{align*} 0 \rightarrow {\mathcal{O}}\rightarrow \mathcal{K} \rightarrow \mathcal{K} / \mathcal{O} \rightarrow 0 .\end{align*} `{=html} of (II, Ex. 1.21d) is a flasque resolution of $\mathcal{O}$. Conclude from (II, Ex. 1.21e) that $H^i(X, \mathcal{O})=0$ for all $i>0$. ### V.2.3. Cohomology with Supports. #to_work Let $X$ be a topological space, let $Y$ be a closed subset, and let $\mathcal{F}$ be a sheaf of abelian groups. Let $\Gamma_Y(X, \mathcal{F})$ denote the group of sections of $\mathcal{F}$ with support in $Y$ (II, Ex. 1.20). a. Show that $\Gamma_Y(X, \cdot)$ is a left exact functor from ${\mathsf{Ab}}(X)$ to ${\mathsf{Ab}}$. We denote the right derived functors of $\Gamma_Y(X, \cdot)$ by $H_Y^i(X, \cdot)$. They are the cohomology groups of $X$ with supports in $Y$, and coefficients in a given sheaf. b. If $0 \rightarrow \mathcal{F}^{\prime} \rightarrow \mathcal{F} \rightarrow \dot{{\mathcal{F}}}^{\prime \prime} \rightarrow 0$ is an exact sequence of sheaves, with $\mathcal{F}^{\prime}$ flasque, show that ` \begin{align*} 0 \rightarrow \Gamma_Y\left(X, \mathcal{F}^{\prime}\right) \rightarrow \Gamma_Y(X, \mathcal{F}) \rightarrow \Gamma_Y\left(X, \mathcal{F}^{\prime \prime}\right) \rightarrow 0 \end{align*} `{=html} is exact. c. Show that if $\mathcal{F}$ is flasque, then $H_Y^i(X, \mathcal{F})=0$ for all $i>0$. d. If $\mathcal{F}$ is flasque, show that the sequence ` \begin{align*} 0 \rightarrow \Gamma_Y(X, \mathcal{F}) \rightarrow \Gamma(X, \mathcal{F}) \rightarrow \Gamma(X-Y, \mathcal{F}) \rightarrow 0 \end{align*} `{=html} is exact. e. Let $U=X-Y$. Show that for any $\mathcal{F}$, there is a long exact sequence of cohomology groups ` \begin{align*} 0 &\rightarrow H_Y^0(X, \mathcal{F}) \rightarrow H^0(X, \mathcal{F}) \rightarrow H^0\left(U,\left.\mathcal{F}\right|_U\right) \rightarrow \\ &\rightarrow H_Y^1(X, \mathcal{F}) \rightarrow H^1(X, \mathcal{F}) \rightarrow H^1\left(U,\left.\mathcal{F}\right|_U\right) \rightarrow \\ &\rightarrow H_Y^2(X, \mathcal{F}) \rightarrow \ldots .\end{align*} `{=html} f. *Excision*. Let $V$ be an open subset of $X$ containing $Y$. Then there are natural functorial isomorphisms, for all $i$ and $\mathcal{F}$, ` \begin{align*} H_Y^i(X, \mathcal{F}) \cong H_Y^i\left(V,\left.\mathcal{F}\right|_V\right) . \end{align*} `{=html} ### V.2.4. Mayer-Vietoris Sequence. #to_work Let $Y_1, Y_2$ be two closed subsets of $X$. Then there is a long exact sequence of cohomology with supports ` \begin{align*} \begin{aligned} \ldots & \rightarrow H_{Y_1 \cap Y_2}^i(X, \mathcal{F}) \rightarrow H_{Y_1}^i(X, \mathcal{F}) \oplus H_{Y_2}^i(X, \mathcal{F}) \rightarrow H_{Y_1 \cup Y_2}^i(X, \mathcal{F}) \rightarrow \\ & \rightarrow H_{Y_1 \cap Y_2}^{i+1}(X, \mathcal{F}) \rightarrow \cdots \end{aligned} \end{align*} `{=html} ### V.2.5. #to_work Let $X$ be a Zariski space (II, Ex. 3.17). Let $P \in X$ be a closed point, and let $X_P$ be the subset of $X$ consisting of all points $Q \in X$ such that $P \in\{Q\}^{-}$. We call $X_P$ the **local space** of $X$ at $P$, and give it the induced topology. Let $j: X_P \rightarrow X$ be the inclusion, and for any sheaf $\mathcal{F}$ on $X$, let $\mathcal{F}_P=j^* \mathcal{F}$. Show that for all $i, \mathcal{F}$, we have ` \begin{align*} H_P^i(X, \mathcal{F})=H_P^i\left(X_P, \mathcal{F}_P\right) . \end{align*} `{=html} ### V.2.6. #to_work Let $X$ be a noetherian topological space, and let $\left\{\mathcal{I}_\alpha\right\}_{\alpha \in A}$ be a direct system of injective sheaves of abelian groups on $X$. Then $\colim\, \mathcal{I}_\alpha$ is also injective.[^88] ### V.2.7. #to_work Let $S^1$ be the circle (with its usual topology), and let $\mathbf{Z}$ be the constant sheaf $\mathbf{Z}$. a. Show that $H^1\left(S^1, \mathbf{Z}\right) \cong \mathbf{Z}$, using our definition of cohomology. b. Now let $\mathcal{R}$ be the sheaf of germs of continuous real-valued functions on $S^1$. Show that $H^1\left(S^1, \mathcal{R}\right)=0$. ## III.3: Cohomology of a Noetherian Affine Scheme ### V.3.1. #to_work Let $X$ be a noetherian scheme. Show that $X$ is affine if and only if $X_{\text {red }}$ (II, Ex. 2.3) is affine.[^89] ### V.3.2. #to_work Let $X$ be a reduced noetherian scheme. Show that $X$ is affine if and only if each irreducible component is affine. ### V.3.3. #to_work Let $A$ be a noetherian ring, and let $\mathfrak{a}$ be an ideal of $A$. a. Show that $\Gamma_{\mathrm{{\mathfrak{a}}}}(\cdot)$ (II, Ex. 5.6) is a left-exact functor from the category of $A$-modules to itself. We denote its right derived functors, calculated in $\mathsf{Mod}(A)$, by $H_{{\mathfrak{a}}}^i(\cdot)$. b. Now let $X=\operatorname{Spec} A, Y=V(\mathfrak{a})$. Show that for any $A$-module $M$, ` \begin{align*} H_{{\mathfrak{a}}}^i(M)=H_Y^i(X, \tilde{M}), \end{align*} `{=html} where $H_Y^i(X, \cdot)$ denotes cohomology with supports in $Y($ Ex. 2.3). c. For any $i$, show that $\Gamma_{{\mathfrak{a}}}\left(H_{\mathfrak{a}}^i(M)\right)=H_{\mathfrak{a}}^i(M)$. ### V.3.4. Cohomological Interpretation of Depth. #to_work If $A$ is a ring, ${\mathfrak{a}}$ an ideal, and $M$ an $A$ module, then $\operatorname{depth}_{\mathfrak{a}}M$ is the maximum length of an $M$-regular sequence $x_1, \ldots, x_r$, with all $x_i \in \mathfrak{a}$. This generalizes the notion of depth introduced in $(II, \S 8)$. a. Assume that $A$ is noetherian. Show that if $\operatorname{depth}_{\mathfrak{a}}M \geqslant 1$, then $\Gamma_{\mathfrak{a}}(M)=0$, and the converse is true if $M$ is finitely generated.[^90] b. Show inductively, for $M$ finitely generated, that for any $n \geqslant 0$, the following conditions are equivalent: (i) $\operatorname{depth}_{\mathfrak{a}} M \geqslant n$; (ii) $H_{\mathfrak{a}}^i(M)=0$ for all $i \begin{align*} \Gamma(U, \tilde{M}) \cong \colim_n \mathop{\mathrm{Hom}}_A\left({\mathfrak{a}}^n, M\right) . \end{align*} `{=html} b. Apply this in the case of an injective $A$-module $I$, to give another proof of (3.4). ### V.3.8. #to_work Without the noetherian hypothesis, (3.3) and (3.4) are false. Let $A=k\left[x_0, x_1, x_2, \ldots\right]$ with the relations $x_0^n x_n=0$ for $n=1,2, \ldots$. Let $I$ be an injective $A$-module containing $A$. Show that $I \rightarrow I_{x_0}$ is not surjective. ## III.4: Čech Cohomology ### V.4.1. #to_work Let $f: X \rightarrow Y$ be an affine morphism of noetherian separated schemes (II, Ex. 5.17). Show that for any quasi-coherent sheaf $\mathcal{F}$ on $X$, there are natural isomorphisms for all $i \geqslant 0$,[^92] ` \begin{align*} H^i(X, \mathcal{F}) \cong H^i\left(Y, f_* \mathcal{F}\right) \end{align*} `{=html} ### V.4.2. #to_work Prove Chevalley's theorem: Let $f: X \rightarrow Y$ be a finite surjective morphism of noetherian separated schemes, with $X$ affine. Then $Y$ is affine. a. Let $f: X \rightarrow Y$ be a finite surjective morphism of integral noetherian schemes. Show that there is a coherent sheaf $\mathcal{M}$ on $X$, and a morphism of sheaves $\alpha: \mathcal{O}_Y^r \rightarrow f_* \mathcal{M}$ for some $r>0$, such that $\alpha$ is an isomorphism at the generic point of $Y$. b. For any coherent sheaf $\mathcal{F}$ on $Y$, show that there is a coherent sheaf $\mathcal{G}$ on $X$, and a morphism $\beta: f_* \mathcal{G} \rightarrow \mathcal{F}^r$ which is an isomorphism at the generic point of $Y$.[^93] c. Now prove Chevalley's theorem. First use (Ex. 3.1) and (Ex. 3.2) to reduce to the case $X$ and $Y$ integral. Then use (3.7), (Ex. 4.1), consider $\operatorname{ker} \beta$ and coker $\beta$, and use noetherian induction on $Y$. ### V.4.3. #to_work Let $X=\mathbf{A}_k^2=\operatorname{Spec} k[x, y]$, and let $U=X-\{(0,0)\}$. Using a suitable cover of $U$ by open affine subsets, show that $H^1\left(U, \mathcal{O}_U\right)$ is isomorphic to the $k$-vector space spanned by $\left\{x^i y^j \mathrel{\Big|}i, j<0\right\}$. In particular, it is infinite-dimensional.[^94] ### V.4.4. #to_work On an arbitrary topological space $X$ with an arbitrary abelian sheaf $\mathcal{F}$, Čech cohomology may not give the same result as the derived functor cohomology. But here we show that for $H^1$, there is an isomorphism if one takes the limit over all coverings. a. Let $\mathfrak{U}=\left(U_i\right)_{i \in I}$ be an open covering of the topological space $X$. A refinement of $\mathfrak{U}$ is a covering $\mathfrak{B}=\left(V_j\right)_{j \in J}$, together with a map $\lambda: J \rightarrow I$ of the index sets, such that for each $j \in J, V_j \subseteq U_{\lambda(j)}$. If $\mathfrak{B}$ is a refinement of $\mathfrak{X}$, show that there is a natural induced map on Čech cohomology, for any abelian sheaf $\mathcal{F}$, and for each $i$, ` \begin{align*} \lambda^i: \check{H}^i(\mathfrak{U}, \mathcal{F}) \rightarrow \check{H}^i(\mathfrak{B}, \mathcal{F}) . \end{align*} `{=html} The coverings of $X$ form a partially ordered set under refinement, so we can consider the Ceech cohomology in the limit ` \begin{align*} \colim_{\mathfrak U} \check{H}^i(\mathfrak{U}, \mathcal{F}) . \end{align*} `{=html} b. For any abelian sheaf $\mathcal{F}$ on $X$, show that the natural maps (4.4) for each covering ` \begin{align*} \check{H}^i(\mathfrak{U}, \mathcal{F}) \rightarrow H^i(X, \mathcal{F}) \end{align*} `{=html} are compatible with the refinement maps above. c. Now prove the following theorem. Let $X$ be a topological space, $\mathcal{F}$ a sheaf of abelian groups. Then the natural map ` \begin{align*} \colim_{\mathfrak U} \check{H}^1(\mathfrak{U}, \mathcal{F}) \rightarrow H^1(X, \mathcal{F}) \end{align*} `{=html} is an isomorphism.[^95] ### V.4.5. #to_work For any ringed space $\left(X, \mathcal{O}_X\right)$, let $\operatorname{Pic}X$ be the group of isomorphism classes of invertible sheaves (II, §6). Show that $\operatorname{Pic}X \cong H^1\left(X, \mathcal{O}_X^*\right)$, where $\mathcal{O}_X^*$ denotes the sheaf whose sections over an open set $U$ are the units in the ring $\Gamma\left(U, \mathcal{O}_X\right)$, with multiplication as the group operation.[^96] ### V.4.6. #to_work Let $\left(X, \mathcal{O}_X\right)$ be a ringed space, let $\mathcal{I}$ be a sheaf of ideals with $\mathcal{I}^2=0$, and let $X_0$ be the ringed space $\left(X, \mathcal{O}_X / \mathcal{I}\right)$. Show that there is an exact sequence of sheaves of abelian groups on $X$, ` \begin{align*} 0 \rightarrow \mathcal{I} \rightarrow \mathcal{O}_X^* \rightarrow \mathcal{O}_{X_0}^* \rightarrow 0, \end{align*} `{=html} where $\mathcal{O}_X^*$ (respectively, $\mathcal{O}_{X_0}^*$ ) denotes the sheaf of (multiplicative) groups of units in the sheaf of rings $\mathcal{O}_X$ (respectively, $\mathcal{O}_{X_0}$ ) the map $\mathcal{I} \rightarrow \mathcal{O}_X^*$ is defined by $a \mapsto$ $1+a$, and $\mathcal{I}$ has its usual (additive) group structure. Conclude there is an exact sequence of abelian groups ` \begin{align*} \ldots \rightarrow H^1(X, \mathcal{I}) \rightarrow \operatorname{Pic} X \rightarrow \operatorname{Pic} X_0 \rightarrow H^2(X, \mathcal{I}) \rightarrow \ldots . \end{align*} `{=html} ### V.4.7. #to_work Let $X$ be a subscheme of $\mathbf{P}_k^2$ defined by a single homogeneous equation $f\left(x_0, x_1, x_2\right)=0$ of degree $d$. (Do not assume $f$ is irreducible.) Assume that $(1,0,0)$ is not on $X$. Then show that $X$ can be covered by the two open affine subsets $U=X \cap\left\{x_1 \neq 0\right\}$ and $V=X \cap\left\{x_2 \neq 0\right\}$. Now calculate the Čech complex ` \begin{align*} \Gamma\left(U, \mathcal{O}_X\right) \oplus \Gamma\left(V, \mathcal{O}_X\right) \rightarrow \Gamma\left(U \cap V, \mathcal{O}_X\right) \end{align*} `{=html} explicitly, and thus show that ` \begin{align*} \operatorname{dim} H^0\left(X, \mathcal{O}_X\right)&=1, \\ \operatorname{dim} H^1\left(X, \mathcal{O}_X\right)&=\frac{1}{2}(d-1)(d-2) . \end{align*} `{=html} ### V.4.8. Cohomological Dimension. #to_work Let $X$ be a noetherian separated scheme. We define the **cohomological dimension** of $X$, denoted $\operatorname{cd}(X)$, to be the least integer $n$ such that $H^i(X, \mathcal{F})=0$ for all quasi-coherent sheaves $\mathcal{F}$ and all $i>n$. Thus for example, Serre's theorem (3.7) says that $\operatorname{cd}(X)=0$ if and only if $X$ is affine. Grothendieck's theorem (2.7) implies that $\operatorname{cd}(X) \leqslant \operatorname{dim} X$. a. In the definition of $\operatorname{cd}(X)$, show that it is sufficient to consider only coherent sheaves on $X$. Use (II, Ex. 5.15) and (2.9). b. If $X$ is quasi-projective over a field $k$, then it is even sufficient to consider only locally free coherent sheaves on $X$. Use (II, 5.18). c. Suppose $X$ has a covering by $r+1$ open affine subsets. Use Čech cohomology to show that $\operatorname{cd}(X) \leqslant r$. d. \* If $X$ is a quasi-projective scheme of dimension $r$ over a field $k$, then $X$ can be covered by $r+1$ open affine subsets. Conclude (independently of (2.7)) that $\operatorname{cd}(X) \leqslant \operatorname{dim} X$. e. Let $Y$ be a set-theoretic complete intersection (I, Ex. 2.17) of codimension $r$ in $X=\mathbf{P}_k^n$. Show that $\operatorname{cd}(X-Y) \leqslant r-1$. ### V.4.9. #to_work Let $X=\operatorname{Spec} k\left[x_1, x_2, x_3, x_4\right]$ be affine four-space over a field $k$. Let $Y_1$ be the plane $x_1=x_2=0$ and let $Y_2$ be the plane $x_3=x_4=0$. Show that $Y=Y_1 \cup Y_2$ is not a set-theoretic complete intersection in $X$. Therefore the projective closure $\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu$ in $\mathbf{P}_k^4$ is also not a set-theoretic complete intersection.[^97] ### V.4.10. #to_work \* Let $X$ be a nonsingular variety over an algebraically closed field $k$, and let $\mathcal{F}$ be a coherent sheaf on $X$. Show that there is a one-to-one correspondence between the set of infinitesimal extensions of $X$ by $\mathcal{F}$ (II, Ex. 8.7) up to isomorphism, and the group $H^1(X, \mathcal{F} \otimes \mathcal{T})$, where $\mathcal{T}$ is the tangent sheaf of $X$, see $(II \S 8)$.[^98] ### V.4.11. #to_work This exercise shows that Cech cohomology will agree with the usual cohomology whenever the sheaf has no cohomology on any of the open sets. More precisely, let $X$ be a topological space, $\mathcal{F}$ a sheaf of abelian groups, and $\mathfrak{U}=\left(U_i\right)$ an open cover. Assume for any finite intersection $V=U_{i_0} \cap \ldots \cap U_{i_p}$ of open sets of the covering, and for any $k>0$, that $H^k\left(V,\left.\mathcal{F}\right|_V\right)=0$. Then prove that for all $p \geqslant 0$, the natural maps ` \begin{align*} \check{H}^p(\mathfrak{U}, \mathcal{F}) \rightarrow H^p(X, \mathcal{F}) \end{align*} `{=html} of (4.4) are isomorphisms. Show also that one can recover (4.5) as a corollary of this more general result. ## III.5: The Cohomology of Projective Space ### III.5.1. #to_work Let $X$ be a projective scheme over a field $k$, and let $\mathcal{F}$ be a coherent sheaf on $X$. We define the Euler characteristic of $\mathcal{F}$ by ` \begin{align*} \chi(\mathcal{F})=\sum(-1)^i \operatorname{dim}_k H^i(X, \mathcal{F}) . \end{align*} `{=html} If ` \begin{align*} 0 \rightarrow \mathcal{F}^{\prime} \rightarrow \mathcal{F} \rightarrow \mathcal{F}^{\prime \prime} \rightarrow 0 \end{align*} `{=html} is a short exact sequence of coherent sheaves on $X$, show that $\chi(\mathcal{F})=\chi\left(\mathcal{F}^{\prime}\right)+$ $\chi\left(\mathcal{F}^{\prime \prime}\right)$. ### III.5.2. #to_work a. Let $X$ be a projective scheme over a field $k$, let $\mathcal{O}_X(1)$ be a very ample invertible sheaf on $X$ over $k$, and let $\mathcal{F}$ be a coherent sheaf on $X$. Show that there is a polynomial $P(z) \in \mathbf{Q}[z]$, such that $\chi(\mathcal{F}(n))=P(n)$ for all $n \in \mathbf{Z}$. We call $P$ the **Hilbert polynomial** of $\mathcal{F}$ with respect to the sheaf $\mathcal{O}_X(1)$.[^99] b. Now let $X=\mathbf{P}_k^r$, and let $M=\Gamma_*(\mathcal{F})$, considered as a graded $S=k\left[x_0, \ldots, x_r\right]-$ module. Use (5.2) to show that the Hilbert polynomial of $\mathcal{F}$ just defined is the same as the Hilbert polynomial of $M$ defined in $(I, \S 7 )$. ### III.5.3. Arithmetic Genus. #to_work Let $X$ be a projective scheme of dimension $r$ over a field $k$. We define the arithmetic genus $p_a$ of $X$ by ` \begin{align*} p_a(X)=(-1)^r\left(\chi\left(\mathcal{O}_X\right)-1\right) . \end{align*} `{=html} Note that it depends only on $X$, not on any projective embedding. a. If $X$ is integral, and $k$ algebraically closed, show that $H^0\left(X, \mathcal{O}_X\right) \cong k$, so that ` \begin{align*} p_a(X)=\sum_{i=0}^{r-1}(-1)^i \operatorname{dim}_k H^{r-i}\left(X, \mathcal{O}_X\right) . \end{align*} `{=html} In particular, if $X$ is a curve, we have[^100] ` \begin{align*} p_a(X)=\operatorname{dim}_k H^1\left(X, \mathcal{O}_X\right) . \end{align*} `{=html} b. If $X$ is a closed subvariety of $\mathbf{P}_k^r$, show that this $p_a(X)$ coincides with the one defined in (I, Ex. 7.2), which apparently depended on the projective embedding. c. If $X$ is a nonsingular projective curve over an algebraically closed field $k$, show that $p_a(X)$ is in fact a birational invariant. Conclude that a nonsingular plane curve of degree $d \geqslant 3$ is not rational.[^101] ### III.5.4. #to_work Recall from (II, Ex. 6.10) the definition of the Grothendieck group $K(X)$ of a noetherian scheme $X$. a. Let $X$ be a projective scheme over a field $k$, and let $\mathcal{O}_X(1)$ be a very ample invertible sheaf on $X$. Show that there is a (unique) additive homomorphism ` \begin{align*} P: K(X) \rightarrow \mathbf{Q}[z] \end{align*} `{=html} such that for each coherent sheaf $\mathcal{F}$ on $X, P(\gamma(\mathcal{F}))$ is the Hilbert polynomial of $\mathcal{F}$ (Ex. 5.2). b. Now let $X=\mathbf{P}_k^r$. For each $i=0,1, \ldots, r$, let $L_i$ be a linear space of dimension $i$ in $X$. Then show that (1) $K(X)$ is the free abelian group generated by $\left\{\gamma\left(\mathcal{O}_{L_i}\right) \mathrel{\Big|}i=0, \ldots, r\right\}$, and (2) the map $P: K(X) \rightarrow \mathbf{Q}[z]$ is injective.[^102] ### III.5.5. #to_work Let $k$ be a field, let $X=\mathbf{P}_k^r$, and let $Y$ be a closed subscheme of dimension $q \geqslant 1$, which is a complete intersection (II, Ex. 8.4). Then: a. for all $n \in \mathbf{Z}$, the natural map ` \begin{align*} H^0\left(X, \mathcal{O}_X(n)\right) \rightarrow H^0\left(Y, \mathcal{O}_Y(n)\right) \end{align*} `{=html} is surjective.[^103] b. $Y$ is connected; c. $H^i\left(Y, \mathcal{O}_Y(n)\right)=0$ for $00$, when $Y$ is a disjoint union of $q$ lines $\mathbf{P}^1$ in $Q$, to show: (1) if $|a-b| \leqslant 1$, then $H^1\left(Q, \mathcal{O}_Q(a, b)\right)=0$; (2) if $a, b<0$, then $H^1\left(Q \mathcal{O}_Q(a, b)\right)=0$ (3) If $a \leqslant-2$, then $H^1\left(Q, \mathcal{O}_Q(a, 0)\right) \neq 0$. b. Now use these results to show: (1) if $Y$ is a locally principal closed subscheme of type $(a, b)$, with $a, b>0$, then $Y$ is connected; (2) now assume $k$ is algebraically closed. Then for any $a, b>0$, there exists an irreducible nonsingular curve $Y$ of type (a,b). Use (II, 7.6.2) and (II, 8.18). (3) an irreducible nonsingular curve $Y$ of type $(a, b), a, b>0$ on $Q$ is projectively normal (II, Ex. 5.14) if and only if $|a-b| \leqslant 1$. In particular, this gives lots of examples of nonsingular, but not projectively normal curves in $\mathbf{P}^3$. The simplest is the one of type $(1,3)$, which is just the rational quartic curve (I, Ex. 3.18). c. If $Y$ is a locally principal subscheme of type $(a, b)$ in $Q$, show that[^105] ` \begin{align*} p_a(Y)= a b-a-b+1 .\end{align*} `{=html} ### III.5.7. #to_work Let $X$ (respectively, $Y$ ) be proper schemes over a noetherian ring $A$. We denote by $\mathcal{L}$ an invertible sheaf. a. If $\mathcal{L}$ is ample on $X$, and $Y$ is any closed subscheme of $X$, then $i^* \mathcal{L}$ is ample on $Y$, where $i: Y \rightarrow X$ is the inclusion. b. $\mathcal{L}$ is ample on $X$ if and only if $\mathcal{L}_{\text {red }}=\mathcal{L} \otimes \mathcal{O}_{X_{\text {red }}}$ is ample on $X_{\text {red }}$. c. Suppose $X$ is reduced. Then $\mathcal{L}$ is ample on $X$ if and only if $\mathcal{L} \otimes \mathcal{O}_{X_i}$ is ample on $X_i$, for each irreducible component $X_i$ of $X$. d. Let $f: X \rightarrow Y$ be a finite surjective morphism, and let $\mathcal{L}$ be an invertible sheaf on $Y$. Then $\mathcal{L}$ is ample on $Y$ if and only if $f^* \mathcal{L}$ is ample on $X$.