--- sort: 040 title: Divisors flashcard: "Orals::Divisors" --- # Divisors - [x] What is a **rational section** of a line bundle? ✅ 2022-12-06 - [ ] A pair $(s, U)$ with $U\subseteq X$ open containing all associated points of $X$, $s\in H^0(U; \mcl)$, and $(s_1, U_1) \sim (s_2, U_2) \iff$ $s_1=s_2$ on $U_{12}$. - [x] How are Weil divisors related to 1Cartier divisors? #syllabus ⏫ ✅ 2022-12-05 - [ ] Coincide when $X$ is integral, separated, Noetherian, locally factorial (local rings are UFDs), and principal divisors coincide as well. In general one has a map $\CDiv(X) \to \Div(X)$ where $D$ defined by $\ts{(U_i, f_i)}$ is mapped to $\sum_{Y\in \Div(X)} v_Y(f_i) Y$ where $i$ is any index where $U_i \intersect Y$ is nonempty. The inverse construction: given $D\in \Div(X)$, it induces $D_x \in \spec \OO_{X, x}$, which is a UFD so $D_x$ is a principal divisor $D_x = (f_x)$ where $f_x \in K(X)$. Lift to a principal divisor $(f_x)$ on $X$; its restriction to $\spec \OO_{X, x}$ coincides with $D$ so they differ only by prime divisors that don't pass through $x$. Only finitely many such primes have nonzero coefficients in $D$ or $(f_x)$, so there is an open $U_x \ni x$ such that $D$ and $(f_x)$ restrict to the same divisor. Now cover $X$ by the $U_x$ and use the $f_x$ to defined a Cartier divisor. - [x] What is the Cartier divisor associated to a rational section of an invertible sheaf? #syllabus ✅ 2022-12-05 - [ ] Rational section: $s: X\to \mcl$ defined on an open dense subset. For any $Y\in \Div(X)$ one can define $v_Y(s)$ for a section, so define $(s) \da \sum_{Y\in \Div(X)} v_Y(s) Y$. The valuation is defined using $\mcl_{X, Y} \to \OO_{X, Y}$ which is a valuation ring when $X$ is integral. - [x] What is $\OO(D)$? ✅ 2022-12-04 - [ ] $\OO(D) \da \ts{f\in K(X)\units \st (f) + D \geq 0}$. - [x] Show that every degree zero divisor on $\PP^1$ is principal. ✅ 2022-12-04 - [ ] Any $p_1, p_2$ are linarly equivalent by taking $f(z) = {z-a\over z-b}$. - [x] Show that not every degree zero divisor on an elliptic curve is principal. What is $\Pic(E)$? ✅ 2022-12-04 - [ ] Take $p_1\neq p_2$, then if $p_1-p_2 = (f)$ for some $f: E\to \PP^1$ this produces an injective holomorphic map and forces $E\cong \PP^1$; however e.g. $g(E) = 1\neq 0 = g(\PP^1)$. In general $\Pic(E) \cong \Pic^0(E) \oplus \ZZ \cong E\oplus \ZZ$. - [x] Show that if $D = V(f)$ is a hypersuface in $\PP^n$ then $D\sim \deg(f) H$, so $\Pic(\PP^n) = \gens{H}_\ZZ$. ✅ 2022-12-06 - [ ] Let $d\da \deg f$ and $H = V(x_0)$, then $D - dH = (g)$ where $g(x_0,\cdots, x_n) = {f(x_0,\cdots, x_n) \over x_0^d}$. - [x] Give a definition of **very ample divisors** in terms of hyperplane sections. ✅ 2022-12-06 - [ ] $D\in \Div(X)$ is very ample iff $D \sim D'$ where $D'$ is the divisor of a hyperplane section under an embedding $X\injects \PP^n$. - [x] What