## 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.[^1] 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 $d\dash$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.[^2] ### 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.[^3] ### 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 $Y \cap H \neq \varnothing$.[^4] ### 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.[^5] ### 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 $\operatorname{dim} {\mathcal{O}}_P =\operatorname{dim} X$.[^6] ### 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.[^7] 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 \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 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. [^1]: This result is generalized by (Ex. 6.7) below. [^2]: Hint: Let $H$ have degree $d$. Then consider the $d$-uple embedding of ${\mathbb{P}}^n$ in ${\mathbb{P}}^{\prime}$ and use the fact that ${\mathbb{P}}^{\prime}$ minus a hyperplane is affine. [^3]: Hint: Show that $((X) \cong k[x, y]$ and use (3.5). See (III, Ex. 4.3) for another proof. [^4]: Hint: Use (Ex. 3.5) and (E). 3.1e). See (7.2) for a generalization. [^5]: This gives an alternate proof of $(3.4 \mathrm{a})$ in the case $Y={\mathbb{P}}^n$. [^6]: Hint: Reduce 10 the affine case and use (3.2c) [^7]: 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 $Z_2$.