[^106] ### III.5.8. #to_work Prove that every one-dimensional proper scheme $X$ over an algebraically closed field $k$ is projective. a. If $X$ is irreducible and nonsingular, then $X$ is projective by (II, 6.7). b. If $X$ is integral, let $\tilde{X}$ be its normalization (II, Ex. 3.8). Show that $\tilde{X}$ is complete and nonsingular, hence projective by (a). Let $f: \tilde{X} \rightarrow X$ be the projection. Let $\mathcal{L}$ be a very ample invertible sheaf on $\tilde{X}$. Show there is an effective divisor $D=\sum P_i$ on $\tilde{X}$ with $\mathcal{L}(D) \cong \mathcal{L}$, and such that $f\left(P_i\right)$ is a nonsingular point of $X$, for each $i$. Conclude that there is an invertible sheaf $\mathcal{L}_0$ on $X$ with $f^* \mathcal{L}_0 \cong$ $\mathcal{L}$. Then use (Ex. 5.7d), (II, 7.6) and (II, 5.16.1) to show that $X$ is projective. c. If $X$ is reduced, but not necessarily irreducible, let $X_1, \ldots, X_r$ be the irreducible components of $X$. Use (Ex. 4.5) to show $\operatorname{Pic}X \rightarrow \bigoplus \operatorname{Pic} X_i$ is surjective. Then use (Ex. 5.7c) to show $X$ is projective. d. Finally, if $X$ is any one-dimensional proper scheme over $k$, use (2.7) and (Ex. 4.6) to show that $\operatorname{Pic}X \rightarrow \operatorname{Pic}X_{\text {red }}$ is surjective. Then use (Ex. 5.7b) to show $X$ is projective. ### III.5.9. A Nonprojective Scheme. #to_work We show the result of (Ex. 5.8) is false in dimension 2. Let $k$ be an algebraically closed field of characteristic 0 , and let $X=\mathbf{P}_k^2$. Let $\omega$ be the sheaf of differential 2-forms (II, §8). Define an infinitesimal extension $X^{\prime}$ of $X$ by $\omega$ by giving the element $\xi \in H^1(X, \omega \otimes \mathcal{T})$ defined as follows (Ex. 4.10). Let $x_0, x_1, x_2$ be the homogeneous coordinates of $X$, let $U_0, U_1, U_2$ be the standard open covering, and let $\xi_{i j}=\left(x_j / x_i\right) d\left(x_i / x_j\right)$. This gives a Čech 1-cocycle with values in $\Omega_X^1$, and since $\operatorname{dim} X=2$, we have $\omega \otimes \mathcal{T} \cong \Omega^1$ (II, Ex. 5.16b). Now use the exact sequence ` \begin{align*} \ldots \rightarrow H^1(X, \omega) \rightarrow \operatorname{Pic} X^{\prime} \rightarrow \operatorname{Pic} X \stackrel{\delta}{\rightarrow} H^2(X, \omega) \rightarrow \ldots \end{align*} `{=html} of (Ex. 4.6) and show $\delta$ is injective. We have $\omega \cong \mathcal{O}_X(-3)$ by (II, 8.20.1), so $H^2(X, \omega) \cong k$. Since char $k=0$, you need only show that $\delta(\mathcal{O}(1)) \neq 0$, which can be done by calculating in Čech cohomology. Since $H^1(X, \omega)=0$, we see that $\operatorname{Pic}X^{\prime}=0$. In particular, $X^{\prime}$ has no ample invertible sheaves, so it is not projective.[^107] ### III.5.10. #to_work Let $X$ be a projective scheme over a noetherian ring $A$, and let $\mathcal{F}^1 \rightarrow \mathcal{F}^2 \rightarrow \ldots \rightarrow$ $\mathcal{F}^r$ be an exact sequence of coherent sheaves on $X$. Show that there is an integer $n_0$, such that for all $n \geqslant n_0$, the sequence of global sections ` \begin{align*} \Gamma\left(X, \mathcal{F}^1(n)\right) \rightarrow \Gamma\left(X, \mathcal{F}^2(n)\right) \rightarrow \ldots \rightarrow \Gamma\left(X, \mathcal{F}^r(n)\right) \end{align*} `{=html} is exact. ## III.6: Ext Groups and Sheaves ### III.6.1. #to_work Let $\left(X, \mathcal{O}_X\right)$ be a ringed space, and let $\mathcal{F}^{\prime}, \mathcal{F}^{\prime \prime} \in {\mathsf{Mod}}(X)$. An **extension** of $\mathcal{F}^{\prime \prime}$ by $\mathcal{F}^{\prime}$ is a short exact sequence ` \begin{align*} 0 \rightarrow \mathcal{F}^{\prime} \rightarrow \mathcal{F} \rightarrow \mathcal{F}^{\prime \prime} \rightarrow 0 \end{align*} `{=html} in ${\mathsf{Mod}}(X)$. Two extensions are isomorphic if there is an isomorphism of the short exact sequences, inducing the identity maps on $\mathcal{F}^{\prime}$ and $\mathcal{F}^{\prime \prime}$. Given an extension as above consider the long exact sequence arising from $\operatorname{Hom}\left(\mathcal{F}^{\prime \prime}, \cdot\right)$, in particular the map ` \begin{align*} \delta: \operatorname{Hom}\left(\mathcal{F}^{\prime \prime}, \mathcal{F}^{\prime \prime}\right) \rightarrow \operatorname{Ext}^1\left(\mathcal{F}^{\prime \prime}, \mathcal{F}^{\prime}\right), \end{align*} `{=html} and let $\xi \in \operatorname{Ext}^1\left(\mathcal{F}^{\prime \prime}, \mathcal{F}^{\prime}\right)$ be $\delta\left(1_{\mathcal{F}^{\prime \prime}}\right)$. Show that this process gives a one-to-one correspondence between isomorphism classes of extensions of $\mathcal{F}^{\prime \prime}$ by $\mathcal{F}^{\prime}$, and elements of the group $\operatorname{Ext}^1\left(\mathcal{F}^{\prime \prime}, \mathcal{F}^{\prime}\right)$. ### III.6.2. #to_work Let $X=\mathbf{P}_k^1$, with $k$ an infinite field. a. Show that there does not exist a projective object $\mathcal{P} \in {\mathsf{Mod}}(X)$, together with a surjective map $\mathcal{P} \rightarrow \mathcal{O}_X \rightarrow 0$.[^108] ```{=html} ``` b. Show that there does not exist a projective object $\mathcal{P}$ in either ${\mathsf{QCoh}}(X)$ or ${\mathsf{Coh}}(X)$ together with a surjection $\mathcal{P} \rightarrow \mathcal{O}_X \rightarrow 0$.[^109] ### III.6.3. #to_work Let $X$ be a noetherian scheme, and let $\mathcal{F}, \mathcal{G} \in {\mathsf{Mod}}(X)$. a. If $\mathcal{F}, \mathcal{G}$ are both coherent, then $\mathcal{E x t}(\mathcal{F}, \mathcal{G})$ is coherent, for all $i \geqslant 0$. b. If $\mathcal{F}$ is coherent and $\mathcal{G}$ is quasi-coherent, then $\mathcal{E} x t^i(\mathcal{F}, \mathcal{G})$ is quasi-coherent, for all $i \geqslant 0$. ### III.6.4. #to_work Let $X$ be a noetherian scheme, and suppose that every coherent sheaf on $X$ is a quotient of a locally free sheaf. In this case we say ${\mathsf{Coh}}(X)$ has enough locally frees. Then for any $\mathcal{G} \in {\mathsf{Mod}}(X)$, show that the $\delta$-functor $\left(\mathcal{E} x t^i(\cdot, \mathcal{G})\right)$, from ${\mathsf{Coh}}(X)$ to ${\mathsf{Mod}}(X)$ is a contravariant universal $\delta$-functor.[^110] ### III.6.5. #to_work Let $X$ be a noetherian scheme, and assume that ${\mathsf{Coh}}(X)$ has enough locally frees (Ex. 6.4). Then for any coherent sheaf $\mathcal{F}$ we define the **homological dimension** of $\mathcal{F}$, denoted $\mathrm{hd} (\mathcal{F})$, to be the least length of a locally free resolution of $\mathcal{F}$ (or $+\infty$ if there is no finite one). Show: a. $\mathcal{F}$ is locally free $\Leftrightarrow \mathcal{E} x t^1(\mathcal{F}, \mathcal{G})=0$ for all $\mathcal{G} \in {\mathsf{Mod}}(X)$; b. $\operatorname{hd}(\mathcal{F}) \leqslant n \Leftrightarrow \mathcal{E x t}(\mathcal{F}, \mathcal{G})=0$ for all $i>n$ and all $\mathcal{G} \in {\mathsf{Mod}}(X)$; c. $\operatorname{hd}(\mathcal{F})=\sup _x \operatorname{pd}_{{\mathcal{O}}_x} \mathcal{F}_x$. ### III.6.6. #to_work Let $A$ be a regular local ring, and let $M$ be a finitely generated $A$-module. In this case, strengthen the result $(6.10 \mathrm{~A})$ as follows. a. $M$ is projective if and only if $\operatorname{Ext}^i(M, A)=0$ for all $i>0$.[^111] b. Use (a) to show that for any $n$, pd $M \leqslant n$ if and only if $\operatorname{Ext}^i(M, A)=0$ for all $i>n$. ### III.6.7. #to_work Let $X=\operatorname{Spec} A$ be an affine noetherian scheme. Let $M, N$ be $A$-modules, with $M$ finitely generated. Then ` \begin{align*} \operatorname{Ext}_X^i(\tilde{M}, \tilde{N}) \cong \operatorname{Ext}_A^i(M, N) \end{align*} `{=html} and ` \begin{align*} \mathcal{E} x t_X^i(\tilde{M}, \tilde{N}) \cong \operatorname{Ext}_A^i(M \cdot N)^{\sim} . \end{align*} `{=html} ### III.6.8. #to_work Prove the following theorem of Kleiman (see Borelli $[1]$): if $X$ is a noetherian, integral, separated, locally factorial scheme, then every coherent sheaf on $X$ is a quotient of a locally free sheaf (of finite rank). a. First show that open sets of the form $X_s$, for various $s \in \Gamma(X, \mathcal{L})$, and various invertible sheaves $\mathcal{L}$ on $X$, form a base for the topology of $X$.[^112] b. Now use (II, 5.14) to show that any coherent sheaf is a quotient of a direct sum $\bigoplus \mathcal{L}_i^{n_i}$ for various invertible sheaves $\mathcal{L}_i$ and various integers $n_i$. ### III.6.9. #to_work Let $X$ be a noetherian, integral, separated, regular scheme. (We say a scheme is regular if all of its local rings are regular local rings.) Recall the definition of the Grothendieck group $K(X)$ from (II, Ex. 6.10). We define similarly another group $K_1(X)$ using locally free sheaves: it is the quotient of the free abelian group generated by all locally free (coherent) sheaves, by the subgroup generated by all expressions of the form $\mathcal{E}-\mathcal{E}^{\prime}-\mathcal{E}^{\prime \prime}$, whenever $0 \rightarrow \mathcal{E}^{\prime} \rightarrow \mathcal{E} \rightarrow \mathcal{E}^{\prime \prime} \rightarrow 0$ is a short exact sequence of locally free sheaves. Clearly there is a natural group homomorphism $\varepsilon: K_1(X) \rightarrow K(X)$. Show that $\varepsilon$ is an isomorphism (Borel and Serre $[1, \S 4])$ as follows. a. Given a coherent sheaf $\mathcal{F}$, use (Ex. 6.8) to show that it has a locally free resolution $\mathcal{E} . \rightarrow \mathcal{F} \rightarrow 0$. Then use (6.11A) and (Ex. 6.5) to show that it has a finite locally free resolution ` \begin{align*} 0 \rightarrow \mathcal{E}_n \rightarrow \ldots \rightarrow \mathcal{E}_1 \rightarrow \mathcal{E}_0 \rightarrow \mathcal{F} \rightarrow 0 . \end{align*} `{=html} b. For each $\mathcal{F}$, choose a finite locally free resolution $\mathcal{E}$. $\rightarrow \mathcal{F} \rightarrow 0$, and let $\delta(\mathcal{F})=\sum(-1)^i \gamma\left(\mathcal{E}_i\right)$ in $K_1(X)$. Show that $\delta(\mathcal{F})$ is independent of the resolution chosen, that it defines a homomorphism of $K(X)$ to $K_1(X)$, and finally, that it is an inverse to $\varepsilon$. ### III.6.10. Duality for a Finite Flat Morphism. #to_work a. Let $f: X \rightarrow Y$ be a finite morphism of noetherian schemes. For any quasicoherent $\mathcal{O}_Y$-module $\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}_Y(f_* {\mathcal{O}}_X, {\mathcal{G}})$ is a quasi-coherent $f_* \mathcal{O}_X$-module, hence corresponds to a quasi-coherent $\mathcal{O}_X$-module, which we call $f ! \mathcal{G}$ (II, Ex. 5.17e). b. Show that for any coherent $\mathcal{F}$ on $X$ and any quasi-coherent $\mathcal{G}$ on $Y$, there is a natural isomorphism ` \begin{align*} f_* \mathop{\mathcal{H}\! \mathit{om}}_X\left(\mathcal{F}, f^{\prime} \mathcal{G}\right) { \, \xrightarrow{\sim}\, }\mathop{\mathcal{H}\! \mathit{om}}_Y\left(f_* \mathcal{F}, \mathcal{G}\right) . \end{align*} `{=html} c. For each $i \geqslant 0$, there is a natural map[^113] ` \begin{align*} \varphi_i: \operatorname{Ext}_X^i\left(\mathcal{F}, f^{!} \mathcal{G}\right) \rightarrow \operatorname{Ext}_Y^i\left(f_* \mathcal{F}, \mathcal{G}\right) . \end{align*} `{=html} d. Now assume that $X$ and $Y$ are separated, ${\mathsf{Coh}}(X)$ has enough locally frees, and assume that $f_* \mathcal{O}_X$ is locally free on $Y$ (this is equivalent to saying $f$ flat-see §9). Show that $\varphi_i$ is an isomorphism for all $i$, all $\mathcal{F}$ coherent on $X$, and all $\mathcal{G}$ quasi-coherent on $Y$.[^114] ## III.7: Serre Duality ### III.7.1. Special case of Kodaira vanishing. #to_work Let $X$ be an integral projective scheme of dimension $\geqslant 1$ over a field $k$, and let $\mathcal{L}$ be an ample invertible sheaf on $X$. Then ` \begin{align*} H^0\left(X, \mathcal{L}^{-1}\right)=0 .\end{align*} `{=html} ### III.7.2. #to_work Let $f: X \rightarrow Y$ be a finite morphism of projective schemes of the same dimension over a field $k$, and let $\omega_Y^{\circ}$ be a dualizing sheaf for $Y$. a. Show that $f^{\prime} \omega_Y^{\circ}$ is a dualizing sheaf for $X$, where $f^{\prime}$ is defined as in (Ex. 6.10). b. If $X$ and $Y$ are both nonsingular, and $k$ algebraically closed, conclude that there is a natural trace map $t: f_* \omega_X \rightarrow \omega_Y$. ### III.7.3. #to_work Let $X=\mathbf{P}_k^n$. Show that $H^q\left(X, \Omega_X^p\right)=0$ for $p \neq q$, $k$ for $p=q, 0 \leqslant p, q \leqslant n$. ### III.7.4. \* The Cohomology Class of a Subvariety. #to_work Let $X$ be a nonsingular projective variety of dimension $n$ over an algebraically closed field $k$. Let $Y$ be a nonsingular subvariety of codimension $p$ (hence dimension $n-p$ ). From the natural map $\Omega_X \otimes$ $\mathcal{O}_Y \rightarrow \Omega_Y$ of $(\mathrm{II}, 8.12)$ we deduce a map $\Omega_X^{n-p} \rightarrow \Omega_Y^{n-p}$. This induces a map on cohomology ` \begin{align*} H^{n-p}\left(X, \Omega_X^{n-p}\right) \rightarrow H^{n-p}\left(Y, \Omega_Y^{n-p}\right) .\end{align*} `{=html} Now $\Omega_Y^{n-p}=\omega_Y$ is a dualizing sheaf for $Y$, so we have the trace map ` \begin{align*} t_Y: H^{n-p}\left(Y, \Omega_Y^{n-p}\right) \rightarrow k .\end{align*} `{=html} Composing, we obtain a linear map $H^{n-p}\left(X, \Omega_X^{n-p}\right) \rightarrow k$. By (7.13) this corresponds to an element $\eta(Y) \in$ $H^p\left(X, \Omega_X^p\right)$, which we call the **cohomology class of $Y$**. a. If $P \in X$ is a closed point, show that $t_X(\eta(P))=1$, where $\eta(P) \in H^n\left(X, \Omega^n\right)$ and $t_X$ is the trace map. b. If $X=\mathbf{P}^n$, identify $H^p\left(X, \Omega^p\right)$ with $k$ by (Ex. 7.3), and show that $\eta(Y)=(\operatorname{deg} Y) \cdot 1$, where deg $Y$ is its degree as a projective variety (I, $\S$ 7).[^115] c. For any scheme $X$ of finite type over $k$, we define a homomorphism of sheaves of abelian groups $d \log : \mathcal{O}_X^* \rightarrow \Omega_X$ by $d \log (f)=f^{-1} \, df$. Here $\mathcal{O}^*$ is a group under multiplication, and $\Omega_X$ is a group under addition. This induces a map on cohomology ` \begin{align*} \operatorname{Pic}X=H^1\left(X, \mathcal{O}_X^*\right) \rightarrow H^1\left(X, \Omega_X\right) \end{align*} `{=html} which we denote by $c$. See (Ex. 4.5). d. Returning to the hypotheses above, suppose $p=1$. Show that $\eta(Y)=c(\mathcal{L}(Y))$, where $\mathcal{L}(Y)$ is the invertible sheaf corresponding to the divisor $Y$. ## III.8: Higher Direct Images of Sheaves ### III.8.1. #to_work Let $f: X \rightarrow Y$ be a continuous map of topological spaces. Let $\mathcal{F}$ be a sheaf of abelian groups on $X$, and assume that $R^i f_*(\mathcal{F})=0$ for all $i>0$. Show that there are natural isomorphisms, for each $i \geqslant 0$,[^116] ` \begin{align*} H^i(X, \mathcal{F}) \cong H^i\left(Y, f_* \mathcal{F}\right) . \end{align*} `{=html} ### III.8.2. #to_work Let $f: X \rightarrow Y$ be an affine morphism of schemes (II, Ex. 5.17) with $X$ noetherian, and let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Show that the hypotheses of (Ex. 8.1) are satisfied, and hence that $H^i(X, \mathcal{F}) \cong H^i\left(Y, f_* \mathcal{F}\right)$ for each $i \geqslant 0$. ### III.8.3. The Projection Formula. #to_work Let $f: X \rightarrow Y$ be a morphism of ringed spaces, let $\mathcal{F}$ be an $\mathcal{O}_X$-module, and let $\mathcal{E}$ be a locally free $\mathcal{O}_Y$-module of finite rank. Prove the projection formula (cf. (II, Ex. 5.1)) ` \begin{align*} R^i f_*\left(\mathcal{F} \otimes f^* \mathcal{E}\right) \cong R^i f_*(\mathcal{F}) \otimes \mathcal{E} . \end{align*} `{=html} ### III.8.4. #to_work Let $Y$ be a noetherian scheme, and let $\mathcal{E}$ be a locally free $\mathcal{O}_Y$-module of rank $n+1$, $n \geqslant 1$. Let $X=\mathbf{P}(\mathcal{E})$ (II, $\S$ ), with the invertible sheaf $\mathcal{O}_X(1)$ and the projection morphism $\pi: X \rightarrow Y$. a. Then - $\pi_*(\mathcal{O}(l)) \cong S^l(\mathcal{E})$ for $l \geqslant 0, \pi_*(\mathcal{O}(l))=0$ for $l<0$ (II, 7.11); - $R^i \pi_*(\mathcal{O}(l))=0$ for $0-n-1$. b. Show there is a natural exact sequence ` \begin{align*} 0 \rightarrow \Omega_{X / Y} \rightarrow\left(\pi^* \mathcal{E}\right)(-1) \rightarrow \mathcal{O} \rightarrow 0, \end{align*} `{=html} cf. (II, 8.13), and conclude that the **relative canonical sheaf** $\omega_{X / Y}=\wedge^n \Omega_{X / Y}$ is isomorphic to $\left(\pi^* \wedge^{n+1} \mathcal{E}\right)(-n-1)$. Show furthermore that there is a natural isomorphism $R^n \pi_*\left(\omega_{X / Y}\right) \cong \mathcal{O}_Y$ (cf. (7.1.1)). c. Now show, for any $l \in \mathbf{Z}$, that ` \begin{align*} R^n \pi_*(\mathcal{O}(l)) \cong \pi_*(\mathcal{O}(-l-n-1)) {}^{ \vee }\otimes\left(\wedge^{n+1} \mathcal{E}\right) {}^{ \vee }. \end{align*} `{=html} d. Show that $p_a(X)=(-1)^n p_a(Y)$ (use (Ex. 8.1)) and $p_g(X)=0$ (use (II, 8.11)). e. In particular, if $Y$ is a nonsingular projective curve of genus $g$, and $\mathcal{E}$ a locally free sheaf of rank 2 , then $X$ is a projective surface with $p_a=-g, p_g=0$, and irregularity $g$ (7.12.3). This kind of surface is called a **geometrically ruled surface** (V, §2). ## III.9: Flat Morphisms ### III.9.1. #to_work A flat morphism $f: X \rightarrow Y$ of finite type of noetherian schemes is open, i.e, for every open subset $U \subseteq X, f(U)$ is open in Y.[^117] ### III.9.2. #to_work Do the calculation of (9.8.4) for the curve of (I, Ex. 3.14). Show that you get an embedded point at the cusp of the plane cubic curve. ### III.9.3. #to_work Some examples of flatness and nonflatness. a. If $f: X \rightarrow Y$ is a finite surjective morphism of nonsingular varieties over an algebraically closed field $k$, then $f$ is flat. b. Let $X$ be a union of two planes meeting at a point, each of which maps isomorphically to a plane $Y$. Show that $f$ is not flat. For example, let $Y=$ $\operatorname{Spec} k[x, y]$ and ` \begin{align*} X=\operatorname{Spec} k[x, y, z, w] /(z, w) \cap(x+z, y+w) .\end{align*} `{=html} c. Again let $Y=\operatorname{Spec} k[x, y]$, but take $X=\operatorname{Spec} k[x, y, z, w] /\left(z^2, z w, w^2, x z-y w\right)$. Show that $X_{\text {red }} \cong Y, X$ has no embedded points, but that $f$ is not flat. ### III.9.4. Open Nature of Flatness. #to_work Let $f: X \rightarrow Y$ be a morphism of finite type of noetherian schemes. Then $\{x \in X \mathrel{\Big|}f$ is flat at $x\}$ is an open subset of $X$ (possibly empty).[^118] ### III.9.5. Very Flat Families. #to_work For any closed subscheme $X \subseteq \mathbf{P}^n$, we denote by $C(X) \subseteq \mathbf{P}^{n+1}$ the projective cone over $X$ (I, Ex. 2.10). If $I \subseteq k\left[x_0, \ldots, x_n\right]$ is the (largest) homogeneous ideal of $X$, then $C(X)$ is defined by the ideal generated by $I$ in $k\left[x_0, \ldots, x_{n+1}\right]$. a. Give an example to show that if $\left\{X_t\right\}$ is a flat family of closed subschemes of $\mathbf{P}^n$, then $\left\{C\left(X_t\right)\right\}$ need not be a flat family in $\mathbf{P}^{n+1}$. b. To remedy this situation, we make the following definition. Let $X \subseteq \mathbf{P}_T^n$ be a closed subscheme, where $T$ is a noetherian integral scheme. For each $t \in T$, let $I_t \subseteq S_t=k(t)\left[x_0, \ldots, x_n\right]$ be the homogeneous ideal of $X_t$ in $\mathbf{P}_{k(t)}^n$. We say that the family $\left\{X_t\right\}$ is **very flat** if for all $d \geqslant 0$, ` \begin{align*} \operatorname{dim}_{k(t)}\left(S_t / I_t\right)_d \end{align*} `{=html} is independent of $t$. Here $(\quad)_d$ means the homogeneous part of degree $d$. c. If $\left\{X_t\right\}$ is a very flat family in $\mathbf{P}^n$, show that it is flat. Show also that $\left\{C\left(X_t\right)\right\}$ is a very flat family in $\mathbf{P}^{n+1}$, and hence flat. d. If $\left\{X_{(t)}\right\}$ is an algebraic family of projectively normal varieties in $\mathbf{P}_k^n$, parametrized by a nonsingular curve $T$ over an algebraically closed field $k$, then $\left\{X_{(t)}\right\}$ is a very flat family of schemes. ### III.9.6. #to_work Let $Y \subseteq \mathbf{P}^n$ be a nonsingular variety of dimension $\geqslant 2$ over an algebraically closed field $k$. Suppose $\mathbf{P}^{n-1}$ is a hyperplane in $\mathbf{P}^n$ which does not contain $Y$, and such that the scheme $Y^{\prime}=Y \cap \mathbf{P}^{n-1}$ is also nonsingular. Prove that $Y$ is a complete intersection in $\mathbf{P}^n$ if and only if $Y^{\prime}$ is a complete intersection in $\mathbf{P}^{n-1}$.[^119] ### III.9.7. #to_work Let $Y \subseteq X$ be a closed subscheme, where $X$ is a scheme of finite type over a field $k$. Let $D=k[t] / t^2$ be the ring of dual numbers, and define an infinitesimal deformation of $Y$ as a closed subscheme of $X$, to be a closed subscheme $Y^{\prime} \subseteq X \underset{\scriptscriptstyle {k} }{\times} D$, which is flat over $D$, and whose closed fibre is $Y$. Show that these $Y^{\prime}$ are classified by $H^0\left(Y, \mathcal{N}_{Y / X}\right)$, where ` \begin{align*} \mathcal{N}_{Y / X}= \mathop{\mathcal{H}\! \mathit{om}}_{{\mathcal{O}}_Y} \left(\mathcal{I}_Y / \mathcal{I}_Y^2, \mathcal{O}_Y\right) . \end{align*} `{=html} ### III.9.8. \* #to_work Let $A$ be a finitely generated $k$-algebra. Write $A$ as a quotient of a polynomial ring $P$ over $k$, and let $J$ be the kernel: ` \begin{align*} 0 \rightarrow J \rightarrow P \rightarrow A \rightarrow 0 . \end{align*} `{=html} Consider the exact sequence of (II, 8.4A) ` \begin{align*} J / J^2 \rightarrow \Omega_{P / k} \otimes_P A \rightarrow \Omega_{A / k} \rightarrow 0 . \end{align*} `{=html} Apply the functor $\operatorname{Hom}_A(\cdot, A)$, and let $T^1(A)$ be the cokernel: ` \begin{align*} \operatorname{Hom}_A\left(\Omega_{P / k} \otimes A, A\right) \rightarrow \operatorname{Hom}_A\left(J / J^2, A\right) \rightarrow T^1(A) \rightarrow 0 . \end{align*} `{=html} Now use the construction of (II, Ex. 8.6) to show that $T^1(A)$ classifies infinitesimal deformations of $A$, i.e., algebras $A^{\prime}$ flat over $D=k[t] / t^2$, with $A^{\prime} \otimes_D k \cong A$. It follows that $T^1(A)$ is independent of the given representation of $A$ as a quotient of a polynomial ring $P$. ### III.9.9. #to_work A $k$-algebra $A$ is said to be **rigid** if it has no infinitesimal deformations, or equivalently, by (Ex. 9.8) if $T^1(A)=0$. Let $A=k[x, y, z, w] /(x, y) \cap(z, w)$, and show that $A$ is rigid. This corresponds to two planes in $\mathbf{A}^4$ which meet at a point. ### III.9.10. #to_work A scheme $X_0$ over a field $k$ is rigid if it has no infinitesimal deformations. a. Show that $\mathbf{P}_k^1$ is rigid, using (9.13.2). b. One might think that if $X_0$ is rigid over $k$, then every global deformation of $X_0$ is locally trivial. Show that this is not so, by constructing a proper, flat morphism $f: X \rightarrow \mathbf{A}^2$ over $k$ algebraically closed, such that $X_0 \cong \mathbf{P}_k^1$, but there is no open neighborhood $U$ of 0 in $\mathbf{A}^2$ for which $f^{-1}(U) \cong U \times \mathbf{P}^1$. c. \* Show, however, that one can trivialize a global deformation of $\mathbf{P}^1$ after a flat base extension, in the following sense: let $f: X \rightarrow T$ be a flat projective morphism, where $T$ is a nonsingular curve over $k$ algebraically closed. Assume there is a closed point $t \in T$ such that $X_t \cong \mathbf{P}_k^1$. Then there exists a nonsingular curve $T^{\prime}$, and a flat morphism $g: T^{\prime} \rightarrow T$, whose image contains $t$, such that if $X^{\prime}=X \times_T T^{\prime}$ is the base extension, then the new family $f^{\prime}: X^{\prime} \rightarrow T^{\prime}$ is isomorphic to $\mathbf{P}_{T^{\prime}}^1 \rightarrow T^{\prime}$. ### III.9.11. #to_work Let $Y$ be a nonsingular curve of degree $d$ in $\mathbf{P}_k^n$, over an algebraically closed field $k$. Show that[^120] ` \begin{align*} 0 \leqslant p_a(Y) \leqslant \frac{1}{2}(d-1)(d-2) . \end{align*} `{=html} ## III.10: Smooth Morphisms ### III.10.1. Smooth $\neq$ Regular. #to_work {#iii.10.1.-smooth-neq-regular.-to_work} Over a nonperfect field, smooth and regular are not equivalent. For example, let $k_0$ be a field of characteristic $p>0$, let $k=k_0(t)$, and let $X \subseteq \mathbf{A}_k^2$ be the curve defined by $y^2=x^p-t$. Show that every local ring of $X$ is a regular local ring, but $X$ is not smooth over $k$. ### III.10.2. #to_work Let $f: X \rightarrow Y$ be a proper, flat morphism of varieties over $k$. Suppose for some point $y \in Y$ that the fibre $X_y$ is smooth over $k(y)$. Then show that there is an open neighborhood $U$ of $y$ in $Y$ such that $f: f^{-1}(U) \rightarrow U$ is smooth. ### III.10.3. Tale Morphisms. #to_work A morphism $f: X \rightarrow Y$ of schemes of finite type over $k$ is **étale**if it is smooth of relative dimension 0 . It is **unramified** if for every $x \in X$, letting $y=f(x)$, we have $\mathrm{m}_y \cdot \mathcal{O}_x=\mathfrak{m}_x$, and $k(x)$ is a separable algebraic extension of $k(y)$. Show that the following conditions are equivalent: (i) $f$ is étale; (ii) $f$ is flat, and $\Omega_{X / Y}=0$; (iii) $f$ is flat and unramified. ### III.10.4. #to_work Show that a morphism $f: X \rightarrow Y$ of schemes of finite type over $k$ is étale if and only if the following condition is satisfied: for each $x \in X$, let $y=f(x)$. Let $\widehat{\mathcal{O}}_x$ and $\widehat{\mathcal{O}}_y$ be the completions of the local rings at $x$ and $y$. Choose fields of representatives (II, 8.25A) $k(x) \subseteq \widehat{\mathcal{O}}_x$ and $k(y) \subseteq \widehat{\mathcal{O}}_y$ so that $k(y) \subseteq k(x)$ via the natural map $\widehat{\mathcal{O}}_y \rightarrow \widehat{\mathcal{O}}_x$. Then our condition is that for every $x \in X, k(x)$ is a separable algebraic extension of $k(y)$, and the natural map is an isomorphism. ` \begin{align*} \widehat{\mathcal{O}}_y \otimes_{k(y)} k(x) \rightarrow \widehat{\mathcal{O}}_x \end{align*} `{=html} ### III.10.5. Étale Neighborhoods. #to_work If $x$ is a point of a scheme $X$, we define an **étale neighborhood** of $x$ to be an étale morphism $f: U \rightarrow X$, together with a point $x^{\prime} \in U$ such that $f\left(x^{\prime}\right)=x$. As an example of the use of étale neighborhoods, prove the following: if $\mathcal{F}$ is a coherent sheaf on $X$, and if every point of $X$ has an étale neighborhood $f: U \rightarrow X$ for which $f^* \mathcal{F}$ is a free $\mathcal{O}_U$-module, then $\mathcal{F}$ is locally free on $X$. ### III.10.6. #to_work Let $Y$ be the plane nodal cubic curve $y^2=x^2(x+1)$. Show that $Y$ has a finite étale covering $X$ of degree 2, where $X$ is a union of two irreducible components, each one isomorphic to the normalization of $Y$ (Fig. 12). ![](figures/2022-10-23_00-23-42.png) ### III.10.7. (Serre). A linear system with moving singularities. #to_work Let $k$ be an algebraically closed field of characteristic 2. Let $P_1, \ldots, P_7 \in \mathbf{P}_k^2$ be the seven points of the projective plane over the prime field $\mathbf{F}_2 \subseteq k$. Let $D$ be the linear system of all cubic curves in $X$ passing through $P_1, \ldots, P_7$. a. $D$ is a linear system of dimension 2 with base points $P_1, \ldots, P_7$, which determines an inseparable morphism of degree 2 from $X-\left\{P_i\right\}$ to $\mathbf{P}^2$. b. Every curve $C \in D$ is singular. More precisely, either $C$ consists of 3 lines all passing through one of the $P_i$, or $C$ is an irreducible cuspidal cubic with cusp $P \neq$ any $P_i$. Furthermore, the correspondence $C \mapsto$ the singular point of $C$ is a $1-1$ correspondence between $D$ and $\mathbf{P}^2$. Thus the singular points of elements of $D$ move all over. ### III.10.8. A linear system with moving singularities contained in the base locus (any characteristic). #to_work In affine 3 -space with coordinates $x, y, z$, let $C$ be the conic $(x-1)^2+$ $y^2=1$ in the $x y$-plane, and let $P$ be the point $(0,0, t)$ on the $z$-axis. Let $Y_t$ be the closure in $\mathbf{P}^3$ of the cone over $C$ with vertex $P$. Show that as $t$ varies, the surfaces $\left\{Y_t\right\}$ form a linear system of dimension 1, with a moving singularity at $P$. The base locus of this linear system is the conic $C$ plus the $z$-axis. ### III.10.9. #to_work Let $f: X \rightarrow Y$ be a morphism of varieties over $k$. Assume that $Y$ is regular, $X$ is Cohen-Macaulay, and that every fibre of $f$ has dimension equal to $\operatorname{dim} X-\operatorname{dim} Y$. Then $f$ is flat.[^121] ## III.11: The Theorem on Formal Functions ### III.11.1. #to_work Show that the result of $(11.2)$ is false without the projective hypothesis. For example, let $X=\mathbf{A}_k^n$, let $P=(0, \ldots, 0)$, let $U=X-P$, and let $f: U \rightarrow X$ be the inclusion. Then the fibres of $f$ all have dimension 0 , but $R^{n-1} f_* \mathcal{O}_U \neq 0$. ### III.11.2. #to_work Show that a projective morphism with finite fibres (= quasi-finite (II, Ex. 3.5)) is a finite morphism. ### III.11.3. Improved Bertini's Theorem. #to_work Let $X$ be a normal, projective variety over an algebraically closed field $k$. Let $D$ be a linear system (of effective Cartier divisors) without base points, and assume that $D$ is **not composite with a pencil**, which means that if $f: X \rightarrow \mathbf{P}_k^n$ is the morphism determined by $\mathrm{D}$, then $\operatorname{dim} f(X) \geqslant 2$. Then show that every divisor in $\mathrm{D}$ is connected.[^122] ### III.11.4. Principle of Connectedness. #to_work Let $\left\{X_t\right\}$ be a flat family of closed subschemes of $\mathbf{P}_k^n$ parametrized by an irreducible curve $T$ of finite type over $k$. Suppose there is a nonempty open set $U \subseteq T$, such that for all closed points $t \in U, X_t$ is connected. Then prove that $X_t$ is connected for all $t \in T$. ### III.11.5. \* #to_work Let $Y$ be a hypersurface in $X=\mathbf{P}_k^N$ with $N \geqslant 4$. Let $\widehat{X}$ be the formal completion of $X$ along $Y$ (II, $\S$ ). Prove that the natural map $\operatorname{Pic}\widehat{X} \rightarrow \operatorname{Pic}Y$ is an isomorphism.[^123] ### III.11.6. #to_work Again let $Y$ be a hypersurface in $X=\mathbf{P}_k^N$, this time with $N \geqslant 2$. a. If $\mathcal{F}$ is a locally free sheaf on $X$, show that the natural map ` \begin{align*} H^0(X, \mathcal{F}) \rightarrow H^0(\widehat{X}, \widehat{\mathcal{F}}) \end{align*} `{=html} is an isomorphism. b. Show that the following conditions are equivalent: (i) For each locally free sheaf $\mathcal{F}$ on $\widehat{X}$, there exists a coherent sheaf $\mathscr{F}$ on $X$ such that $\mathcal{F} \cong \widehat{\mathscr{F}}$ (i.e., $\mathcal{F}$ is algebraizable); (ii) For each locally free sheaf $\mathcal{F}$ on $\widehat{X}$, there is an integer $n_0$ such that $\mathcal{F}(n)$ is generated by global sections for all $n \geqslant n_0$.[^124] c. Show that the conditions (i) and (ii) of (b) imply that the natural map $\operatorname{Pic}X \rightarrow \operatorname{Pic}\widehat{X}$ is an isomorphism.[^125] ### III.11.7. #to_work Now let $Y$ be a curve in $X=\mathbf{P}_k^2$. a. Use the method of (Ex. 11.5) to show that Pic $\widehat{X} \rightarrow$ Pic $Y$ is surjective, and its kernel is an infinite-dimensional vector space over $k$. b. Conclude that there is an invertible sheaf ${\mathcal{L}}$ on $\widehat{X}$ which is not algebraizable. c. Conclude also that there is a locally free sheaf $\mathcal{F}$ on $\widehat{X}$ so that no twist $\mathcal{F}(n)$ is generated by global sections. Cf. (II, 9.9.1) ### III.11.8. #to_work Let $f: X \rightarrow Y$ be a projective morphism, let $\mathcal{F}$ be a coherent sheaf on $X$ which is flat over $Y$, and assume that $H^i\left(X_y, \mathcal{F}_y\right)=0$ for some $i$ and some $y \in Y$. Then show that $R^i f_*(\mathcal{F})$ is $0$ in a neighborhood of $y$. ## III.12: The Semicontinuity Theorem ### III.12.1. #to_work Let $Y$ be a scheme of finite type over an algebraically closed field $k$. Show that the function ` \begin{align*} \varphi(y)=\operatorname{dim}_k\left(m_y / m_y^2\right) \end{align*} `{=html} is upper semicontinuous of the set of closed points $Y$. ### III.12.2. #to_work Let $\left\{X_t\right\}$ be a family of hypersurfaces of the same degree in $\mathbf{P}_k^n$. Show that for each $i$, the function $h^i\left(X_t, \mathcal{O}_{X_t}\right)$ is a constant function of $t$. ### III.12.3. #to_work Let $X_1 \subseteq \mathbf{P}_k^4$ be the **rational normal quartic curve** (which is the 4-uple embedding of $\mathbf{P}^1$ in $\mathbf{P}^4$ ). Let $X_0 \subseteq \mathbf{P}_k^3$ be a nonsingular rational quartic curve, such as the one in (I, Ex. 3.18b). Use (9.8.3) to construct a flat family $\left\{X_t\right\}$ of curves in $\mathbf{P}^4$, parametrized by $T=\mathbf{A}^1$, with the given fibres $X_1$ and $X_0$ for $t=1$ and $t=0$. Let $\mathcal{I} \subseteq \mathcal{O}_{\mathbf{P}^4 \times T}$ be the ideal sheaf of the total family $X \subseteq \mathbf{P}^4 \times T$. Show that $\mathcal{I}$ is flat over $T$. Then show that ` \begin{align*} h^0(t, \mathcal{I})= \begin{cases}0 & \text { for } t \neq 0 \\ 1 & \text { for } t=0\end{cases} \end{align*} `{=html} and also ` \begin{align*} h^1(t, \mathcal{I})= \begin{cases}0 & \text { for } t \neq 0 \\ 1 & \text { for } t=0 .\end{cases} \end{align*} `{=html} This gives another example of cohomology groups jumping at a special point. ### III.12.4. #to_work Let $Y$ be an integral scheme of finite type over an algebraically closed field $k$. Let $f: X \rightarrow Y$ be a flat projective morphism whose fibres are all integral schemes. Let $\mathcal{L}, \mathcal{M}$ be invertible sheaves on $X$, and assume for each $y \in Y$ that $\mathcal{L}_y \cong \mathcal{M}_y$ on the fibre $X_y$. Then show that there is an invertible sheaf $\mathcal{N}$ on $Y$ such that $\mathcal{L} \cong \mathcal{M} \otimes f^* \mathcal{N}$.[^126] ### III.12.5. #to_work Let $Y$ be an integral scheme of finite type over an algebraically closed field $k$. Let $\mathcal{E}$ be a locally free sheaf on $Y$, and let $X=\mathbf{P}(\mathcal{E})$ -- see $(II, \S 7)$. Then show that $\operatorname{Pic}X \cong( \operatorname{Pic}Y) \times \mathbf{Z}$. This strengthens (II, Ex. 7.9). ### III.12.6. \* #to_work Let $X$ be an integral projective scheme over an algebraically closed field $k$, and assume that $H^1\left(X, \mathcal{O}_X\right)=0$. Let $T$ be a connected scheme of finite type over $k$. a. If $\mathcal{L}$ is an invertible sheaf on $X \times T$, show that the invertible sheaves $\mathcal{L}_t$ on $X=X \times\{t\}$ are isomorphic, for all closed points $t \in T$. b. Show that $\operatorname{Pic}(X \times T)=$ Pic $X \times$ Pic $T$. (Do not assume that $T$ is reduced!)[^127] Cf. (IV, Ex. 4.10) and (V, Ex. 1.6) for examples where $\operatorname{Pic}(X \times T) \neq \operatorname{Pic} X \times$ Pic T. # IV: Curves ## IV.1: Riemann-Roch ### 1.1. Let $X$ be a curve, and let $P \in X$ be a point. Then there exists a nonconstant rational function $f \in K(X)$, which is regular everywhere except at $P$. ### 1.2. Again let $X$ be a curve, and let $P_1, \ldots P_r \in X$ be points. Then there is a rational function $f \in K(X)$ having poles (of some order) at each of the $P_1$, and regular elsewhere. ### 1.3. Let $X$ be an integral, separated, regular, one-dimensional scheme of finite type over $k$, which is **not** proper over $k$. Then $X$ is affine.[^128] ### 1.4. Show that a separated, one-dimensional scheme of finite type over $k$, none of whose irreducible components is proper over $k$, is affine.[^129] ### 1.5. For an effective divisor $D$ on a curve $X$ of genus $g$, show that $\operatorname{dim}|D| \leqslant \operatorname{deg} D$. Furthermore, equality holds if and only if $D=0$ or $g=0$. ### 1.6. Let $X$ be a curve of genus $g$. Show that there is a finite morphism $f: X \rightarrow \mathbf{P}^1$ of degree[^130] $\leqslant g+1$. ### 1.7. A curve $X$ is called **hyperelliptic** if $g \geqslant 2$ and there exists a finite morphism $f: X \rightarrow \mathbf{P}^1$ of degree 2. a. If $X$ is a curve of genus $g=2$, show that the canonical divisor defines a complete linear system $|K|$ of degree 2 and dimension 1, without base points. Use (II, 7.8.1) to conclude that $X$ is hyperelliptic. b. Show that the curves constructed in (1.1.1) all admit a morphism of degree 2 to $\mathbf{P}^1$. Thus there exist hyperelliptic curves of any genus $g \geqslant 2$.[^131] ### 1.8. $p_a$ of a Singular Curve. {#p_a-of-a-singular-curve.} Let $X$ be an integral projective scheme of dimension 1 over $k$, and let $\tilde{X}$ be its normalization (II, Ex. 3.8). Then there is an exact sequence of sheaves on $X$, ` \begin{align*} 0 \rightarrow {\mathcal{O}}_X \rightarrow f_* {\mathcal{O}}_X \rightarrow \sum_{P \in X} \tilde{\mathcal{O}}_P/{\mathcal{O}}_P \rightarrow 0 \end{align*} `{=html} where $\tilde{\mathcal{O}}_P$ is the integral closure of ${\mathcal{O}}_P$. For each $P \in X$, let $\delta_P=\operatorname{length}(\tilde{\mathcal{O}}_P/{\mathcal{O}}_P)$. a. Show that $p_a(X)=p_a(\tilde{X})+\sum_{p \in X} \delta_p$.[^132] b. If $p_a(X)=0$, show that $X$ is already nonsingular and in fact isomorphic to $\mathbf{P}^1$.[^133] c. \* If $P$ is a node or an ordinary cusp (I, Ex. 5.6, Ex. 5.14), show that $\delta_P=1$.[^134] ### 1.9. \* Riemann-Roch for Singular Curves. Let $X$ be an integral projective scheme of dimension 1 over $k$. Let $X_{\mathrm{reg}}$ be the set of regular points of $X$. a. Let $D=\sum n_i P_i$ be a divisor with support in $X_{\mathrm{reg}}$, i.e., all $P_i \in X_{\mathrm{reg}}$. Then define deg $D=\sum n_i$. Let $\mathscr{L}(D)$ be the associated invertible sheaf on $X$, and show that ` \begin{align*} \chi(\mathscr{L}(D))=\operatorname{deg} D+1-p_a . \end{align*} `{=html} b. Show that any Cartier divisor on $X$ is the difference of two very ample Cartier divisors. [^135] c. Conclude that every invertible sheaf $\mathscr{L}$ on $X$ is isomorphic to $\mathscr{L}(D)$ for some divisor $D$ with support in $X_{\mathrm{reg}}$. d. Assume furthermore that $X$ is a locally complete intersection in some projective space. Then by (III, 7.11) the dualizing sheaf $\omega_X$ is an invertible sheaf on $X$, so we can define the canonical divisor $K$ to be a divisor with support in $X_{\mathrm{reg}}$ corresponding to $\omega_X$. Then the formula of a. becomes ` \begin{align*} l(D)-l(K-D)=\operatorname{deg} D+1-p_a \end{align*} `{=html} ### 1.10. Let $X$ be an integral projective scheme of dimension 1 over $k$, which is locally complete intersection, and has $p_a=1$. Fix a point $P_0 \in X_{\text {reg. }}$. Imitate (1.3.7) to show that the map $P \rightarrow \mathscr{L}\left(P-P_0\right)$ gives a one-to-one correspondence between the points of $X_{\mathrm{reg}}$ and the elements of the group $\operatorname{Pic}X$.[^136] ## IV.2: Hurwitz ### 2.1 Use (2.5.3) to show that $\mathbf{P}^n$ is simply connected. ### 2.2 Classification of Curves of Genus 2 . Fix an algebraically closed field $k$ of characteristic $\neq 2$. a. If $X$ is a curve of genus 2 over $k$, the canonical linear system $|K|$ determines a finite morphism $f: X \rightarrow \mathbf{P}^1$ of degree 2 (Ex. 1.7). Show that it is ramified at exactly 6 points, with ramification index 2 at each one. Note that $f$ is uniquely determined, up to an automorphism of $\mathbf{P}^1$, so $X$ determines an (unordered) set of 6 points of $\mathbf{P}^1$, up to an automorphism of $\mathbf{P}^1$. b. Conversely, given six distinct elements $\alpha_1, \ldots, \alpha_6 \in k$, let $K$ be the extension of $k(x)$ determined by the equation $z^2=\left(x-\alpha_1\right) \cdots\left(x-\alpha_6\right)$. Let $f: X \rightarrow \mathbf{P}^1$ be the corresponding morphism of curves. Show that $g(X)=2$, the map $f$ is the same as the one determined by the canonical linear system, and $f$ is ramified over the six points $x=\alpha_i$ of $\mathbf{P}^1$, and nowhere else. (Cf. (II, Ex. 6.4).) c. Using (I, Ex. 6.6), show that if $P_1, P_2, P_3$ are three distinct points of $\mathbf{P}^1$, then there exists a unique $\varphi \in$ Aut $\mathbf{P}^1$ such that $\varphi\left(P_1\right)=0, \varphi\left(P_2\right)=1, \varphi\left(P_3\right)=\infty$. Thus in (a), if we order the six points of $\mathbf{P}^1$, and then normalize by sending the first three to $0,1, x$, respectively, we may assume that $X$ is ramified over $0,1, \infty, \beta_1, \beta_2, \beta_3$, where $\beta_1, \beta_2, \beta_3$ are three distinct elements of $k, \neq 0,1$. d. Let $\Sigma_6$ be the symmetric group on 6 letters. Define an action of $\Sigma_6$ on sets of three distinct elements $\beta_1, \beta_2, \beta_3$ of $k, \neq 0,1$, as follows: reorder the set $0,1, \infty, \beta_1, \beta_2, \beta_3$ according to a given element $\sigma \in \Sigma_6$, then renormalise as in (c) so that the first three become $0,1, \infty$ again. Then the last three are the new $\beta_1^{\prime}, \beta_2^{\prime}, \beta_3^{\prime}$. e. Summing up, conclude that there is a one-to-one correspondence between the set of isomorphism classes of curves of genus 2 over $k$, and triples of distinct elements $\beta_1, \beta_2, \beta_3$ of $k, \neq 0,1$, modulo the action of $\Sigma_6$ described in (d). In particular, there are many non-isomorphic curves of genus 2 . We say that curves of genus 2 depend on three parameters, since they correspond to the points of an open subset of $\mathbf{A}_k^3$ modulo a finite group. ### 2.3 Plane Curves. Let $X$ be a curve of degree $d$ in $\mathbf{P}^2$. For each point $P \in X$, let $T_P(X)$ be the tangent line to $X$ at $P$ (I, Ex. 7.3). Considering $T_P(X)$ as a point of the dual projective plane $\left(\mathbf{P}^2\right)^*$, the map $P \rightarrow T_P(X)$ gives a morphism of $X$ to its **dual curve** $X^*$ in $\left(\mathbf{P}^2\right)^*$ (I, Ex. 7.3). Note that even though $X$ is nonsingular, $X^*$ in general will have singularities. We assume char $k=0$ below. a. Fix a line $L \subseteq \mathbf{P}^2$ which is not tangent to $X$. Define a morphism $\varphi: X \rightarrow L$ by $\varphi(P)=T_P(X) \cap L$, for each point $P \in X$. Show that $\varphi$ is ramified at $P$ if and only if either (1) $P \in L$, or (2) $P$ is an inflection point of $X$, which means that the intersection multiplicity (I, Ex. 5.4) of $T_P(X)$ with $X$ at $P$ is $\geqslant 3$. Conclude that $X$ has only finitely many inflection points. b. A line of $\mathbf{P}^2$ is a **multiple tangent** of $X$ if it is tangent to $X$ at more than one point. It is a **bitangent** if it is tangent to $X$ at exactly two points. If $L$ is a multiple tangent of $X$, tangent to $X$ at the points $P_1, \ldots, P_r$, and if none of the $P_i$ is an inflection point, show that the corresponding point of the dual curve $X^*$ is an ordinary $r$-fold point, which means a point of multiplicity $r$ with distinct tangent directions (I, Ex. 5.3). Conclude that $X$ has only finitely many multiple tangents. c. Let $O \in \mathbf{P}^2$ be a point which is not on $X$, nor on any inflectional or multiple tangent of $X$. Let $L$ be a line not containing $O$. Let $\psi: X \rightarrow L$ be the morphism defined by projection from $O$. Show that $\psi$ is ramified at a point $\mathrm{P} \in X$ if and only if the line $O P$ is tangent to $X$ at $P$, and in that case the ramification index is 2. Use Hurwitz's theorem and (I, Ex. 7.2) to conclude that there are exactly $d(d-1)$ tangents of $X$ passing through $O$. Hence the degree of the dual curve (sometimes called the **class** of $X)$ is $d(d-1)$. d. Show that for all but a finite number of points of $X$, a point $O$ of $X$ lies on exactly $(d+1)(d-2)$ tangents of $X$, not counting the tangent at $O$. e. Show that the degree of the morphism $\varphi$ of a. is $d(d-1)$. Conclude that if $d \geqslant 2$, then $X$ has $3 d(d-2)$ inflection points, properly counted. (If $T_P(X)$ has intersection multiplicity $r$ with $X$ at $P$, then $P$ should be counted $r-2$ times as an inflection point. If $r=3$ we call it an ordinary inflection point.) Show that an ordinary inflection point of $X$ corresponds to an ordinary cusp of the dual curve $X^*$. f. Now let $X$ be a plane curve of degree $d \geqslant 2$, and assume that the dual curve $X^*$ has only nodes and ordinary cusps as singularities (which should be true for sufficiently general $X)$. Then show that $X$ has exactly $\frac{1}{2} d(d-2)(d-3)(d+3)$ bitangents.[^137] g. For example, a plane cubic curve has exactly 9 inflection points, all ordinary. The line joining any two of them intersects the curve in a third one. h. A plane quartic curve has exactly 28 bitangents. (This holds even if the curve has a tangent with four-fold contact, in which case the dual curve $X^*$ has a tacnode.) ### 2.4 A Funny Curve in Characteristic $p$. {#a-funny-curve-in-characteristic-p.} Let $X$ be the plane quartic curve $x^3 y+y^3 z+ z^3 x = 0$ over a field of characteristic 3 . Show that $X$ is nonsingular, every point of $X$ is an inflection point, the dual curve $X^*$ is isomorphic to $X$, but the natural map $X \rightarrow X^*$ is purely inseparable. ### 2.5 Automorphisms of a Curve of Genus $\geqslant 2$. {#automorphisms-of-a-curve-of-genus-geqslant-2.} Prove the theorem of Hurwitz that a curve $X$ of genus $g \geqslant 2$ over a field of characteristic 0 has at most $84(g-1)$ automorphisms. We will see later (Ex. 5.2) or (V, Ex. 1.11) that the group $G=$ Aut $X$ is finite. So let $G$ have order $n$. Then $G$ acts on the function field $K(X)$. Let $L$ be the fixed field. Then the field extension $L \subseteq K(X)$ corresponds to a finite morphism of curves $f: X \rightarrow Y$ of degree $n$. a. If $P \in X$ is a ramification point, and $e_P=r$, show that $f^{-1} f(P)$ consists of exactly $n / r$ points, each having ramification index $r$. Let $P_1, \ldots, P_s$ be a maximal set of ramification points of $X$ lying over distinct points of $Y$, and let $e_{P_1}=r_i$. Then show that Hurwitz's theorem implies that ` \begin{align*} {2 g-2 \over n } =2 g(Y)-2+\sum_{i=1}^s\left(1- {1\over r_i}\right) \end{align*} `{=html} b. Since $g \geqslant 2$, the left hand side of the equation is $>0$. Show that if $g(Y) \geqslant 0$, $s \geqslant 0, r_i \geqslant 2, i=1, \ldots, s$ are integers such that ` \begin{align*} 2g(Y)-2+\sum_{i=1}^s\left(1- {1\over r_i}\right)>0, \end{align*} `{=html} then the minimum value of this expression is $1 / 42$. Conclude that $n \leqslant 84(g-1)$.[^138] ### 2.6 $f_*$ for Divisors. {#f_-for-divisors.} Let $f: X \rightarrow Y$ be a finite morphism of curves of degree $n$. We define a homomorphism $f_*: \operatorname{Div} X \rightarrow$ Div $Y$ by $f_*\left(\sum n_i P_i\right)=\sum n_i f\left(P_i\right)$ for any divisor $D=\sum n_i P_i$ on $X$. a. For any locally free sheaf $\mathscr{E}$ on $Y$, of rank $r$, we define $\operatorname{det}\mathscr{E}=\wedge^r \mathscr{E} \in \operatorname{Pic}Y$ (II, Ex. 6.11). In particular, for any invertible sheaf $\mathscr{M}$ on $X, f_* \mathscr{M}$ is locally free of rank $n$ on $Y$, so we can consider $\operatorname{det}f_* \mathscr{M} \in \operatorname{Pic}Y$. Show that for any divisor $D$ on $X$,[^139] ` \begin{align*} \operatorname{det}\left(f_* \mathscr{L}(D)\right) \cong\left(\operatorname{det} f_*\left({\mathcal{O}}_X\right) \otimes \mathscr{L}\left(f_* D\right) .\right. \end{align*} `{=html} Note in particular that $\operatorname{det}\left(f_* \mathscr{L}(D)\right) \neq \mathscr{L}\left(f_* D\right)$ in general! b. Conclude that $f_* D$ depends only on the linear equivalence class of $D$, so there is an induced homomorphism $f_*: \operatorname{Pic}X \rightarrow \operatorname{Pic}Y$. Show that $f_* f^*: \operatorname{Pic}Y \rightarrow \operatorname{Pic}Y$ is just multiplication by $n$. c. Use duality for a finite flat morphism (III, Ex. 6.10) and (III, Ex. 7.2) to show that ` \begin{align*}\operatorname{det}f_* \Omega_X \cong \qty{ \operatorname{det}f_* {\mathcal{O}}_X}^{-1}\otimes \Omega_Y^{\otimes n}.\end{align*} `{=html} d. Now assume that $f$ is separable, so we have the ramification divisor $R$. We define the **branch divisor** $B$ to be the divisor $f_* R$ on $Y$. Show that ` \begin{align*}\left(\operatorname{det} f_* \mathcal{O}_X\right)^2 \cong \mathscr{L}(-B).\end{align*} `{=html} ### 2.7 Etale Covers of Degree 2. Let $Y$ be a curve over a field $k$ of characteristic $\neq 2$. We show there is a one-to-one correspondence between finite étale morphisms $f: X \rightarrow Y$ of degree 2, and 2-torsion elements of $\operatorname{Pic}Y$, i.e., invertible sheaves $\mathscr{L}$ on $Y$ with $\mathscr{L}^2 \cong \mathcal{O}_Y$. a. Given an étale morphism $f: X \rightarrow Y$ of degree 2, there is a natural map ${\mathcal{O}}_Y \rightarrow$ $f_* {\mathcal{O}}_X$. Let $\mathscr{L}$ be the cokernel. Then $\mathscr{L}$ is an invertible sheaf on $Y, \mathscr{L} \cong \operatorname{det} f_* {\mathcal{O}}_X$, and so $\mathscr{L}^2 \cong \mathcal{O}_Y$ by (Ex. 2.6). Thus an étale cover of degree 2 determines a 2-torsion element in $\operatorname{Pic}Y$. b. Conversely, given a 2-torsion element $\mathscr{L}$ in $\operatorname{Pic}Y$, define an ${\mathcal{O}}_Y{\hbox{-}}$algebra structure on $\mathcal{O}_Y \oplus \mathscr{L}$ by $\langle a, b\rangle \cdot\left\langle a^{\prime}, b^{\prime}\right\rangle=\left\langle a a^{\prime}+\varphi\left(b \otimes b^{\prime}\right), a b^{\prime}+a^{\prime} b\right\rangle$, where $\varphi$ is an isomorphism of $\mathscr{L} \otimes \mathscr{L} \rightarrow \mathcal{O}_Y$. Then take $X=\operatorname{Spec}\left({\mathcal{O}}_Y \oplus \mathscr{L}\right)$ (II, Ex. 5.17). Show that $X$ is an étale cover of $Y$. c. Show that these two processes are inverse to each other.[^140][^141] ## IV.3: Embeddings in Projective Space ### II.3.1. #to_work If $X$ is a curve of genus 2 , show that a divisor $D$ is very ample $\Leftrightarrow \operatorname{deg} D \geqslant 5$. This strengthens (3.3.4). ### II.3.2. #to_work Let $X$ be a plane curve of degree 4 . a. Show that the effective canonical divisors on $X$ are exactly the divisors $X.L$, where $L$ is a line in $\mathbf{P}^2$. b. If $D$ is any effective divisor of degree 2 on $X$, show that $\operatorname{dim}|D|=0$. c. Conclude that $X$ is not hyperelliptic (Ex. 1.7). ### II.3.3. #to_work If $X$ is a curve of genus $\geqslant 2$ which is a complete intersection (II, Ex. 8.4) in some $\mathbf{P}^n$, show that the canonical divisor $K$ is very ample. Conclude that a curve of genus 2 can never be a complete intersection in any $\mathbf{P}^n$. Cf. (Ex. 5.1). ### II.3.4. The Rational Normal Curve. #to_work Let $X$ be the $d$-uple embedding (I, Ex. 2.12) of $\mathbf{P}^1$ in $\mathbf{P}^d$, for any $d \geqslant 1$. We call $X$ the **rational normal curve** of degree $d$ in $\mathbf{P}^d$. a. Show that $X$ is projectively normal, and that its homogeneous ideal can be generated by forms of degree 2 . b. If $X$ is any curve of degree $d$ in $\mathbf{P}^n$, with $d \leqslant n$, which is not contained in any $\mathbf{P}^{n-1}$, show that in fact $d=n, g(X)=0$, and $X$ differs from the rational normal curve of degree $d$ only by an automorphism of $\mathbf{P}^d$. Cf. (II. 7.8.5). c. In particular, any curve of degree 2 in any $\mathbf{P}^n$ is a conic in some $\mathbf{P}^2$. d. A curve of degree 3 in any $\mathbf{P}^n$ must be either a plane cubic curve, or the twisted cubic curve in $\mathbf{P}^3$. ### II.3.5. #to_work Let $X$ be a curve in $\mathbf{P}^3$, which is not contained in any plane. a. If $O \notin X$ is a point, such that the projection from $O$ induces a birational morphism $\varphi$ from $X$ to its image in $\mathbf{P}^2$, show that $\varphi(X)$ must be singular.[^142] b. If $X$ has degree $d$ and genus $g$, conclude that $g<\frac{1}{2}(d-1)(d-2)$. (Use (Ex. 1.8).) c. Now let $\left\{X_t\right\}$ be the flat family of curves induced by the projection (III, 9.8.3) whose fibre over $t=1$ is $X$, and whose fibre $X_0$ over $t=0$ is a scheme with support $\varphi(X)$. Show that $X_0$ always has nilpotent elements. Thus the example (III, 9.8.4) is typical. ### II.3.6. #to_work Curves of Degree 4. a. If $X$ is a curve of degree 4 in some $\mathbf{P}^n$, show that either (1) $g=0$, in which case $X$ is either the rational normal quartic in $\mathbf{P}^4$ (Ex. 3.4) or the rational quartic curve in $\mathbf{P}^3$ (II, 7.8.6), or (2) $X \subseteq \mathbf{P}^2$, in which case $g=3$, or (3) $X \subseteq \mathbf{P}^3$ and $g=1$. b. In the case $g=1$, show that $X$ is a complete intersection of two irreducible quadric surfaces in $\mathbf{P}^3$ (I, Ex. 5.11).[^143] ### II.3.7. #to_work In view of (3.10), one might ask conversely, is every plane curve with nodes a projection of a nonsingular curve in $\mathbf{P}^3$ ? Show that the curve $x y+x^4+y^4=0$ (assume char $k \neq 2$ ) gives a counterexample. ### II.3.8. #to_work We say a (singular) integral curve in $\mathbf{P}^n$ is **strange** if there is a point which lies on all the tangent lines at nonsingular points of the curve. a. There are many singular strange curves, e.g., the curve given parametrically by $x=t, y=t^p, z=t^{2 p}$ over a field of characteristic $p>0$. b. Show, however, that if char $k=0$, there aren't even any singular strange curves besides $\mathbf{P}^1$. ### II.3.9. Bertini's Lemma. #to_work Prove the following lemma of Bertini: if $X$ is a curve of degree $d$ in $\mathbf{P}^3$, not contained in any plane, then for almost all planes $H \subseteq \mathbf{P}^3$ (meaning a Zariski open subset of the dual projective space $\left.\left(\mathbf{P}^3\right)^*\right)$, the intersection $X \cap H$ consists of exactly $d$ distinct points, no three of which are collinear. ### II.3.10. Not every secant is a multisecant. #to_work Generalize the statement that "not every secant is a multisecant" as follows. If $X$ is a curve in $\mathbf{P}^n$, not contained in any $\mathbf{P}^{n-1}$, and if char $k=0$, show that for almost all choices of $n-1$ points $P_1, \ldots, P_{n-1}$ on $X$, the linear space $L^{n-2}$ spanned by the $P_i$ does not contain any further points of $X$. ### II.3.11 #to_work a. If $X$ is a nonsingular variety of dimension $r$ in $\mathbf{P}^n$, and if $n>2 r+1$, show that there is a point $O \notin X$, such that the projection from $O$ induces a closed immersion of $X$ into $\mathbf{P}^{n-1}$. b. If $X$ is the Veronese surface in $\mathbf{P}^5$, which is the 2-uple embedding of $\mathbf{P}^2$ (I, Ex. 2.13), show that each point of every secant line of $X$ lies on infinitely many secant lines. Therefore, the secant variety of $X$ has dimension 4, and so in this case there is a projection which gives a closed immersion of $X$ into $\mathrm{P}^4$ (II, Ex. 7.7).[^144] ### II.3.12. #to_work For each value of $d=2,3,4,5$ and $r$ satisfying $0 \leqslant r \leqslant \frac{1}{2}(d-1)(d-2)$, show that there exists an irreducible plane curve of degree $d$ with $r$ nodes and no other singularities. ## IV.4: Elliptic Curves ### II.4.1. #to_work Let $X$ be an elliptic curve over $k$, with char $k \neq 2$, let $P \in X$ be a point, and let $R$ be the graded ring $R=\bigoplus_{n \geqslant 0} H^0\left(X, \mathcal{O}_X(n P)\right)$. Show that for suitable choice of $t, x, y$ ` \begin{align*} R \cong k[t, x, y] /\left(y^2-x\left(x-t^2\right)\left(x-\lambda t^2\right)\right), \end{align*} `{=html} as a graded ring, where $k[t, x, y]$ is graded by setting $\operatorname{deg} t=1, \operatorname{deg} x=2$, $\operatorname{deg} y=3$. ### II.4.2. #to_work If $D$ is any divisor of degree $\geqslant 3$ on the elliptic curve $X$, and if we embed $X$ in $\mathbf{P}^n$ by the complete linear system $|D|$, show that the image of $X$ in $\mathbf{P}^n$ is projectively normal.[^145] ### II.4.3. #to_work Let the elliptic curve $X$ be embedded in $\mathbf{P}^2$ so as to have the equation $y^2=$ $x(x-1)(x-\lambda)$. Show that any automorphism of $X$ leaving $P_0=(0,1,0)$ fixed is induced by an automorphism of $\mathbf{P}^2$ coming from the automorphism of the affine $(x, y)$-plane given by ` \begin{align*} \left\{\begin{array}{l} x^{\prime}=a x+b \\ y^{\prime}=c y . \end{array}\right. \end{align*} `{=html} In each of the four cases of (4.7), describe these automorphisms of $\mathbf{P}^2$ explicitly, and hence determine the structure of the group $G=\operatorname{Aut}\left(X, P_0\right)$. ### II.4.4. #to_work Let $X$ be an elliptic curve in $\mathbf{P}^2$ given by an equation of the form ` \begin{align*} y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6 \text {. } \end{align*} `{=html} Show that the $j$-invariant is a rational function of the $a_i$, with coefficients in $\mathbf{Q}$. In particular, if the $a_i$ are all in some field $k_0 \subseteq k$, then $j \in k_0$ also. Furthermore, for every $\alpha \in k_0$, there exists an elliptic curve defined over $k_0$ with $j$-invariant equal to $\alpha$. ### II.4.5. #to_work Let $X, P_0$ be an elliptic curve having an endomorphism $f: X \rightarrow X$ of degree 2 . a. If we represent $X$ as a 2-1 covering of $\mathbf{P}^1$ by a morphism $\pi: X \rightarrow \mathbf{P}^1$ ramified at $P_0$, then as in (4.4), show that there is another morphism $\pi^{\prime}: X \rightarrow \mathbf{P}^1$ and a morphism $g: \mathbf{P}^1 \rightarrow \mathbf{P}^1$, also of degree 2 , such that $\pi \circ f=g \circ \pi^{\prime}$. b. For suitable choices of coordinates in the two copies of $\mathbf{P}^1$, show that $g$ can be taken to be the morphism $x \rightarrow x^2$. c. Now show that $g$ is branched over two of the branch points of $\pi$, and that $g^{-1}$ of the other two branch points of $\pi$ consists of the four branch points of $\pi^{\prime}$. Deduce a relation involving the invariant $\lambda$ of $X$. d. Solving the above, show that there are just three values of $j$ corresponding to elliptic curves with an endomorphism of degree 2 , and find the corresponding values of $\lambda$ and $j$.[^146] ### II.4.6. #to_work a. Let $X$ be a curve of genus $g$ embedded birationally in $\mathbf{P}^2$ as a curve of degree $d$ with $r$ nodes. Generalize the method of (Ex. 2.3) to show that $X$ has ` \begin{align*} 6(g-1)+ 3d \end{align*} `{=html} inflection points. A node does not count as an inflection point. Assume char $k=0$. b. Now let $X$ be a curve of genus $g$ embedded as a curve of degree $d$ in $\mathbf{P}^n, n \geqslant 3$, not contained in any $\mathbf{P}^{n-1}$. For each point $P \in X$, there is a hyperplane $H$ containing $P$, such that $P$ counts at least $n$ times in the intersection $H \cap X$. This is called an **osculating** hyperplane at $P$. It generalizes the notion of tangent line for curves in $\mathbf{P}^2$. If $P$ counts at least $n+1$ times in $H \cap X$, we say $H$ is a **hyperosculating hyperplane**, and that $P$ is a **hyperosculation point**. Use Hurwitz's theorem as above, and induction on $n$, to show that $X$ has ` \begin{align*} n(n+1)(g-1)+(n+1) d \end{align*} `{=html} hyperosculation points. c. If $X$ is an elliptic curve, for any $d \geqslant 3$, embed $X$ as a curve of degree $d$ in $\mathbf{P}^{d-1}$, and conclude that $X$ has exactly $d^2$ points of order $d$ in its group law. ### II.4.7. The Dual of a Morphism. #to_work Let $X$ and $X^{\prime}$ be elliptic curves over $k$, with base points $P_0, P_0^{\prime}$. a. If $f: X \rightarrow X^{\prime}$ is any morphism, use (4.11) to show that $f^*:$ Pic $X^{\prime} \rightarrow$ Pic $X$ induces a homomorphism $\widehat{f}:\left(X^{\prime}, P_0^{\prime}\right) \rightarrow\left(X, P_0\right)$. We call this the **dual** of $f$. b. If $f: X \rightarrow X^{\prime}$ and $g: X^{\prime} \rightarrow X^{\prime \prime}$ are two morphisms, then $(g \circ f)\widehat{} = \widehat{f} \circ \widehat{g}$. c. Assume $f\left(P_0\right)=P_0^{\prime}$, and let $n=\operatorname{deg} f$. Show that if $Q \in X$ is any point, and $f(Q)=Q^{\prime}$, then $\widehat{f}\left(Q^{\prime}\right)=n_X(Q)$. (Do the separable and purely inseparable cases separately, then combine.) Conclude that $f \circ \widehat{f}=n_{X^{\prime}}$ and $\widehat{f} \circ f=n_X$. d. \* If $f, g: X \rightarrow X^{\prime}$ are two morphisms preserving the base points $P_0, P_0^{\prime}$, then $(f+g) \widehat{} = \widehat{f}+\widehat{g}$.[^147] ```{=html} ``` e. Using (d), show that for any $n \in \mathbf{Z}, \widehat{n}_X=n_X$. Conclude that deg $n_X=n^2$. f. Show for any $f$ that $\operatorname{deg} \widehat{f}=\operatorname{deg} f$. ### II.4.8. The Algebraic Fundamental Group. #to_work For any curve $X$, the **algebraic fundamental group** $\pi_1(X)$ is defined as $\cocolim \operatorname{Gal}\left(K^{\prime} / K\right)$, where $K$ is the function field of $X$, and $K^{\prime}$ runs over all Galois extensions of $K$ such that the corresponding curve $X^{\prime}$ is étale over $X$ (III, Ex. 10.3). Thus, for example, $\pi_1\left(\mathbf{P}^1\right)=1$. (See 2.5.3) Show that for an elliptic curve $X$, ` \begin{align*} \pi_1(X) = \begin{cases} \displaystyle\prod_{\ell \text{ prime}} {\mathbf{Z}}_\ell \times {\mathbf{Z}}_\ell & \operatorname{ch}k = 0, \\ \\ \displaystyle\prod_{\ell\neq p} {\mathbf{Z}}_\ell \times {\mathbf{Z}}_\ell & \operatorname{ch}k = p \text{ and } {\mathrm{Hasse}}X = 0, \\ \\ {\mathbf{Z}}_p \times \displaystyle\prod_{\ell\neq p} {\mathbf{Z}}_\ell \times {\mathbf{Z}}_\ell & \operatorname{ch}k = p \text{ and } {\mathrm{Hasse}}X \neq 0 \end{cases} ,\end{align*} `{=html} where $\mathbf{Z}_l=\lim \mathbf{Z} / l^n$ is the $l$-adic integers.[^148][^149] ### II.4.9. Isogenies. #to_work We say two elliptic curves $X, X^{\prime}$ are **isogenous** if there is a finite morphism $f: X \rightarrow X^{\prime}$. a. Show that isogeny is an equivalence relation. b. For any elliptic curve $X$, show that the set of elliptic curves $X^{\prime}$ isogenous to $X$, up to isomorphism, is countable.[^150] ### II.4.10. #to_work If $X$ is an elliptic curve, show that there is an exact sequence ` \begin{align*} 0 \rightarrow p_1^* \operatorname{Pic} X \oplus p_2^* \operatorname{Pic} X \rightarrow \operatorname{Pic}(X \times X) \rightarrow R \rightarrow 0, \end{align*} `{=html} where $R=\operatorname{End}\left(X, P_0\right)$. In particular, we see that $\operatorname{Pic}(X \times X)$ is bigger than the sum of the Picard groups of the factors.[^151] ### II.4.11. #to_work Let $X$ be an elliptic curve over $\mathbf{C}$, defined by the elliptic functions with periods $1, \tau$. Let $R$ be the ring of endomorphisms of $X$. a. If $f \in R$ is a nonzero endomorphism corresponding to complex multiplication by $\alpha$, as in (4.18), show that $\operatorname{deg} f=|\alpha|^2$. b. If $f \in R$ corresponds to $\alpha \in \mathbf{C}$ again, show that the dual $\widehat{f}$ of (Ex. 4.7) corresponds to the complex conjugate $\mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5mu$ of $\alpha$. c. If $\tau \in \mathbf{Q}(\sqrt{-d})$ happens to be integral over $\mathbf{Z}$, show that $R=\mathbf{Z}[\tau]$. ### II.4.12. #to_work Again let $X$ be an elliptic curve over $\mathbf{C}$ determined by the elliptic functions with periods $1, \tau$, and assume that $\tau$ lies in the region $G$ of (4.15B). a. If $X$ has any automorphisms leaving $P_0$ fixed other than $\pm 1$, show that either $\tau=i$ or $\tau=\omega$, as in (4.20.1) and (4.20.2). This gives another proof of the fact (4.7) that there are only two curves, up to isomorphism, having automorphisms other than $\pm 1$. b. Now show that there are exactly three values of $\tau$ for which $X$ admits an endomorphism of degree 2. Can you match these with the three values of $j$ determined in (Ex. 4.5)?[^152] ### II.4.13. #to_work If $p=13$, there is just one value of $j$ for which the Hasse invariant of the corresponding curve is 0 . Find it.[^153] ### II.4.14. #to_work The Fermat curve $X: x^3+y^3=z^3$ gives a nonsingular curve in characteristic $p$ for every $p \neq 3$. Determine the set $\mathfrak{P}=\left\{p \neq 3 \mathrel{\Big|}X_{(p)}\right.$ has Hasse invariant 0$\}$, and observe (modulo Dirichlet's theorem) that it is a set of primes of density $\frac{1}{2}$. ### II.4.15. #to_work Let $X$ be an elliptic curve over a field $k$ of characteristic $p$. Let $F^{\prime}: X_p \rightarrow X$ be the $k$-linear Frobenius morphism (2.4.1). Use (4.10.7) to show that the dual morphism $\widehat{F}^{\prime}: X \rightarrow X_p$ is separable if and only if the Hasse invariant of $X$ is 1 . Now use (Ex. 4.7) to show that if the Hasse invariant is 1, then the subgroup of points of order $p$ on $X$ is isomorphic to $\mathbf{Z} / p$; if the Hasse invariant is 0 , it is 0 . ### II.4.16. #to_work Again let $X$ be an elliptic curve over $k$ of characteristic $p$, and suppose $X$ is defined over the field $\mathbf{F}_q$ of $q=p^r$ elements, i.e., $X \subseteq \mathbf{P}^2$ can be defined by an equation with coefficients in $\mathbf{F}_q$. Assume also that $X$ has a rational point over $\mathbf{F}_q$. Let $F^{\prime}: X_q \rightarrow X$ be the $k$-linear Frobenius with respect to $q$. a. Show that $X_q \cong X$ as schemes over $k$, and that under this identification, $F^{\prime}: X \rightarrow X$ is the map obtained by the $q$ th-power map on the coordinates of points of $X$, embedded in $\mathbf{P}^2$. b. Show that $1_X-F^{\prime}$ is a separable morphism and its kernel is just the set $X\left(\mathbf{F}_q\right)$ of points of $X$ with coordinates in $\mathbf{F}_q$. c. Using (Ex. 4.7), show that $F^{\prime}+\widehat{F}^{\prime}=a_X$ for some integer $a$, and that $N=$ $q-a+1$, where $N={\sharp}X\left(\mathbf{F}_q\right)$. d. Use the fact that $\operatorname{deg}\left(m+n F^{\prime}\right)>0$ for all $m, n \in \mathbf{Z}$ to show that $|a| \leqslant 2 \sqrt{q}$. This is Hasse's proof of the analogue of the Riemann hypothesis for elliptic curves (App. C, Ex. 5.6). e. Now assume $q=p$, and show that the Hasse invariant of $X$ is 0 if and only if $a \equiv 0(\bmod p)$. Conclude for $p \geqslant 5$ that $X$ has Hasse invariant 0 if and only if $N=p+1$. ### II.4.17. #to_work Let $X$ be the curve $y^2+y=x^3-x$ of $(4.23 .8)$. a. If $Q=(a, b)$ is a point on the curve, compute the coordinates of the point $P+Q$, where $P=(0,0)$, as a function of $a, b$. Use this formula to find the coordinates of $n P, n=1,2, \ldots, 10$.[^154] b. This equation defines a nonsingular curve over $\mathbf{F}_p$ for all $p \neq 37$. ### II.4.18. #to_work Let $X$ be the curve $y^2=x^3-7 x+10$. This curve has at least 26 points with integer coordinates. Find them (use a calculator), and verify that they are all contained in the subgroup (maybe equal to all of $X(\mathbf{Q})$?) generated by $P=(1,2)$ and $Q=(2,2)$. ### II.4.19. #to_work Let $X, P_0$ be an elliptic curve defined over $\mathbf{Q}$, represented as a curve in $\mathbf{P}^2$ defined by an equation with integer coefficients. Then $X$ can be considered as the fibre over the generic point of a scheme $\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu$ over Spec $\mathbf{Z}$. Let $T \subseteq \operatorname{Spec} \mathbf{Z}$ be the open subset consisting of all primes $p \neq 2$ such that the fibre $X_{(p)}$ of $\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu$ over $p$ is nonsingular. - For any $n$, show that $n_X: X \rightarrow X$ is defined over $T$, and is a flat morphism. - Show that the kernel of $n_X$ is also flat over $T$. - Conclude that for any $p \in T$, the natural map $X(\mathbf{Q}) \rightarrow X_{(p)}\left(\mathbf{F}_p\right)$ induced on the groups of rational points, maps the $n$-torsion points of $X(\mathbf{Q})$ injectively into the torsion subgroup of $X_{(p)}\left(\mathbf{F}_p\right)$, for any $(n, p)=1$. By this method one can show easily that the groups $X(\mathbf{Q})$ in (Ex. 4.17) and (Ex. 4.18) are torsion-free. ### II.4.20. #to_work Let $X$ be an elliptic curve over a field $k$ of characteristic $p>0$, and let $R=$ $\operatorname{End}\left(X, P_0\right)$ be its ring of endomorphisms. a. Let $X_p$ be the curve over $k$ defined by changing the $k$-structure of $X$ (2.4.1). Show that $j\left(X_p\right)=j(X)^{1 / p}$. Thus $X \cong X_p$ over $k$ if and only if $j \in \mathbf{F}_p$. b. Show that $p_X$ in $R$ factors into a product $\pi \widehat{\pi}$ of two elements of degree $p$ if and only if $X \cong X_p$. In this case, the Hasse invariant of $X$ is 0 if and only if $\pi$ and $\widehat{\pi}$ are associates in $R$ (i.e., differ by a unit). (Use (2.5).) c. If $\operatorname{Hasse}(X)=0$ show in any case $j \in \mathbf{F}_{p^2}$. d. For any $f \in R$, there is an induced map $f^*: H^1\left(\mathcal{O}_X\right) \rightarrow H^1\left(\mathcal{O}_X\right)$. This must be multiplication by an element $\lambda_f \in k$. So we obtain a ring homomorphism $\varphi: R \rightarrow k$ by sending $f$ to $\lambda_f$. Show that any $f \in R$ commutes with the (nonlinear) Frobenius morphism $F: X \rightarrow X$, and conclude that if Hasse $(X) \neq 0$, then the image of $\varphi$ is $\mathbf{F}_p$. Therefore, $R$ contains a prime ideal $\mathfrak{p}$ with $R / p \cong { \mathbf{F} }_p$. ### II.4.21. #to_work Let $O$ be the ring of integers in a quadratic number field $\mathbf{Q}(\sqrt{-d})$. Show that any subring $R \subseteq O, R \neq \mathbf{Z}$, is of the form $R=\mathbf{Z}+f \cdot O$, for a uniquely determined integer $f \geqslant 1$. This integer $f$ is called the **conductor** of the ring $R$. ### II.4.22 \*. #to_work If $X \rightarrow \mathbf{A}_{\mathbf{C}}^1$ is a family of elliptic curves having a section, show that the family is trivial.[^155] ## IV.5: The Canonical Embedding ### IV.5.1. #to_work Show that a hyperelliptic curve can never be a complete intersection in any projective space. Cf. (Ex. 3.3). ### IV.5.2. #to_work If $X$ is a curve of genus $\geqslant 2$ over a field of characteristic 0 , show that the group $\mathop{\mathrm{Aut}}X$ of automorphisms of $X$ is finite.[^156] ### IV.5.3. Moduli of Curves of Genus 4. #to_work The hyperelliptic curves of genus 4 form an irreducible family of dimension 7 . The nonhyperelliptic ones form an irreducible family of dimension 9. The subset of those having only one $g_3^1$ is an irreducible family of dimension 8.[^157] ### IV.5.4. #to_work Another way of distinguishing curves of genus $g$ is to ask, what is the least degree of a birational plane model with only nodes as singularities (3.11)? Let $X$ be nonhyperelliptic of genus 4 . Then: a. if $X$ has two $g_3^1$,s, it can be represented as a plane quintic with two nodes, and conversely; b. if $X$ has one $g_3^1$, then it can be represented as a plane quintic with a tacnode (I, Ex. 5.14d), but the least degree of a plane representation with only nodes is 6 . ### IV.5.5. Curves of Genus 5. #to_work Assume $X$ is not hyperelliptic. a. The curves of genus 5 whose canonical model in $\mathbf{P}^4$ is a complete intersection $F_2 . F_2 . F_2$ form a family of dimension 12 . b. $X$ has a $g_3^1$ if and only if it can be represented as a plane quintic with one node. These form an irreducible family of dimension 11.[^158] c. \* In that case, the conics through the node cut out the canonical system (not counting the fixed points at the node). Mapping $\mathbf{P}^2 \rightarrow \mathbf{P}^4$ by this linear system of conics, show that the canonical curve $X$ is contained in a cubic surface $V \subseteq \mathbf{P}^4$, with $V$ isomorphic to $\mathbf{P}^2$ with one point blown up (II, Ex. 7.7). Furthermore, $V$ is the union of all the trisecants of $X$ corresponding to the $g_3^1(5.5 .3)$, so $V$ is contained in the intersection of all the quadric hypersurfaces containing $X$. Thus $V$ and the $g_3^1$ are unique.[^159] ### IV.5.6. #to_work Show that a nonsingular plane curve of degree 5 has no $g_3^1$. Show that there are nonhyperelliptic curves of genus 6 which cannot be represented as a nonsingular plane quintic curve. ### IV.5.7. #to_work a. Any automorphism of a curve of genus 3 is induced by an automorphism of $\mathbf{P}^2$ via the canonical embedding. b. \* Assume char $k \neq 3$. If $X$ is the curve given by ` \begin{align*} x^3 y+y^3 z+z^3 x=0 \end{align*} `{=html} the group $\mathop{\mathrm{Aut}}X$ is the simple group of order 168 , whose order is the maximum $84(g-1)$ allowed by (Ex. 2.5).[^160] c. \* Most curves of genus 3 have no automorphisms except the identity.[^161][^162] ## IV.6: Classification of Curves in ${\mathbf{P}}^3$ {#iv.6-classification-of-curves-in-mathbfp3} ### IV.6.1. #to_work A rational curve of degree 4 in $\mathbf{P}^3$ is contained in a unique quadric surface $Q$, and $Q$ is necessarily nonsingular. ### IV.6.2. #to_work A rational curve of degree 5 in $\mathbf{P}^3$ is always contained in a cubic surface, but there are such curves which are not contained in any quadric surface. ### IV.6.3. #to_work A curve of degree 5 and genus 2 in $\mathbf{P}^3$ is contained in a unique quadric surface $Q$. Show that for any abstract curve $X$ of genus 2 , there exist embeddings of degree 5 in $\mathbf{P}^3$ for which $Q$ is nonsingular, and there exist other embeddings of degree 5 for which $Q$ is singular. ### IV.6.4. #to_work There is no curve of degree 9 and genus 11 in $\mathbf{P}^3$.[^163] ### IV.6.5. #to_work If $X$ is a complete intersection of surfaces of degrees $a, b$ in $\mathbf{P}^3$, then $X$ does not lie on any surface of degree $<\min (a, b)$. ### IV.6.6. #to_work Let $X$ be a projectively normal curve in $\mathbf{P}^3$, not contained in any plane. If $d=6$, then $g=3$ or 4 . If $d=7$, then $g=5$ or 6 . Cf. (II, Ex. 8.4) and (III, Ex. 5.6). ### IV.6.7. #to_work The line, the conic, the twisted cubic curve and the elliptic quartic curve in $\mathbf{P}^3$ have no multisecants. Every other curve in $\mathbf{P}^3$ has infinitely many multisecants.[^164] ### IV.6.8. #to_work A curve $X$ of genus $g$ has a nonspecial divisor $D$ of degree $d$ such that $|D|$ has no base points if and only if $d \geqslant g+1$. ### IV.6.9. #to_work \* Let $X$ be an irreducible nonsingular curve in $\mathbf{P}^3$. Then for each $m \gg>0$, there is a nonsingular surface $F$ of degree $m$ containing $X$.[^165] # V: Surfaces ## V.1: Geometry on a Surface ### V.1.1. #to_work Let $C, D$ be any two divisors on a surface $X$, and let the corresponding invertible sheaves be $\mathcal{L}, \mathcal{M}$. Show that ` \begin{align*} C . D=\chi\left(\mathcal{O}_X\right)-\chi\left(\mathcal{L}^{-1}\right)-\chi\left(\mathcal{M}^{-1}\right)+\chi\left(\mathcal{L}^{-1} \otimes \mathcal{M}^{-1}\right) . \end{align*} `{=html} ### V.1.2. #to_work Let $H$ be a very ample divisor on the surface $X$, corresponding to a projective embedding $X \subseteq \mathbf{P}^N$. If we write the Hilbert polynomial of $X$ (III, Ex. 5.2) as ` \begin{align*} P(z)=\frac{1}{2} a z^2+b z+c \end{align*} `{=html} show that $a=H^2, b=\frac{1}{2} H^2+1-\pi$, where $\pi$ is the genus of a nonsingular curve representing $H$, and $c=1+p_a$. Thus the degree of $X$ in $\mathbf{P}^N$, as defined in (I, §7), is just $H^2$. Show also that if $C$ is any curve in $X$, then the degree of $C$ in $\mathbf{P}^N$ is just $C . H$ ### V.1.3. #to_work Recall that the arithmetic genus of a projective scheme $D$ of dimension 1 is defined as ` \begin{align*} p_a=1-\chi\left(\mathcal{O}_D\right) .\end{align*} `{=html} See III, Ex. 5.3. a. If $D$ is an effective divisor on the surface $X$, use (1.6) to show that ` \begin{align*} 2 p_a-2= D.(D+K) .\end{align*} `{=html} b. $p_a(D)$ depends only on the linear equivalence class of $D$ on $X$. c. More generally, for any divisor $D$ on $X$, we define the virtual arithmetic genus (which is equal to the ordinary arithmetic genus if $D$ is effective) by the same formula: $2 p_a-2=D .(D+K)$. Show that for any two divisors $C, D$ we have ` \begin{align*} p_a(-D)=D^2-p_a(D)+2 \end{align*} `{=html} and ` \begin{align*} p_a(C+D)=p_a(C)+p_a(D)+C . D-1 . \end{align*} `{=html} ### V.1.4. #to_work a. If a surface $X$ of degree $d$ in $\mathbf{P}^3$ contains a straight line $C=\mathbf{P}^1$, show that $C^2=2-d$ b. Assume char $k=0$, and show for every $d \geqslant 1$, there exists a nonsingular surface $X$ of degree $d$ in $\mathbf{P}^3$ containing the line $x=y=0$. ### V.1.5. #to_work a. If $X$ is a surface of degree $d$ in $\mathbf{P}^3$, then ` \begin{align*} K^2=d(d-4)^2 .\end{align*} `{=html} b. If $X$ is a product of two nonsingular curves $C, C^{\prime}$, of genus $g, g^{\prime}$ respectively, then ` \begin{align*} K^2=8(g-1)\left(g^{\prime}-1\right) .\end{align*} `{=html} Cf. (II, Ex. 8.3). ### V.1.6. #to_work a. If $C$ is a curve of genus $g$, show that the diagonal $\Delta \subseteq C \times C$ has self-intersection $\Delta^2=2-2 g$. (Use the definition of $\Omega_{C / k}$ in (II, §8).) b. Let $l=C \times \mathrm{pt}$ and $m=\mathrm{pt} \times C$. If $g \geqslant 1$, show that $l, m$, and $\Delta$ are linearly independent in $\operatorname{Num}(C \times C)$. Thus $\operatorname{Num}(C \times C)$ has rank $\geqslant 3$, and in particular, ` \begin{align*} \operatorname{Pic}(C \times C) \neq p_1^* \operatorname{Pic} C \oplus p_2^* \operatorname{Pic}C .\end{align*} `{=html} Cf. (III, Ex. 12.6), (V, Ex. 4.10). ### V.1.7. Algebraic Equivalence of Divisors. #to_work Let $X$ be a surface. Recall that we have defined an algebraic family of effective divisors on $X$, parametrized by a nonsingular curve $T$, to be an effective Cartier divisor $D$ on $X \times T$, flat over $T$ (III, 9.8.5). In this case, for any two closed points $0,1 \in T$, we say the corresponding divisors $D_0, D_1$ on $X$ are prealgebraically equivalent. Two arbitrary divisors are prealgebraically equivalent if they are differences of prealgebraically equivalent effective divisors. Two divisors $D, D^{\prime}$ are algebraically equivalent if there is a finite sequence $D=D_0, D_1, \ldots, D_n=D^{\prime}$ with $D_i$ and $D_{i+1}$ prealgebraically equivalent for each $i$. a. Show that the divisors algebraically equivalent to 0 form a subgroup of Div $X$. b. Show that linearly equivalent divisors are algebraically equivalent.[^166] c. Show that algebraically equivalent divisors are numerically equivalent.[^167][^168] ### V.1.8. Cohomology Class of a Divisor. #to_work For any divisor $D$ on the surface $X$, we define its cohomology class $c(D) \in H^1\left(X, \Omega_X\right)$ by using the isomorphism Pic $X \cong$ $H^1\left(X, \mathcal{O}_X^*\right.$ ) of (III, Ex. 4.5) and the sheaf homomorphism $d \log : \mathcal{O}^* \rightarrow \Omega_X$ (III, Ex. 7.4c). Thus we obtain a group homomorphism $c: \operatorname{Pic} X \rightarrow H^1\left(X, \Omega_X\right)$. On the other hand, $H^1(X, \Omega)$ is dual to itself by Serre duality (III, 7.13), so we have a nondegenerate bilinear map ` \begin{align*} \langle\quad, \quad\rangle: H^1(X, \Omega) \times H^1(X, \Omega) \rightarrow k . \end{align*} `{=html} a. Prove that this is compatible with the intersection pairing, in the following sense: for any two divisors $D, E$ on $X$, we have ` \begin{align*} \langle c(D), c(E)\rangle=(D . E) \cdot 1 \end{align*} `{=html} in $k$.[^169] ```{=html} ``` b. If char $k=0$, use the fact that $H^1\left(X, \Omega_X\right)$ is a finite-dimensional vector space to show that $\operatorname{Num}X$ is a finitely generated free abelian group. ### V.1.9. #to_work a. If $H$ is an ample divisor on the surface $X$, and if $D$ is any divisor, show that ` \begin{align*} \left(D^2\right)\left(H^2\right) \leqslant(D . H)^2 \text {. } \end{align*} `{=html} b. Now let $X$ be a product of two curves $X=C \times C^{\prime}$. Let $l=C \times \mathrm{pt}$, and $m=$ pt $\times C^{\prime}$. For any divisor $D$ on $X$, let $a=D . l, b=D . m$. Then we say $D$ has type $(a, b)$. If $D$ has type $(a, b)$, with $a, b \in \mathbf{Z}$, show that ` \begin{align*} D^2 \leqslant 2 a b \text {, } \end{align*} `{=html} and equality holds if and only if $D \equiv b l+a m$.[^170] ### V.1.10. Weil's Proof of the Analogue of the Riemann Hypothesis for Curves. #to_work Let $C$ be a curve of genus $g$ defined over the finite field $\mathbf{F}_q$, and let $N$ be the number of points of $C$ rational over $\mathbf{F}_q$. Then $N=1-a+q$, with $|a| \leqslant 2 g \sqrt{q}$. To prove this, we consider $C$ as a curve over the algebraic closure $k$ of $\mathbf{F}_q$. Let $f: C \rightarrow C$ be the $k$-linear Frobenius morphism obtained by taking $q$ th powers, which makes sense since $C$ is defined over $\mathbf{F}_q$, so $X_q \cong X$ (See $V, 2.4 .1)$. Let $\Gamma \subseteq C \times C$ be the graph of $f$, and let $\Delta \subseteq C \times C$ be the diagonal. Show that $\Gamma^2=q(2-2 g)$, and $\Gamma . \Delta=N$. Then apply (Ex. 1.9) to $D=r \Gamma+s \Delta$ for all $r$ and $s$ to obtain the result.[^171] ### V.1.11. #to_work In this problem, we assume that $X$ is a surface for which $\operatorname{Num}X$ is finitely generated (i.e., any surface, if you accept the Néron-Severi theorem (Ex. 1.7)). a. If $H$ is an ample divisor on $X$, and $d \in \mathbf{Z}$, show that the set of effective divisors $D$ with $D . H=d$, modulo numerical equivalence, is a finite set.[^172] ```{=html} ``` b. Now let $C$ be a curve of genus $g \geqslant 2$, and use (a) to show that the group of automorphisms of $C$ is finite, as follows. Given an automorphism $\sigma$ of $C$, let $\Gamma \subseteq X=C \times C$ be its graph. First show that if $\Gamma \equiv \Delta$, then $\Gamma=\Delta$, using the fact that $\Delta^2<0$, since $g \geqslant 2$ (Ex. 1.6). Then use (a). Cf. (V, Ex. 2.5). ### V.1.12. #to_work If $D$ is an ample divisor on the surface $X$, and $D^{\prime} \equiv D$, then $D^{\prime}$ is also ample. Give an example to show, however, that if $D$ is very ample, $D^{\prime}$ need not be very ample. ## V.2: Ruled Surfaces ### V.2.1. #to_work If $X$ is a birationally ruled surface, show that the curve $C$, such that $X$ is birationally equivalent to $C \times \mathbf{P}^1$, is unique (up to isomorphism). ### V.2.2. #to_work Let $X$ be the ruled surface $\mathbf{P}(\mathcal{E})$ over a curve $C$. Show that $\mathcal{E}$ is decomposable if and only if there exist two sections $C^{\prime}, C^{\prime \prime}$ of $X$ such that $C^{\prime} \cap C^{\prime \prime}=\varnothing$. ### V.2.3. #to_work a. If $\mathcal{E}$ is a locally free sheaf of rank $r$ on a (nonsingular) curve $C$, then there is a sequence ` \begin{align*} 0=\mathcal{E}_0 \subseteq \mathcal{E}_1 \subseteq \ldots \subseteq \mathcal{E}_r=\mathcal{E} \end{align*} `{=html} of subsheaves such that $\mathcal{E}_i / \mathcal{E}_{i-1}$ is an invertible sheaf for each $i=1, \ldots, r$. We say that $\mathcal{E}$ is a successive extension of invertible sheaves.[^173] b. Show that this is false for varieties of dimension $\geqslant 2$. In particular, the sheaf of differentials $\Omega$ on $\mathbf{P}^2$ is not an extension of invertible sheaves. ### V.2.4. #to_work Let $C$ be a curve of genus $g$, and let $X$ be the ruled surface $C \times \mathbf{P}^1$. We consider the question, for what integers $s \in \mathbf{Z}$ does there exist a section $D$ of $X$ with $D^2=s$ ? First show that $s$ is always an even integer, say $s=2 r$. a. Show that $r=0$ and any $r \geqslant g+1$ are always possible. Cf. (V, Ex. 6.8). b. If $g=3$, show that $r=1$ is not possible, and just one of the two values $r=2,3$ is possible, depending on whether $C$ is hyperelliptic or not. ### V.2.5. Values of $e$. #to_work {#v.2.5.-values-of-e.-to_work} Let $C$ be a curve of genus $g \geqslant 1$. a. Show that for each $0 \leqslant e \leqslant 2 g-2$ there is a ruled surface $X$ over $C$ with invariant $e$, corresponding to an indecomposable $\mathcal{E}$. Cf. (2.12). b. Let $e<0$, let $D$ be any divisor of degree $d=-e$, and let $\xi \in H^1(\mathcal{L}(-D))$ be a nonzero element defining an extension ` \begin{align*} 0 \rightarrow \mathcal{O}_C \rightarrow \mathcal{E} \rightarrow \mathcal{L}(D) \rightarrow 0 . \end{align*} `{=html} Let $H \subseteq|D+K|$ be the sublinear system of codimension 1 defined by ker $\xi$, where $\xi$ is considered as a linear functional on $H^0(\mathcal{L}(D+K))$. For any effective divisor $E$ of degree $d-1$, let $L_E \subseteq|D+K|$ be the sublinear system $|D+K-E|+E$. Show that $\mathcal{E}$ is normalized if and only if for each $E$ as above, $L_E \nsubseteq H$. Cf. proof of $(2.15)$. c. Now show that if $-g \leqslant e<0$, there exists a ruled surface $X$ over $C$ with invariant $e$.[^174] d. For $g=2$, show that $e \geqslant-2$ is also necessary for the existence of $X$.[^175] ### V.2.6. #to_work Show that every locally free sheaf of finite rank on $\mathbf{P}^1$ is isomorphic to a direct sum of invertible sheaves.[^176] ### V.2.7. #to_work On the elliptic ruled surface $X$ of (2.11.6), show that the sections $C_0$ with $C_0^2=1$ form a one-dimensional algebraic family, parametrized by the points of the base curve $C$, and that no two are linearly equivalent. ### V.2.8. #to_work A locally free sheaf $\mathcal{E}$ on a curve $C$ is said to be **stable** if for every quotient locally free sheaf ` \begin{align*} \mathcal{E} \rightarrow \mathcal{F} \rightarrow 0, \qquad \mathcal{F} \neq \mathcal{E}, \mathcal{F} \neq 0 ,\end{align*} `{=html} we have ` \begin{align*} (\operatorname{deg} \mathcal{F}) / \operatorname{rank} \mathcal{F}>(\operatorname{deg} \mathcal{E}) / \operatorname{rank} \mathcal{E} .\end{align*} `{=html} Replacing $>$ by $\geqslant$ defines **semistable**. a. A decomposable $\mathcal{E}$ is never stable. b. If $\mathcal{E}$ has rank 2 and is normalized, then $\mathcal{E}$ is stable (respectively, semistable) if and only if $\deg \mathcal{E}>0$ (respectively, $\geqslant 0$ ). c. Show that the indecomposable locally free sheaves $\mathcal{E}$ of rank 2 that are not semistable are classified, up to isomorphism, by giving (1) an integer $0e \geqslant 0$, let $X$ be the rational scroll of degree $d=2 n-e$ in $\mathbf{P}^{d+1}$ given by (2.19). If $n \geqslant 2 e-2$, show that $X$ contains a nonsingular curve $Y$ of genus $g=d+2$ which is a canonical curve in this embedding. Conclude that for every $g \geqslant 4$, there exists a nonhyperelliptic curve of genus $g$ which has a $g_3^1$. Cf. (V, $\S 5$). ### V.2.11. #to_work Let $X$ be a ruled surface over the curve $C$, defined by a normalized bundle $\mathcal{E}$, and let $\mathfrak e$ be the divisor on $C$ for which $\mathcal{L}(\mathfrak{e}) \cong \bigwedge^2 \mathcal{E}$ (See 2.8 .1). Let $\mathfrak{b}$ be any divisor on $C$. a. If $|{\mathfrak{b}}|$ and $|{\mathfrak{b}}+ {\mathfrak{e}}|$ have no base points, and if ${\mathfrak{b}}$ is nonspecial, then there is a section $D \sim C_0+{\mathfrak{b}}f$, and $|D|$ has no base points. b. If ${\mathfrak{b}}$ and ${\mathfrak{b}}+ {\mathfrak{e}}$ are very ample on $C$, and for every point $P \in C$, we have $\mathfrak{b}-P$ and $\mathfrak{b}+\mathfrak{e}-P$ nonspecial, then $C_0+\mathfrak{b} f$ is very ample. ### V.2.12. #to_work Let $X$ be a ruled surface with invariant $e$ over an elliptic curve $C$, and let $\mathfrak{b}$ be a divisor on $C$. a. If $\deg \mathfrak{b} \geqslant e+2$, then there is a section $D \sim C_0+\mathfrak{b} f$ such that $|D|$ has no base points. b. The linear system $\left|C_0+\mathfrak{b} f\right|$ is very ample if and only if $\deg \mathfrak{b} \geqslant e+3$. Note. The case $e=-1$ will require special attention. ### V.2.13. #to_work For every $e \geqslant-1$ and $n \geqslant e+3$, there is an elliptic scroll of degree $d=2 n-e$ in $\mathbf{P}^{d-1}$. In particular, there is an elliptic scroll of degree 5 in $\mathbf{P}^4$. ### V.2.14. #to_work Let $X$ be a ruled surface over a curve $C$ of genus $g$, with invariant $e<0$, and assume that char $k=p>0$ and $g \geqslant 2$. a. If $Y \equiv a C_0+b f$ is an irreducible curve $\neq C_0, f$, then either - $a=1, b \geqslant 0$, or - $2 \leqslant a \leqslant p-1, b \geqslant \frac{1}{2} a e$, or - $a \geqslant p, b \geqslant \frac{1}{2} a e+1-g$. b. If $a>0$ and $b>a\left(\frac{1}{2} e+(1 / p)(g-1)\right)$, then any divisor $D \equiv a C_0+b f$ is ample. On the other hand, if $D$ is ample, then $a>0$ and $b>\frac{1}{2} a e$. ### V.2.15. Funny behavior in characteristic $p$. #to_work {#v.2.15.-funny-behavior-in-characteristic-p.-to_work} Let $C$ be the plane curve $x^3 y+y^3 z+z^3 x=0$ over a field $k$ of characteristic 3 (V, Ex. 2.4). a. Show that the action of the $k$-linear Frobenius morphism $f$ on $H^1\left(C, \mathcal{O}_C\right)$ is identically 0 (Cf. (V, 4.21)). b. Fix a point $P \in C$, and show that there is a nonzero $\xi \in H^1(\mathcal{L}(-P))$ such that $f^* \xi=0$ in $H^1(\mathcal{L}(-3 P))$. c. Now let $\mathcal{E}$ be defined by $\xi$ as an extension ` \begin{align*} 0 \rightarrow \mathcal{O}_C \rightarrow \mathcal{E} \rightarrow \mathcal{L}(P) \rightarrow 0, \end{align*} `{=html} and let $X$ be the corresponding ruled surface over $C$. Show that $X$ contains a nonsingular curve $Y \equiv 3 C_0-3 f$, such that $\pi: Y \rightarrow C$ is purely inseparable. Show that the divisor $D=2 C_0$ satisfies the hypotheses of (2.21.b), but is not ample. ### V.2.16. #to_work Let $C$ be a nonsingular affine curve. Show that two locally free sheaves $\mathcal{E}, \mathcal{E}^{\prime}$ of the same rank are isomorphic if and only if their classes in the Grothendieck group $K(X)$ (II, Ex. 6.10) and (II, Ex. 6.11) are the same. This is false for a projective curve. ### V.2.17 \*. #to_work a. Let $\varphi: \mathbf{P}_k^1 \rightarrow \mathbf{P}_k^3$ be the 3-uple embedding (I, Ex. 2.12). Let $\mathcal{I}$ be the sheaf of ideals of the twisted cubic curve $C$ which is the image of $\varphi$. Then $\mathcal{I} / \mathcal{I}^2$ is a locally free sheaf of rank 2 on $C$, so $\varphi^*\left(\mathcal{I} / \mathcal{I}^2\right)$ is a locally free sheaf of rank 2 on $\mathbf{P}^1$. By (2.14), therefore, ` \begin{align*} \varphi^*\left(\mathcal{I} / \mathcal{I}^2\right) \cong \mathcal{O}(l) \oplus \mathcal{O}(m) .\end{align*} `{=html} for some $l, m \in \mathbf{Z}$. Determine $l$ and $m$. b. Repeat part (a) for the embedding $\varphi: \mathbf{P}^1 \rightarrow \mathbf{P}^3$ given by $x_0=t^4, x_1=t^3 u$, $x_2=t u^3, x_3=u^4$, whose image is a nonsingular rational quartic curve.[^178] ## V.3: Monoidal Transformations ### V.3.1. #to_work Let $X$ be a nonsingular projective variety of any dimension, let $Y$ be a nonsingular subvariety, and let $\pi: \tilde{X} \rightarrow X$ be obtained by blowing up $Y$. Show that $p_a(\tilde{X})=$ $p_a(X)$ ### V.3.2. #to_work Let $C$ and $D$ be curves on a surface $X$, meeting at a point $P$. Let $\pi: \tilde{X} \rightarrow X$ be the monoidal transformation with center $P$. Show that ` \begin{align*} \tilde{C} \cdot \tilde{D}=C . D-\mu_P(C) \cdot \mu_P(D) .\end{align*} `{=html} Conclude that $C . D=\sum \mu_P(C) \cdot \mu_P(D)$, where the sum is taken over all intersection points of $C$ and $D$, including infinitely near intersection points. ### V.3.3. #to_work Let $\pi: \tilde{X} \rightarrow X$ be a monoidal transformation, and let $D$ be a very ample divisor on $X$. Show that $2 \pi^* D-E$ is ample on $\tilde{X}$.[^179] ### V.3.4. Multiplicity of a Local Ring. #to_work Let $A$ be a noetherian local ring with maximal ideal ${\mathfrak{m}}$. For any $l>0$, let $\psi(l)=$ length $\left(A / {\mathfrak{m}}^l\right)$. We call $\psi$ the **Hilbert-Samuel function of $A$**. a. Show that there is a polynomial $P_A(z) \in \mathbf{Q}[z]$ such that $P_A(l)=\psi(l)$ for all $l \gg 0$. This is the Hilbert-Samuel polynomial of $A$.[^180][^181] b. Show that $\operatorname{deg} P_A=\operatorname{dim} A$. c. Let $n=\operatorname{dim} A$. Then we define the multiplicity of $A$, denoted $\mu(A)$, to be $(n !)$. (leading coefficient of $P_A$ ). If $P$ is a point on a noetherian scheme $X$, we define the multiplicity of $P$ on $X, \mu_P(X)$, to be $\mu\left(\mathcal{O}_{P, X}\right)$. d. Show that for a point $P$ on a curve $C$ on a surface $X$, this definition of $\mu_P(C)$ coincides with the one in the text just before (3.5.2). e. If $Y$ is a variety of degree $d$ in $\mathbf{P}^n$, show that the vertex of the cone over $Y$ is a point of multiplicity $d$. ### V.3.5. #to_work Let $a_1, \ldots, a_r, r \geqslant 5$, be distinct elements of $k$, and let $C$ be the curve in $\mathbf{P}^2$ given by the (affine) equation $y^2=\prod_{i=1}^r\left(x-a_i\right)$. Show that the point $P$ at infinity on the $y$-axis is a singular point. Compute $\delta_P$ and $g(\tilde{Y})$, where $\tilde{Y}$ is the normalization of $Y$. Show in this way that one obtains hyperelliptic curves of every genus $g \geqslant 2$. ### V.3.6. #to_work Show that analytically isomorphic curve singularities (I, 5.6.1) are equivalent in the sense of (3.9.4), but not conversely. ### V.3.7. #to_work For each of the following singularities at $(0,0)$ in the plane, give an embedded resolution, compute $\delta_P$, and decide which ones are equivalent. a. $x^3+y^5=0$. b. $x^3+x^4+y^5=0$. c. $x^3+y^4+y^5=0$. d. $x^3+y^5+y^6=0$. e. $x^3+x y^3+y^5=0$. ### V.3.8. #to_work Show that the following two singularities have the same multiplicity, and the same configuration of infinitely near singular points with the same multiplicities, hence the same $\delta_P$, but are not equivalent. a. $x^4-x y^4=0$. b. $x^4-x^2 y^3-x^2 y^5+y^8=0$. ## V.4: The Cubic Surface in ${\mathbf{P}}^3$ {#v.4-the-cubic-surface-in-mathbfp3} ### V.4.1. #to_work The linear system of conics in $\mathbf{P}^2$ with two assigned base points $P_1$ and $P_2$ (4.1) determines a morphism $\psi$ of $X^{\prime}$ (which is $\mathbf{P}^2$ with $P_1$ and $P_2$ blown up) to a nonsingular quadric surface $Y$ in $\mathbf{P}^3$, and furthermore $X^{\prime}$ via $\psi$ is isomorphic to $Y$ with one point blown up. ### V.4.2. #to_work Let $\varphi$ be the quadratic transformation of $(4.2 .3)$, centered at $P_1, P_2, P_3$. If $C$ is an irreducible curve of degree $d$ in $\mathbf{P}^2$, with points of multiplicity $r_1, r_2, r_3$ at $P_1, P_2, P_3$, then the strict transform $C^{\prime}$ of $C$ by $\varphi$ has degree ` \begin{align*} d^{\prime}=2 d-r_1-r_2-r_3 \end{align*} `{=html} and has points of multiplicity - $d-r_2-r_3$ at $Q_1$, - $d-r_1-r_3$ at $Q_2$ and - $d-$ $r_1-r_2$ at $Q_3$. The curve $C$ may have arbitrary singularities.[^182] ### V.4.3. #to_work Let $C$ be an irreducible curve in $\mathbf{P}^2$. Then there exists a finite sequence of quadratic transformations, centered at suitable triples of points, so that the strict transform $C^{\prime}$ of $C$ has only ordinary singularities, i.e., multiple points with all distinct tangent directions (I, Ex. 5.14). Use (3.8). ### V.4.4. #to_work a. Use (4.5) to prove the following lemma on cubics: If $C$ is an irreducible plane cubic curve, if $L$ is a line meeting $C$ in points $P, Q, R$, and $L^{\prime}$ is a line meeting $C$ in points $P^{\prime}, Q^{\prime}, R^{\prime}$, let $P^{\prime \prime}$ be the third intersection of the line $P P^{\prime}$ with $C$, and define $Q^{\prime \prime}, R^{\prime \prime}$ similarly. Then $P^{\prime \prime}, Q^{\prime \prime}, R^{\prime \prime}$ are collinear. b. Let $P_0$ be an inflection point of $C$, and define the group operation on the set of regular points of $C$ by the geometric recipe \"let the line $P Q$ meet $C$ at $R$, and let $P_0 R$ meet $C$ at $T$, then $P+Q=T^{\prime \prime}$ as in (II, 6.10.2) and (II, 6.11.4). Use (a) to show that this operation is associative. ### V.4.5. #to_work Prove Pascal's theorem: if $A, B, C, A^{\prime}, B^{\prime}, C^{\prime}$ are any six points on a conic, then the points $P=A B^{\prime} \cdot A^{\prime} B, Q=A C^{\prime} \cdot A^{\prime} C$, and $R=B C^{\prime} \cdot B^{\prime} C$ are collinear (Fig. 22). ![](figures/2022-10-22_21-13-10.png) ### V.4.6. #to_work Generalize (4.5) as follows: given 13 points $P_1, \ldots, P_{13}$ in the plane, there are three additional determined points $P_{14}, P_{15}, P_{16}$, such that all quartic curves through $P_1, \ldots, P_{13}$ also pass through $P_{14}, P_{15}, P_{16}$. What hypotheses are necessary on $P_1, \ldots, P_{13}$ for this to be true? ### V.4.7. #to_work If $D$ is any divisor of degree $d$ on the cubic surface (4.7.3), show that ` \begin{align*} p_a(D) \leqslant \begin{cases}\frac{1}{6}(d-1)(d-2) & \text { if } d \equiv 1,2\, (\bmod 3) \\ \frac{1}{6}(d-1)(d-2)+\frac{2}{3} & \text { if } d \equiv 0\, (\bmod 3)\end{cases} \end{align*} `{=html} Show furthermore that for every $d>0$, this maximum is achieved by some irreducible nonsingular curve. ### V.4.8. \* #to_work Show that a divisor class $D$ on the cubic surface contains an irreducible curve $\iff$ if it contains an irreducible nonsingular curve $\iff$ it is either a. one of the 27 lines, or b. a conic (meaning a curve of degree 2) with $D^2=0$, or c. $D . L \geqslant 0$ for every line $L$, and $D^2>0$.[^183] ### V.4.9. #to_work If $C$ is an irreducible non-singular curve of degree $d$ on the cubic surface, and if the genus $g>0$, then ` \begin{align*} g \geqslant \begin{cases}\frac{1}{2}(d-6) & \text { if } d \text { is even, } d \geqslant 8 \\ \frac{1}{2}(d-5) & \text { if } d \text { is odd }, d \geqslant 13\end{cases} \end{align*} `{=html} and this minimum value of $g>0$ is achieved for each $d$ in the range given. ### V.4.10. #to_work A curious consequence of the implication (iv) $\Rightarrow$ (iii) of (4.11) is the following numerical fact: Given integers $a, b_1, \ldots, b_6$ such that $b_i>0$ for each $i, a-b_i-$ $b_j>0$ for each $i, j$ and $2 a-\sum_{i \neq j} b_i>0$ for each $j$, we must necessarily have $a^2-\sum b_i^2>0$. Prove this directly (for $a, b_1, \ldots, b_6 \in \mathbf{R}$ ) using methods of freshman calculus. ### V.4.11. The Weyl Groups. #to_work Given any diagram consisting of points and line segments joining some of them, we define an abstract group, given by generators and relations, as follows: - Each point represents a generator $x_i$. The relations are - $x_i^2=1$ for each $i$; - $\left(x_i x_j\right)^2=1$ if $i$ and $j$ are not joined by a line segment, and - $\left(x_i x_j\right)^3=1$ if $i$ and $j$ are joined by a line segment. a. The Weyl group $\mathbf{A}_n$ is defined using the following diagram of $n-1$ points, each joined to the next: ![](figures/2022-10-22_21-16-28.png) Show that it is isomorphic to the symmetric group $\Sigma_n$ as follows: - Map the generators of $\mathbf{A}_n$ to the elements $(12),(23), .., (n-1,n)$ of $\Sigma_n$, to get a surjective homomorphism $\mathbf{A}_n \rightarrow \Sigma_n$. - Then estimate the number of elements of $\mathbf{A}_n$ to show in fact it is an isomorphism. b. The Weyl group $\mathbf{E}_6$ is defined using the diagram ![](figures/2022-10-22_21-17-05.png) Call the generators $x_1, \ldots, x_5$ and $y$. Show that one obtains a surjective homomorphism $\mathbf{E}_6 \rightarrow G$, the group of automorphisms of the configuration of 27 lines $(4.10 .1)$, by sending $x_1, \ldots, x_5$ to the permutations $(12),(23), \ldots,(56)$ of the $E_i$, respectively, and $y$ to the element associated with the quadratic transformation based at $P_1, P_2, P_3$. c. \* Estimate the number of elements in $\mathbf{E}_6$, and thus conclude that $\mathbf{E}_6 \cong G$.[^184] ### V.4.12. #to_work Use (4.11) to show that if $D$ is any ample divisor on the cubic surface $X$, then $H^1\left(X, \mathcal{O}_X(-D)\right)=0$. This is Kodaira's vanishing theorem for the cubic surface (III, 7.15). ### V.4.13. #to_work Let $X$ be the Del Pezzo surface of degree 4 in $\mathbf{P}^4$ obtained by blowing up 5.points of $\mathbf{P}^2(4.7)$ a. Show that $X$ contains 16 lines. b. Show that $X$ is a complete intersection of two quadric hypersurfaces in $\mathbf{P}^4$ (the converse follows from (4.7.1)). ### V.4.14. #to_work Using the method of (4.13.1), verify that there are nonsingular curves in $\mathbf{P}^3$ with $d=8, g=6,7 ; d=9, g=7,8,9 ; d=10, g=8,9,10,11$. Combining with (IV, §6), this completes the determination of all posible $g$ for curves of degree $d \leqslant 10$ in $\mathbf{P}^3$. ### V.4.15. #to_work Let $P_1, \ldots, P_r$ be a finite set of (ordinary) points of $\mathbf{P}^2$, no 3 collinear. We define an **admissible transformation** to be a quadratic transformation (4.2.3) centered at some three of the $P_i$ (call them $P_1, P_2, P_3$ ). This gives a new $\mathbf{P}^2$, and a new set of $r$ points, namely $Q_1, Q_2, Q_3$, and the images of $P_4, \ldots, P_r$. We say that $P_1, \ldots, P_r$ are **in general position** if no three are collinear, and furthermore after any finite sequence of admissible transformations, the new set of $r$ points also has no three collinear. a. A set of 6 points is in general position if and only if no three are collinear and not all six lie on a conic. b. If $P_1, \ldots, P_r$ are in general position, then the $r$ points obtained by any finite sequence of admissible transformations are also in general position. c. Assume the ground field $k$ is uncountable. Then given $P_1, \ldots, P_r$ in general position, there is a dense subset $V \subseteq \mathbf{P}^2$ such that for any $P_{r+1} \in V, P_1, \ldots, P_{r+1}$ will be in general position.[^185] d. Now take $P_1, \ldots, P_r \in \mathbf{P}^2$ in general position, and let $X$ be the surface obtained by blowing up $P_1, \ldots, P_r$. If $r=7$, show that $X$ has exactly 56 irreducible nonsingular curves $C$ with $g=0, C^2=-1$, and that these are the only irreducible curves with negative self-intersection. Ditto for $r=8$, the number being 240 . e. \* For $r=9$, show that the surface $X$ defined in (d) has infinitely many irreducible nonsingular curves $C$ with $g=0$ and $C^2=-1$.[^186] ### V.4.16. #to_work For the Fermat cubic surface $x_0^3+x_1^3+x_2^3+x_3^3=0$, find the equations of the 27 lines explicitly, and verify their incidence relations. What is the group of automorphisms of this surface? ## V.5: Birational Transformations ### V.5.1. #to_work Let $f$ be a rational function on the surface $X$. Show that it is possible to \"resolve the singularities of $f^{\prime \prime}$ in the following sense: there is a birational morphism $g$ : $X^{\prime} \rightarrow X$ so that $f$ induces a morphism of $X^{\prime}$ to $\mathbf{P}^1$.[^187] ### V.5.2. #to_work Let $Y \cong \mathbf{P}^1$ be a curve in a surface $X$, with $Y^2<0$. Show that $Y$ is contractible (5.7.2) to a point on a projective variety $X_0$ (in general singular). ### V.5.3. #to_work If $\pi: \tilde{X} \rightarrow X$ is a monoidal transformation with center $P$, show that $H^1\left(\tilde{X}, \Omega_{\tilde{X}}\right) \cong$ $H^1\left(X, \Omega_X\right) \oplus k$. This gives another proof of (5.8).[^188] ### V.5.4. #to_work Let $f: X \rightarrow X^{\prime}$ be a birational morphism of nonsingular surfaces. a. If $Y \subseteq X$ is an irreducible curve such that $f(Y)$ is a point, then $Y \cong \mathbf{P}^1$ and $Y^2<0$ b. Let $P^{\prime} \in X^{\prime}$ be a fundamental point of $f^{-1}$, and let $Y_1, \ldots, Y_r$ be the irreducible components of $f^{-1}\left(P^{\prime}\right)$. Show that the matrix $\left|Y_i . Y_j\right|$ is negative definite. ### V.5.5. #to_work Let $C$ be a curve, and let $\pi: X \rightarrow C$ and $\pi^{\prime}: X^{\prime} \rightarrow C$ be two geometrically ruled surfaces over $C$. Show that there is a finite sequence of elementary transformations (5.7.1) which transform $X$ into $X^{\prime}$.[^189] ### V.5.6. #to_work Let $X$ be a surface with function field $K$. Show that every valuation $\operatorname{ring} R$ of $K / k$ is one of the three kinds described in (II, Ex. 4.12).[^190] ### V.5.7. #to_work Let $Y$ be an irreducible curve on a surface $X$, and suppose there is a morphism $f: X \rightarrow X_0$ to a projective variety $X_0$ of dimension 2 , such that $f(Y)$ is a point $P$ and $f^{-1}(P)=Y$. Then show that $Y^2<0$.[^191] ### V.5.8. A surface singularity. #to_work Let $k$ be an algebraically closed field, and let $X$ be the surface in $\mathbf{A}_k^3$ defined by the equation $x^2+y^3+z^5=0$. It has an isolated singularity at the origin $P=(0,0,0)$. a. Show that the affine ring $A=k[x, y, z] /\left(x^2+y^3+z^5\right)$ of $X$ is a unique factorization domain, as follows. Let $t=z^{-1} ; u=t^3 x$, and $v=t^2 y$. Show that $z$ is irreducible in $A ; t \in k[u, v]$, and $A\left[z^{-1}\right]=k\left[u, v, t^{-1}\right]$. Conclude that $A$ is a UFD. b. Show that the singularity at $P$ can be resolved by eight successive blowings-up. If $\tilde{X}$ is the resulting nonsingular surface, then the inverse image of $P$ is a union of eight projective lines, which intersect each other according to the Dynkin $\operatorname{diagram} \mathbf{E}_8$ : ![](figures/2022-10-22_21-57-34.png) ## V.6: Classification of Surfaces ### V.6.1. #to_work Let $X$ be a surface in $\mathbf{P}^n, n \geqslant 3$, defined as the complete intersection of hypersurfaces of degrees $d_1, \ldots, d_{n-2}$, with each $d_i \geqslant 2$. Show that for all but finitely many choices of $\left(n, d_1, \ldots, d_{n-2}\right)$, the surface $X$ is of general type. List the exceptional cases, and where they fit into the classification picture. ### V.6.2. #to_work Prove the following theorem of Chern and Griffiths. Let $X$ be a nonsingular surface of degree $d$ in $\mathbf{P}_{\mathbf{C}}^{n+1}$, which is not contained in any hyperplane. If $d<2 n$, then $p_g(X)=0$. If $d=2 n$, then either $p_g(X)=0$, or $p_g(X)=1$ and $X$ is a K3 surface.[^192] [^1]: Use (b) above. [^2]: Hint: Interpret the problem in terms of the affine $(n+1)$-space whose affine coordinate ring is $S$, and use the usual Nullstellensatz, (1.3A). [^3]: Note: Since $S_{+}$does not occur in this correspondence, it is sometimes called the **irrelevant maximal ideal of $S$**. [^4]: Hint: Let $\varphi_i: U_i \rightarrow \mathbf{A}^n$ be the homeomorphism of (2.2), let $Y_t$ be the affine variety $\varphi_t\left(Y \cap U_i\right)$, and let $A\left(Y_i\right)$ be its affine coordinate ring. Show that $A\left(Y_t\right)$ can be identified with the subring of elements of degree 0 of the localized ring $S(Y)_{x_i}$. Then show that $S(Y)_{x_i} \cong A\qty{Y_i}[x_i, x_i^{-1}]$. Now use (1.7), (1.8A), and (Ex 1.10), and look at transcendence degrees. Conclude also that $\operatorname{dim} Y=\operatorname{dim} Y_i$ whenever $Y_i$ is nonempty. [^5]: Hint: Use (Ex. 2.6) to reduce to (1.10). [^6]: Think of $\mathbf{A}^{n+1}$ as a vector space over $k$, and work with its subspaces. [^7]: One inclusion is easy. The other will require some calculation. [^8]: Hint: Let the homogeneous coordinates of $\mathbf{P}^N$ be $\left\{{z_{ij} {~\mathrel{\Big\vert}~}0\leq i, j\leq r}\right\}$ and let $a$ be the kernel of the homomorphism $k[\left\{{z_{ij}}\right\}] \to k[x_0,\cdots, x_r, y_0, \cdots, y_s]$, which sends $z_{ij}$ to $x_i y_j$. Then show that $\operatorname{im}\psi = Z(a)$. [^9]: Use (b) above. [^10]: Use (c) above. [^11]: Use induction on $\operatorname{dim} X$. [^12]: Note: We will give another proof of this result using sheaves of ideals later (V.10). [^13]: This gives an alternate proof of $(3.4 \mathrm{a})$ in the case $Y={\mathbb{P}}^n$. [^14]: Hint: Reduce 10 the affine case and use (3.2c) [^15]: Hint: Suppose that $X \times Y$ is a union of two closed subsets $Z_1 \cup Z_2$. Let $X_1=$ $\left\{x \in X \mathrel{\Big|}x \times Y \subseteq Z_1\right\}, i=1,2$. Show that $X=X_1 \cup X_2$ and $X_1, X_2$ are closed. Then $X=X_1$ or $X_2$ so $X \times Y=Z_1$ or $Z_2$. [^16]: Use (b) above. [^17]: Use (b) above. [^18]: Use (c) above. [^19]: Use induction on $\operatorname{dim} X$. [^20]: Note: We will give another proof of this result using sheaves of ideals later (V.10). [^21]: Answer: $d^n$. [^22]: Answer: ${r+s\choose s}$. [^23]: Hint: Show that the set of lines in $\left(\mathbf{P}^2\right)^*$ which are either tangent to $Y$ or pass through a singular point of $Y$ is contained in a proper closed subset. [^24]: Hint: First, use (7.7) and treat the case $\operatorname{dim} Y=1$. Then do the general case by cutting with a hyperplane and using induction. [^25]: Hint: Use induction on $\operatorname{dim} Y$. [^26]: This exercise is included only to establish the connection between our definition of a sheaf and another definition often found in the literature. See for example Godement (1. Ch. II, 1.2). [^27]: This is an analogue of what is called the *first Cousin problem* in several complex variables. See Gunning and Rossi (1, p. 248) [^28]: Hint: Use (Ex. 2.4) and (Ex. 2.16d) above. [^29]: Hint: Consider $X^{\prime}=\operatorname{Spec}(A / \operatorname{ker} \varphi)$ and use (b) and (c). [^30]: Hint: First show that the function field of $X$ is a finite field extension of the function field of $Y$. [^31]: Hints: First show that $Y$ can be covered by a finite number of open affine subsets of the form $D\left(f_i\right) \cap Y$, with $f_i \in A$. By adding some more $f_i$ with $D\left(f_i\right) \cap Y=\varnothing$, if necessary, show that we may assume that the $D\left(f_i\right)$ cover $X$. Next show that $f_1, \ldots, f_r$ generate the unit ideal of $A$. Then use (Ex. 2.17b) to show that $Y$ is affine, and (Ex. 2.18d) to show that $Y$ comes from an ideal $\mathfrak{a} \subseteq A$.\] [^32]: Note: We will give another proof of this result using sheaves of ideals later (V.10). [^33]: We will see later (5.16) that every closed subscheme of $X$ comes from a homogeneous ideal $I$ of $S$ (at least in the case where $S$ is a polynomial ring over $S_0$ ). [^34]: By abuse of notation, we write $X \underset{\scriptscriptstyle {k} }{\times} \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu$ to denote $X \underset{\scriptscriptstyle {\operatorname{Spec}k} }{\times} \operatorname{Spec}\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu$. [^35]: The real importance of the notion of constructible subsets derives from the following theorem of Chevalley-see Cartan and Chevalley (1, exposé 7) and see also Matsumura (2, Ch. 2, 6). [^36]: Hint: Prove this algebraic result by induction on the number of generators of $B$ over $A$. For the case of one generator, prove the result directly. In the application, take $b=1$. [^37]: Hint: Let $Y^{\prime}=\{y\}^{-}$, and use (a) and (Ex. 3.20b). [^38]: Hint: First reduce to the case where $X$ and $Y$ are affine, say $X=\operatorname{Spec} A$ and $Y=\operatorname{Spec} B$. Then $A$ is a finitely generated $B$-algebra. Take $t_1, \ldots, t_e \in A$ which form a transcendence base of $K(X)$ over $K(Y)$, and let $X_1=\operatorname{Spec} B\left[t_1, \ldots, t_e\right]$. Then $X_1$ is isomorphic to affine $e$-space over $Y$, and the morphism $X \rightarrow X_1$ is generically finite. Now use (Ex. 3.7) above. [^39]: Use (b) above. [^40]: Use (c) above. [^41]: Use induction on $\operatorname{dim} X$. [^42]: Hint: Consider the map $h: X \rightarrow Y \times{ }_s Y$ obtained from $f$ and $g$. [^43]: Hint: Factor $f$ into the graph morphism $\Gamma_f: X \rightarrow X \times_s Y$ followed by the second projection $p_2$, and show that $\Gamma_f$ is a closed immersion. [^44]: Note: if $X$ is a variety over $k$, the criterion of (b) is sometimes taken as the definition of a complete variety. [^45]: Hint: While parts (a) and (b) follow quite easily from (4.3) and (4.7), their converses will require some comparison of valuations in different fields. [^46]: Hint: Let $a \in \Gamma\left(X, \mathcal{O}_X\right)$, with $a \notin k$. Show that there is a valuation ring $R$ of $K / k$ with $a^{-1} \in \mathfrak{m}_R$. Then use (b) to get a contradiction. [^47]: Hint: Use (4.11A). [^48]: Hint: For (e), consider the graph morphism $\Gamma_f: X \rightarrow X \times{ }_Z Y$ and note that it is obtained by base extension from the diagonal morphism $\Delta: Y \rightarrow Y \times_Z Y$. [^49]: Hint: Use the Segre embedding defined in (I, Ex. 2.14) and show that it gives a closed immersion $\left.\mathbf{P}^r \times \mathbf{P}^s \rightarrow \mathbf{P}^{r+r+s}.\right.$. [^50]: See Ex. 3.11d. [^51]: See Nagata 7, p. 115. [^52]: Note. We will see later (V, Ex. 5.6) that in fact the $R_0$ of $(3)$ is already a valuation ring itself, so $R_0=R$. Furthermore, every valuation ring of $K / k$ (except for $K$ itself) is one of the three kinds just described. [^53]: Hint: Use (Ex. 1.20) and (5.8) to show a priori that $\mathcal{H}_Z^0(\mathcal{F})$ is quasi-coherent. Then show that $\Gamma_{\mathrm{a}}(M) \cong \Gamma_Z(\mathcal{F})$. [^54]: This justifies the terminology invertible: it means that $\mathcal{F}$ is an invertible element of the monoid of coherent sheaves under the operation $\otimes$. [^55]: Hint: Use the methods of the proof of (5.19). [^56]: Hint: Use a Segre embedding. [^57]: Hint: First show that $X$ is integral. Then regard $S^{\prime}$ as the global sections of the sheaf of rings $\mathcal{S}=\bigoplus_{n \geq 0} {\mathcal{O}}_X(n)$ on $X$, and show that $\mathcal{S}$ is a sheaf of integrally closed domains. [^58]: Hint: Let $i: U \rightarrow X$ be the inclusion map. Show that $i_* \mathcal{F}$ is quasi-coherent, then use (a). [^59]: Hint: Use the same method, but replace $i_* \mathcal{F}$ by $\rho^{-1}\left(i_* \mathcal{F}\right)$, where $\rho$ is the natural map $\mathcal{G} \rightarrow i_*\left(\left.\mathcal{G}\right|_U\right)$. [^60]: Hint: Cover $X$ with open affines, and extend over one of them at a time. [^61]: Hint: If $s$ is a section of $\mathcal{F}$ over an open set $U$, apply (d) to the subsheaf of $\left.\mathcal{F}\right|_U$ generated by $s$. [^62]: Hint: Reduce to the case $Y$ affine, and use (Ex. 2.17). [^63]: Hint: Construct $X$ by glueing together the schemes $\operatorname{Spec} \mathcal{A}(V)$, for $V$ open affine in $Y$. [^64]: Hint: For any quasi-coherent $\mathcal{A}$-module $\mathcal{M}$, construct a quasi-coherent ${\mathcal{O}}_X$-module $\tilde{\mathcal{M}}$, and show that the functors $f_*$ and $\sim$ are inverse to each other. [^65]: Hint: It is enough to define the module structure locally, so we can assume $Y=\operatorname{Spec} A$ is affine, and $X=\mathbf{A}_Y^n$. Then a section $s: Y \rightarrow X$ comes from an $A$-algebra homomorphism $\theta: A\left[x_1, \ldots, x_n\right] \rightarrow$ $A$, which in turn determines an ordered $n$-tuple $\left\langle\theta\left(x_1\right), \ldots, \theta\left(x_n\right)\right\rangle$ of elements of $A$. Use this correspondence between sections $s$ and ordered $n$-tuples of elements of $A$ to define the module structure.\] [^66]: This gives another proof of (6.10), since any complete nonsingular curve is projective. [^67]: Hint: The quotient field $K$ of $A$ is just $k\left(x_1, \ldots, x_n\right)[z] /\left(z^2-f\right)$. It is a Galois extension of $k\left(x_1, \ldots, x_n\right)$ with Galois group $\mathbf{Z} / 2 \mathbf{Z}$ generated by $z \mapsto-z$. If $\alpha=g+h z \in K$, where $g, h \in k\left(x_1, \ldots, x_n\right)$, then the minimal polynomial of $\alpha$ is ` \begin{align*} X^2-2 g X+\left(g^2-h^2 f\right) .\end{align*} `{=html} Now show that $\alpha$ is integral over $k\left[x_1, \ldots, x_n\right]$ if and only if $g, h \in k\left[x_1, \ldots, x_n\right]$. [^68]: cf. (I, Ex. 5.12). [^69]: An inflection point of a plane curve is a nonsingular point $P$ of the curve, whose tangent line (I, Ex. 7.3) has intersection multiplicity $\geqslant 3$ with the curve at $P$. [^70]: Hint: Represent $\operatorname{Pic}X$ and Pic $\tilde{X}$ as the groups of Cartier divisors modulo principal divisors, and use the exact sequence of sheaves on $X$` \begin{align*} 0 \rightarrow \pi_* {\mathcal{O}}_{\tilde{X}}^* / {\mathcal{O}}_X^* \rightarrow \mathcal{K}^* / {\mathcal{O}}_X^* \rightarrow \mathcal{K}^* / \pi_* {\mathcal{O}}_{\tilde{X}}^* \rightarrow 0 \end{align*} `{=html} [^71]: Hint: For exactness in the middle, show that if ${\mathcal{F}}$ is a coherent sheaf on $X$, whose support is contained in $Y$, then there is a finite filtration ${\mathcal{F}}={\mathcal{F}}_0 \supseteq$ ${\mathcal{F}}_1 \supseteq \ldots \supseteq {\mathcal{F}}_n=0$, such that each ${\mathcal{F}}_i / {\mathcal{F}}_{i+1}$ is an ${\mathcal{O}}_Y$-module. To show surjectivity on the right, use (Ex. 5.15).\ For further information about $K(X)$, and its applications to the generalized Riemann-Roch theorem, see Borel-Serre $[1]$, Manin $[1]$, and Appendix A. [^72]: Hint: Reduce to a question of modules over a local ring by looking at the stalks. [^73]: See (IV, 1.3.2), $(\mathrm{V}, 1.6)$, and Appendix A. [^74]: I, Ex. 2.13. [^75]: Hint: Let $U \subseteq X$ be an open set such that $\pi^{-1}(U) \cong U \times \mathbf{P}^n$, and let $\mathcal{L}_0$ be the invertible sheaf $\mathcal{C}(1)$ on $U \times \mathbf{P}^n$. Show that $\mathcal{L}_0$ extends to an invertible sheaf $\mathcal{L}$ on $P$. Then show that $\pi_* \mathcal{L}=\mathcal{E}$ is a locally free sheaf on $X$ and that $P \cong \mathbf{P}(\mathcal{E})$.\] Can you weaken the hypothesis \" $X$ regular\"? [^76]: Hint: If $A$ is a domain and if $*$ denotes the group of units, then $(A[u])^* \cong A^*$ and $\left(A\left[u, u^{-1}\right]\right)^* \cong A^* \times \mathbf{Z}$. [^77]: Hint: Use (7.10) and (Ex. 5.12) [^78]: Hint: In copying the proof of (8.7), first pass to $B / \mathrm{m}^2$, which is a complete local ring, and then use (8.25A) to choose a field of representatives for $B / \mathrm{m}^2$. [^79]: Hint: Use a method similar to the proof of Bertini's theorem (8.18).\] [^80]: Hint: Use the fact that the uniqueness theorem holds in $S$ (Matsumura $[2, p. 107]$). [^81]: Hint: Apply (8.23) to the affine cone over $Y$. [^82]: Hint: By (a) we can write in any case ` \begin{align*} \omega_{\tilde{X}} \cong f^* \mathcal{M} \otimes \mathcal{L}\left(q Y^{\prime}\right) \end{align*} `{=html} for some invertible sheaf $\mathcal{M}$ on $X$, and some integer q. By restricting to $\tilde{X}-Y^{\prime} \cong X-Y$, show that $\mathcal{M} \cong \omega_X$. To determine $q$, proceed as follows. - First show that $\omega_{Y^{\prime}} \cong f^* \omega_X \otimes \mathcal{O}_{Y^{\prime}}(-q-1)$. - Then take a closed point $y \in Y$ and let $Z$ be the fibre of $Y^{\prime}$ over $y$. - Then show that $\omega_Z \cong$ $\mathcal{O}_Z(-q-1)$. But since $Z \cong \mathbf{P}^{r-1}$, we have $\omega_Z \cong \mathcal{O}_Z(-r)$, so $q=r-1$. [^83]: Use 8.21Ae. [^84]: Actually, the same result holds without the hypothesis $Y$ nonsingular, but the proof is more difficult-see Hartshorne $[3,(7.3)]$. [^85]: Be careful, because the $\varphi_n$ may not be compatible with the maps in the two inverse systems $\left(\mathrm{F} / \mathrm{J}^n \mathrm{F}\right)$ and $\left(\mathcal{O}_{Y_n}\right)$! [^86]: Again be careful, because even though each $\mathcal{L}_n$ is locally free of rank 1, the open sets needed to make them free might get smaller and smaller with $n$. [^87]: See (III, Ex. 11.5-11.7) for further examples and applications. [^88]: Hints: First show that a sheaf $\mathcal{I}$ is injective if and only if for every open set $U \subseteq X$, and for every subsheaf $\mathcal{R} \subseteq \mathbf{Z}_U$, and for every map $f: \mathcal{R} \rightarrow \mathcal{I}$, there exists an extension of $f$ to a map of $\mathbf{Z}_U \rightarrow \mathcal{I}$. Secondly, show that any such sheaf $\mathcal{R}$ is finitely generated, so any map $\mathcal{R} \rightarrow \colim\, \mathcal{I}_\alpha$ factors through one of the $\mathcal{I}_\alpha$. [^89]: Hint: Use (3.7), and for any coherent sheaf $\mathcal{F}$ on $X$, consider the filtration $\mathcal{F} \supseteq \mathcal{N} \cdot \mathcal{F} \supseteq \mathcal{N}^2 \cdot \mathcal{F} \supseteq \ldots$, where $\mathcal{N}$ is the sheaf of nilpotent elements on $X$. [^90]: Hint: When $M$ is finitely generated, both conditions are equivalent to saying that $\mathfrak{a}$ is not contained in any associated prime of $M$. [^91]: Hints: The method of proof of (2.4) will not work, because $\mathcal{O}_U$ is not quasi-coherent on $X$ in general. Instead, use (II, Ex. 5.15) to show that if $\mathcal{I} \in \mathbb{Q} \mathrm{co}(X)$ is injective, and if $U \subseteq X$ is an open subset, then $\left.\mathcal{I}\right|_U$ is an injective object of $\operatorname{Qco}(U)$. Then cover $X$ with open affines $\cdots$. [^92]: Hint: Use (II, 5.8). [^93]: Hint: Apply $\mathop{\mathcal{H}\! \mathit{om}}(\cdot, \mathcal{F})$ to $\alpha$ and use (II, Ex. 5.17e). [^94]: Using (3.5), this provides another proof that $U$ is not affine-cf. (I, Ex. 3.6). [^95]: Hint: Embed $\mathcal{F}$ in a flasque sheaf $\mathcal{G}$, and let $\mathcal{R}=\mathcal{G}/\mathcal{F}$, so that we have an exact sequence ` \begin{align*} 0 \rightarrow \mathcal{F} \rightarrow \mathcal{G} \rightarrow \mathcal{R} \rightarrow 0 . \end{align*} `{=html} Define a complex $D^{\cdot}(\mathfrak{U})$ by ` \begin{align*} 0 \rightarrow C^{\cdot}(\mathfrak{U}, \mathcal{F}) \rightarrow C^{\cdot}(\mathfrak{U}, \mathcal{G}) \rightarrow D^{\prime}(\mathfrak{U}) \rightarrow 0 . \end{align*} `{=html} Then use the exact cohomology sequence of this sequence of complexes, and the natural map of complexes ` \begin{align*} D^*(\mathfrak{U}) \rightarrow C^*(\mathfrak{U}, \mathcal{R}), \end{align*} `{=html} and see what happens under refinement. [^96]: Hint: For any invertible sheaf $\mathcal{L}$ on $X$, cover $X$ by open sets $U_i$ on which $\mathcal{L}$ is free, and fix isomorphisms $\phi_i: {\mathcal{O}}_{U_i} { \, \xrightarrow{\sim}\, }{ \left.{{{\mathcal{L}}}} \right|_{{U_i}} }$. Then on $U_i \cap U_j$, we get an isomorphism $\varphi_i^{-1} \circ \varphi_j$ of $\mathcal{O}_{U_i \cap U_j}$ with itself. These isomorphisms give an element of $\check{H}^1\left(\mathfrak{U}, \mathcal{O}_X^*\right)$. Now use (Ex. 4.4). [^97]: Hints: Use an affine analogue of (Ex. 4.8e). Then show that $H^2\left(X-Y, \mathcal{O}_X\right) \neq 0$, by using (Ex. 2.3) and (Ex. 2.4). If $P=Y_1 \cap Y_2$, imitate (Ex. 4.3) to show $H^3\left(X-P, \mathcal{O}_X\right) \neq 0$. [^98]: Hint: Use (II, Ex. 8.6) and (4.5). [^99]: Hints: Use induction on dim Supp $\mathcal{F}$, general properties of numerical polynomials (I, 7.3), and suitable exact sequences ` \begin{align*} 0 \rightarrow \mathcal{R} \rightarrow \mathcal{F}(-1) \rightarrow \mathcal{F} \rightarrow {\mathcal{L}}\rightarrow 0 \end{align*} `{=html} [^100]: Hint: Use (I, 3.4). [^101]: This gives another proof of (II, 8.20.3) where we used the geometric genus. [^102]: Hint: Show that (1) $\Rightarrow$ (2). Then prove (1) and (2) simultaneously, by induction on $r$, using (II, Ex. 6.10c). [^103]: This gives a generalization and another proof of (II, Ex. 8.4c), where we assumed $Y$ was normal. [^104]: Hint: Use exact sequences and induction on the codimension, starting from the case $Y=X$ which is (5.1). [^105]: Hint: Calculate Hilbert polynomials of suitable sheaves, and again use the special case $(q, 0)$ which is a disjoint union of $q$ copies of $\mathbf{P}^1$. See $(\mathrm{V}, 1.5 .2)$ for another method. [^106]: Hints: Use (5.3) and compare (Ex. 3.1, Ex. 3.2, Ex. 4.1, Ex. 4.2). See also Hartshorne $[5, Ch. I \S 4]$ for more details. [^107]: Note. In fact, this result can be generalized to show that for any nonsingular projective surface $X$ over an algebraically closed field $k$ of characteristic 0 , there is an infinitesimal extension $X^{\prime}$ of $X$ by $\omega$, such that $X^{\prime}$ is not projective over $k$.\ Indeed, let $D$ be an ample divisor on $X$. Then $D$ determines an element $c_1(D) \in$ $H^1\left(X, \Omega^1\right)$ which we use to define $X^{\prime}$, as above. Then for any divisor $E$ on $X$ one can show that $\delta(\mathcal{L}(E))=(D . E)$, where $(D . E)$ is the intersection number (Chapter V), considered as an element of $k$. Hence if $E$ is ample, $\delta(\mathcal{L}(E)) \neq 0$. Therefore $X^{\prime}$ has no ample divisors.\ On the other hand, over a field of characteristic $p>0$, a proper scheme $X$ is projective if and only if $X_{\text {red }}$ is! [^108]: Hint: Consider surjections of the form $\mathcal{O}_V \rightarrow$ $k(x) \rightarrow 0$, where $x \in X$ is a closed point, $V$ is an open neighborhood of $x$, and $\mathcal{O}_V=j_{!}\left(\left.\mathcal{O}_X\right|_V\right)$, where $j: V \rightarrow X$ is the inclusion. [^109]: Hint: Consider surjections of the form $\mathcal{L} \rightarrow \mathcal{L} \otimes k(x) \rightarrow 0$, where $x \in X$ is a closed point, and $\mathcal{L}$ is an invertible sheaf on $X$. [^110]: Hint: Show $\mathcal{E} x t^i(\cdot, \mathcal{G})$ is coeffaceable for $i>0$. [^111]: Hint: Use (6.11A) and descending induction on $i$ to show that $\operatorname{Ext}^i(M, N)=0$ for all $i>0$ and all finitely generated $A$-modules $N$. Then show $M$ is a direct summand of a free $A$-module (Matsumura $[2, p. 129]$). [^112]: Hint: Given a closed point $x \in X$ and an open neighborhood $U$ of $x$, to show there is an $\mathcal{L}, s$ such that $x \in X_s \subseteq U$, first reduce to the case that $Z=X-U$ is irreducible. Then let $\zeta$ be the generic point of $Z$. Let $f \in K(X)$ be a rational function with $f \in \mathcal{O}_x, f \notin \mathcal{O}_\zeta$. Let $D=(f)_{\infty}$, and let $\mathcal{L}=\mathcal{L}(D), s \in \Gamma(X, \mathcal{L}(D))$ correspond to $D$ (II, §6). [^113]: Hint: First construct a map ` \begin{align*} \operatorname{Ext}_X^i\left(\mathcal{F}, f^{!} \mathcal{G}\right) \rightarrow \operatorname{Ext}_Y^i\left(f_* \mathcal{F}, f_* f^{!} \mathcal{G}\right) .\end{align*} `{=html} Then compose with a suitable map from $f_* f^{!} \mathcal{G}$ to $\mathcal{G}$. [^114]: Hints: First do $i=0$. Then do $\mathcal{F}=\mathcal{O}_X$, using (Ex. 4.1). Then do $\mathcal{F}$ locally free. Do the general case by induction on $i$, writing $\mathcal{F}$ as a quotient of a locally free sheaf. [^115]: Hint: Cut with a hyperplane $H \subseteq X$, and use Bertini's theorem (II, 8.18) to reduce to the case $Y$ is a finite set of points. [^116]: This is a degenerate case of the Leray spectral sequence-see Godement $[1, II, 4.17.1]$. [^117]: Hint: Show that $f(U)$ is constructible and stable under generization (II, Ex. 3.18) and (II, Ex. 3.19). [^118]: See Grothendieck EGA $IV_3,11.1.1$. [^119]: Hint: See (II, Ex. 8.4) and use (9.12) applied to the affine cones over $Y$ and $Y^{\prime}$. [^120]: Hint: Compare $Y$ to a suitable projection of $Y$ into $\mathbf{P}^2$, as in (9.8.3) and (9.8.4). [^121]: Hint: Imitate the proof of (10.4), using (II, 8.21A). [^122]: See (10.9.1). Hints: Use (11.5), (Ex. 5.7) and (7.9). [^123]: Hint: Use (II, Ex. 9.6), and then study the maps $\operatorname{Pic}X_{n+1} \rightarrow \operatorname{Pic}X_n$ for each $n$ using (Ex. 4.6) and (Ex. 5.5). [^124]: Hint: For (ii) $\Rightarrow$ (i), show that one can find sheaves $\mathcal{E}_0, \mathcal{E}_1$ on $X$, which are direct sums of sheaves of the form $\mathcal{O}\left(-q_i\right)$, and an exact sequence $\widehat{\mathcal{E}}_1 \rightarrow \widehat{\mathcal{E}}_0 \rightarrow \tilde{{\mathcal{F}}} \rightarrow 0$ on $\widehat{X}$. Then apply (a) to the sheaf $\mathop{\mathcal{H}\! \mathit{om}}\left(\mathcal{E}_1, \mathcal{E}_0\right)$. [^125]: Note. In fact, (i) and (ii) always hold if $N \geqslant 3$. This fact, coupled with (Ex. 11.5) leads to Grothendieck's proof $[SGA \, 2]$ of the Lefschetz theorem which says that if $Y$ is a hypersurface in $\mathbf{P}_k^N$ with $N \geqslant 4$, then $\operatorname{Pic}Y \cong \mathbf{Z}$, and it. is generated by $\mathcal{O}_Y(1)$. See Hartshorne $[ 5, Ch. IV]$ for more details. [^126]: Hint: Use the results of this section to show that $f_*\left(\mathcal{L} \otimes \mathcal{M}^{-1}\right)$ is locally free of rank 1 on $Y$. [^127]: Hint: Apply (12.11) with $i=0,1$ for suitable invertible sheaves on $X \times T$. [^128]: Hint: Embed $X$ in a (proper) curve $\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu$ over $k$, and use (Ex. 1.2) to construct a morphism $f: \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu \rightarrow \mathbf{P}^1$ such that $f^{-1}\left(\mathbf{A}^1\right)=X$ [^129]: Hint: Combine (Ex. 1.3) with (III, Ex. 3.1, Ex. 3.2, Ex. 4.2). [^130]: Recall that the degree of a finite morphism of curves $f: X \rightarrow Y$ is defined as the degree of the field extension $[K(X): K(Y)]$ (II.6). [^131]: Note: we will see later (Ex. 3.2) that there exist non-hyperelliptic curves. See also (V, Ex. 2.10). [^132]: Hint: Use (III, Ex. 4.1) and (III, Ex. 5.3). [^133]: This strengthens (1.3.5). [^134]: Hint: Show first that $\delta_P$ depends only on the analytic isomorphism class of the singularity at $P$. Then compute $\delta_P$ for the node and cusp of suitable plane cubic curves. See $(\mathrm{V}, 3.9 .3)$ for another method. [^135]: Use (II, Ex. 7.5). [^136]: This generalizes (II, 6.11.4) and (II, Ex. 6.7). [^137]: Hint: Show that $X$ is the normalization of $X^*$. Then calculate $p_a\left(X^*\right)$ two ways: once as a plane curve of degree $d(d-1)$, and once using (Ex. 1.8). [^138]: See (Ex. 5.7) for an example where this maximum is achieved. Note: It is known that this maximum is achieved for infinitely many values of $g$ (Macbeath \[1\]). Over a field of characteristic $p>0$, the same bound holds, provided $p>g+1$, with one exception, namely the hyperelliptic curve $y^2=x^p-x$, which has $p=2 g+1$ and $2 p\left(p^2-1\right)$ automorphisms (Roquette). For other bounds on the order of the group of automorphisms in characteristic $p$, see Singh and Stichtenoth. [^139]: Hint: First consider an effective divisor $D$, apply $f_*$ to the exact sequence $0 \rightarrow \mathscr{L}(-D) \rightarrow$ ${\mathcal{O}}_X \rightarrow \mathcal{O}_D \rightarrow 0$, and use (II, Ex. 6.11).\] [^140]: Note. This is a special case of the more general fact that for $(n$, char $k)=1$, the étale Galois covers of $Y$ with group $\mathbf{Z} / n \mathbf{Z}$ are classified by the étale cohomology group $H_{\mathrm{et}}^1(Y, \mathbf{Z} / n \mathbf{Z})$, which is equal to the group of $n$-torsion points of $\operatorname{Pic}Y$. See Serre \[6\]. [^141]: Hint: Let $\tau: X \rightarrow X$ be the involution which interchanges the points of each fibre of $f$. Use the trace map $a \mapsto a+\tau(a)$ from $f_* \mathcal{O}_X \rightarrow \mathcal{O}_Y$ to show that the sequence of $\mathcal{O}_{Y^{-}}$ modules in a. is split exact. ` \begin{align*} 0 \rightarrow \mathcal{O}_Y \rightarrow f_* \mathcal{O}_X \rightarrow \mathscr{L} \rightarrow 0 \end{align*} `{=html} [^142]: Hint: Calculate $\operatorname{dim} H^0\left(X, \mathcal{O}_X(1)\right)$ two ways. [^143]: Hint: Use the exact sequence $0 \rightarrow \mathcal{I}_X \rightarrow$ $\mathcal{O}_{\mathbf{P}^3} \rightarrow \mathcal{O}_X \rightarrow 0$ to compute $\operatorname{dim} H^0\left(\mathbf{P}^3, \mathcal{I}_X(2)\right)$, and thus conclude that $X$ is contained in at least two irreducible quadric surfaces. [^144]: A theorem of Severi $[1]$ states that the Veronese surface is the only surface in $\mathbf{P}^5$ for which there is a projection giving a closed immersion into $\mathbf{P}^4$. Usually one obtains a finite number of double points with transversal tangent planes. [^145]: Note. It is true more generally that if $D$ is a divisor of degree $\geqslant 2 g+1$ on a curve of genus $g$, then the embedding of $X$ by $|D|$ is projectively normal (Mumford $[4$, p. 55$])$ [^146]: Answers: $j=2^6 \cdot 3^3 ; j=2^6 \cdot 5^3 ; j=-3^3 \cdot 5^3$. [^147]: Hints: It is enough to show for any $\mathcal{L} \in$ Pic $X^{\prime}$, that $(f+g)^* \mathcal{L} \cong f^* \mathcal{L} \otimes g^* \mathcal{L}$. For any $f$, let $\Gamma_f: X \rightarrow X \times X^{\prime}$ be the graph morphism. Then it is enough to show (for $\mathcal{L}^{\prime}=p_2^* \mathcal{L}$ ) that ` \begin{align*} \Gamma_{f+g}^*\left(\mathcal{L}^{\prime}\right)=\Gamma_f^* \mathcal{L}^{\prime} \otimes \Gamma_g^* \mathcal{L}^{\prime} . \end{align*} `{=html} Let $\sigma: X \rightarrow X \times X^{\prime}$ be the section $x \rightarrow\left(x, P_0^{\prime}\right)$. Define a subgroup of $\operatorname{Pic}\left(X \times X^{\prime}\right)$ as follows: ` \begin{align*} \tsm{\operatorname{Pic}(X\times X)} { {\mathcal{L}}\text{ has degree 0 along each fiber of } p_1, \\ \sigma^* {\mathcal{L}}= 0 \in \operatorname{Pic}(X) } .\end{align*} `{=html} Note that this subgroup is isomorphic to the group $\operatorname{Pic}^{\circ}\left(X^{\prime} / X\right)$ used in the definition of the Jacobian variety. Hence there is a 1-1 correspondence between morphisms $f: X \rightarrow X^{\prime}$ and elements $\mathcal{L}_f \in$ Pic $_\sigma$ (this defines $\mathcal{L}_f$ ). Now compute explicitly to show that $\Gamma_g^*\left(\mathcal{L}_f\right)=\Gamma_f^*\left(\mathcal{L}_g\right)$ for any $f, g$.\ \ Use the fact that $\mathcal{L}_{f+g}=\mathcal{L}_f \otimes \mathcal{L}_g$, and the fact that for any $\mathcal{L}$ on $X^{\prime}$, $p_2^* \mathcal{L} \in \mathrm{Pic}_\sigma^{\circ}$ to prove the result. [^148]: Hints: Any Galois étale cover $X^{\prime}$ of an elliptic curve is again an elliptic curve. If the degree of $X^{\prime}$ over $X$ is relatively prime to $p$, then $X^{\prime}$ can be dominated by the cover $n_X: X \rightarrow X$ for some integer $n$ with $(n, p)=1$. The Galois group of the covering $n_X$ is $\mathbf{Z} / n \times \mathbf{Z} / n$. Étale covers of degree divisible by $p$ can occur only if the Hasse invariant of $X$ is not zero. [^149]: Note: More generally, Grothendieck has shown $[\mathrm{SGA} 1, X, 2.6, p. 272]$ that the algebraic fundamental group of any curve of genus $g$ is isomorphic to a quotient of the completion, with respect to subgroups of finite index, of the ordinary topological fundamental group of a compact Riemann surface of genus $g$, i.e., a group with $2 g$ generators $a_1, \ldots, a_g, b_1, \ldots, b_g$ and the relation $\left(a_1 b_1 a_1^{-1} b_1^{-1}\right) \cdots$ $\left(a_g b_g a_g^{-1} b_g^{-1}\right)=1$. [^150]: Hint: $X^{\prime}$ is uniquely determined by $X$ and $\operatorname{ker} f$. [^151]: Cf. (III, Ex. 12.6), (V, Ex. 1.6). [^152]: Answers: $\tau=i ; \tau=\sqrt{-2} ; \tau=\frac{1}{2}(-1+\sqrt{-7})$. [^153]: Answer: $j=5(\bmod 13)$. [^154]: Check: $6P = (6,14)$ [^155]: Hints: Use the section to fix the group structure on the fibres. Show that the points of order 2 on the fibres form an étale cover of $\mathbf{A}_{\mathbf{C}}^1$, which must be trivial, since $\mathbf{A}_{\mathbf{C}}^1$ is simply connected. This implies that $\lambda$ can be defined on the family, so it gives a map $\mathbf{A}_{\mathbf{C}}^1 \rightarrow \mathbf{A}_{\mathbf{C}}^1-\{0,1\}$. Any such map is constant, so $\lambda$ is constant, so the family is trivial. [^156]: Hint: If $X$ is hyperelliptic, use the unique $g_2^1$ and show that $\mathop{\mathrm{Aut}}X$ permutes the ramification points of the 2 -fold covering $X \rightarrow \mathbf{P}^1$. If $X$ is not hyperelliptic, show that $\mathop{\mathrm{Aut}}X$ permutes the hyperosculation points (Ex. 4.6) of the canonical embedding. Cf. (Ex. 2.5). [^157]: Hint: Use (5.2.2) to count how many complete intersections $Q \cap F_3$ there are. [^158]: Hint: If $D \in g_3^1$, use $K-D$ to $\operatorname{map} X \rightarrow \mathbf{P}^2$. [^159]: Note. Conversely, if $X$ does not have a $g_3^1$, then its canonical embedding is a complete intersection, as in (a). More generally, a classical theorem of Enriques and Petri shows that for any nonhyperelliptic curve of genus $g \geqslant 3$, the canonical model is projectively normal, and it is an intersection of quadric hypersurfaces unless $X$ has a $g_3^1$ or $g=6$ and $X$ has a $g_5^2$. See Saint-Donat $[1]$. [^160]: See Burnside $[1, \S 232]$ or Klein $[1]$. [^161]: Hint: For each $n$, count the dimension of the family of curves with an automorphism $T$ of order $n$. For example, if $n=2$, then for suitable choice of coordinates, $T$ can be written as $x \rightarrow-x, y \rightarrow y, z \rightarrow z$. Then there is an 8-dimensional family of curves fixed by $T$; changing coordinates there is a 4-dimensional family of such $T$, so the curves having an automorphism of degree 2 form a family of dimensional 12 inside the 14-dimensional family of all plane curves of degree 4. [^162]: More generally it is true (at least over $\mathbf{C}$ ) that for any $g \geqslant 3$, a "sufficiently general" curve of genus $g$ has no automorphisms except the identity-see Baily \[1\]. [^163]: Hint: Show that it would have to lie on a quadric surface, then use (6.4.1). [^164]: Hint: Consider a projection from a point of the curve to $\mathbf{P}^2$. [^165]: Hint: Let $\pi: \tilde{\mathbf{P}} \rightarrow \mathbf{P}^3$ be the blowing-up of $X$ and let $Y=\pi^{-1}(X)$. Apply Bertini's theorem to the projective embedding of $\tilde{\mathbf{P}}$ corresponding to $\mathcal{I}_Y \otimes \pi^* \mathcal{O}_{\mathbf{P}^3}(m)$. [^166]: Hint: If $(f)$ is a principal divisor on $X$, consider the principal divisor $(t f-u)$ on $X \times \mathbf{P}^1$, where $t, u$ are the homogeneous coordinates on $\mathbf{P}^1$. [^167]: Hint: Use (III, 9.9) to show that for any very ample $H$, if $D$ and $D^{\prime}$ are algebraically equivalent, then $D . H=D^{\prime} . H$. [^168]: Note. The theorem of Néron and Severi states that the group of divisors modulo algebraic equivalence, called the Néron-Severi group, is a finitely generated abelian group. Over $\mathbf{C}$ this can be proved easily by transcendental methods (App. B, $\S 5$ ) or as in (Ex. 1.8) below. Over a field of arbitrary characteristic, see Lang and Néron \[1\] for a proof, and Hartshorne \[6\] for further discussion. Since $\operatorname{Num}X$ is a quotient of the Néron-Severi group, it is also finitely generated, and hence free, since it is torsion-free by construction. [^169]: Hint: Reduce to the case where $D$ and $E$ are nonsingular curves meeting transversally. Then consider the analogous map $c: \operatorname{Pic} D \rightarrow H^1\left(D, \Omega_D\right)$, and the fact (III, Ex. 7.4) that $c$ (point) goes to 1 under the natural isomorphism of $H^1\left(D, \Omega_D\right)$ with $k$. [^170]: Hint: Show that $H=l+m$ is ample, let $E=l-m$, let $D^{\prime}=\left(H^2\right)\left(E^2\right) D-\left(E^2\right)(D \cdot H) H-\left(H^2\right)(D \cdot E) E$, and apply (1.9). This inequality is due to Castelnuovo and Severi. See Grothendieck $[2]$. [^171]: See (App. C, Ex. 5.7) for another interpretation of this result. [^172]: Hint: Use the adjunction formula, the fact that $p_a$ of an irreducible curve is $\geqslant 0$, and the fact that the intersection pairing is negative definite on $H^{\perp}$ in $\operatorname{Num} X$. [^173]: Hint: Use (II, Ex. 8.2). [^174]: Hint: For any given $D$ in (b), show that a suitable $\xi$ exists, using an argument similar to the proof of (II, 8.18). [^175]: Note. It has been shown that $e \geqslant-g$ for any ruled surface (Nagata $[8]$). [^176]: Hint: Choose a subinvertible sheaf of maximal degree, and use induction on the rank. [^177]: Cf. (V, 6.4.1). Hint: Use (2.11.4). [^178]: Answer: If char $k \neq 2$, then $l=m=-7$; if char $k=2$, then $l, m=-6,-8$. [^179]: Hint: Use a suitable generalization of (I, Ex. 7.5) to curves in $\mathbf{P}^n$. [^180]: Hint: Consider the graded ring $\mathrm{gr}_{{\mathfrak{m}}} A=\bigoplus_{d \geqslant 0} {\mathfrak{m}}^d / {\mathfrak{m}}^{d+1}$, and apply $(\mathrm{I}, 7.5)$ [^181]: See Nagata $[7, \text{Ch} III, \S 23]$ or Zariski-Samuel $[1, \text{vol} 2 , \text{Ch} VIII, \S 10]$. [^182]: Hint: Use (Ex. 3.2). [^183]: Hint: Generalize (4.11) to the surfaces obtained by blowing up $2,3,4$, or 5 points of $\mathbf{P}^2$, and combine with our earlier results about curves on $\mathbf{P}^1 \times \mathbf{P}^1$ and the rational ruled surface $X_1,(2.18)$. [^184]: Note: See Manin $[3, \S 25,26]$ for more about Weyl groups, root systems, and exceptional curves. [^185]: Hint: Prove a lemma that when $k$ is uncountable, a variety cannot be equal to the union of a countable family of proper closed subsets. [^186]: Hint: Let $L$ be the line joining $P_1$ and $P_2$. Show that there exist finite sequences of admissible transformations such that the strict transform of $L$ becomes a plane curve of arbitrarily high degree. This example is apparently due to Kodaira-see Nagata $[5, II, p. 283]$. [^187]: Hints: Write the divisor of $f$ as $(f)=\sum n_i C_i$. Then apply embedded resolution (3.9) to the curve $Y=\bigcup C_i$. Then blow up further as necessary whenever a curve of zeros meets a curve of poles until the zeros and poles of $f$ are disjoint. [^188]: Hints: Use the projection formula (III, Ex. 8.3) and (III, Ex. 8.1) to show that $H^i\left(X, \Omega_X\right) \cong H^i\left(\tilde{X}, \pi^* \Omega_X\right)$ for each i. Next use the exact sequence ` \begin{align*} 0 \rightarrow \pi^* \Omega_X \rightarrow \Omega_{\tilde{X}} \rightarrow \Omega_{\tilde{X} / X} \rightarrow 0 \end{align*} `{=html} and a local calculation with coordinates to show that there is a natural isomorphism $\Omega_{\tilde{X} / X} \cong \Omega_E$, where $E$ is the exceptional curve. Now use the cohomology sequence of the above sequence (you will need every term) and Serre duality to get the result. [^189]: Hints: First show if $D \subseteq X$ is a section of $\pi$ containing a point $P$, and if $\tilde{D}$ is the strict transform of $D$ by $\mathrm{elm}_P$, then $\widetilde{D}^2=D^2-1$ (Fig. 23).\ Next show that $X$ can be transformed into a geometrically ruled surface $X^{\prime \prime}$ with invariant $e \gg 0$. Then use (2.12), and study how the ruled surface $\mathbf{P}(\mathcal{E})$ with $\mathcal{E}$ decomposable behaves under $\operatorname{elm}_P$. [^190]: Hint: In case (3), let $f \in R$. Use (Ex. 5.1) to show that for all $i \gg 0, f \in \mathcal{O}_{X_i}$, so in fact $f \in R_0$. [^191]: Hint: Let $|H|$ be a very ample (Cartier) divisor class on $X_0$, let $H_0 \in|H|$ be a divisor containing $P$, and let $H_1 \in|H|$ be a divisor not containing $P$. Then consider $f^* H_0, f^* H_1$ and $\tilde{H}_0=f^*\left(H_0-P\right)^{-}$. [^192]: Hint: Cut $X$ with a hyperplane and use Clifford's theorem (IV, 5.4). For the last statement, use the Riemann-Roch theorem on $X$ and the Kodaira vanishing theorem (III, 7.15).