is a **prime** divisor? ✅ 2022-11-26 - [ ] A closed integral subscheme $Y\leq X$ of dimension 1. - [x] What is a **Weil** divisor? ✅ 2022-11-26 - [ ] The free $\ZZ\dash$module on prime divisors, so $D = \sum_{i\in I} n_i Y_i$ where $I$ indexes prime divisors. - [x] What is an **effective** divisor? ✅ 2022-11-26 - [ ] $n_i\geq 0$ for all $i$. - [x] What is the **divisor of a rational function** $f\in K(X)$? ✅ 2022-11-26 - [ ] $(f) \da \sum_{i\in I} v_{Y_i}(f) Y_i$ where $v_Y$ is a valuation: for $Y$ prime, let $\eta$ be its generic point, then $\OO_{Y, \eta}$ is a DVR $R$ with $\ff(R) = K(X)$ and discrete valuation $v_Y$. It's a theorem that $v_Y(f) = 0$ for almost all $Y$. - [x] What is a **principal** divisor? ✅ 2022-11-26 - [ ] Any $D$ of the form $D = (f)$ for $f\in K(X)$. - [x] What is **linear equivalence** of divisors? ✅ 2022-11-26 - [ ] $D \sim D'$ iff $D-D' = (f)$ for some $f\in K(X)\units$. - [x] What is the **divisor class group** $\Cl(X)$? #syllabus ✅ 2022-11-26 - [ ] $\Cl(X) = \coker(K(X)\units \mapsvia{ (\wait)} \Div(X))$, i.e. Weil divisors modulo linear equivalence. - [x] Discuss $\Cl(X)$. ✅ 2022-11-26 - [ ] For $A$ a Noetherian domain, $A$ is a UFD iff $\spec A$ is normal and $\Cl(\spec A) = 0$. - [ ] This follows from the equivalence $A$ is a UFD iff every prime ideal of height 1 is principal. - [ ] $\Cl(\AA^n)$ is zero since $\kxn$ is a UFD, $\Cl(\PP^n\slice k) \cong k$ generated by a hyperplane, induced by the degree morphism. - [ ] For $Z\leq X$ proper closed and $U\da X\sm Z$, $\Cl(X) \surjects \Cl(U)$ by taking $\sum n_i Y_i \mapsto \sum n_i (Y_i \intersect Z)$ and is injective if $\codim_X(Z) \geq 2$. If $Z$ is irreducible of codimension 1, then one gets a right-SES $\ZZ\to \Cl(X) \surjects \Cl(U)$. - [x] Is $\Cl(X)$ always free? ✅ 2022-11-26 - [ ] No, it can contain torsion: let $Y\subseteq \PP^2$ be an irreducible curve of degree $d$ and let $X\da \PP^2\sm Y$, then $\ZZ \mapsvia{\times d} \Cl(\PP^2)\cong \ZZ \surjects \Cl(\PP^2\sm Y) \implies \Cl(X) \cong C_d$ - [x] What are minimal conditions for which it makes sense to discuss Weil divisors? ✅ 2022-11-26 - [ ] Hartshorne's condition $(*)$: Noetherian, integral, separated, regular in codimension 1. - [x] Discuss **complete divisors** on a smooth complete curve. ✅ 2022-11-26 - [ ] Every principal divisor has degree zero: use the general result that if $\Phi: X\to Y$ is a finite morpism of curves, $\deg(\Phi^* D) = \deg(\Phi) \deg D$ for any $D\in \Div(Y)$. - [ ] If $f\in k$ then $(f) = 0$, otherwise $f\not\in k$ so consider the extension $L \da k(f)$. The inclusion $L \injects K(X)$ induces a finite morphism $\phi: X\to \PP^1$ and $(f) = \phi\inv(D)$ where $D\da 1[0] - 1[\infty]$, and $\deg D = 0$ on $\PP^1$. - [ ] Since $\phi$ is a finite morphism of curves, we have $\deg (f) = \deg(\phi^* D) = \deg(\phi)\cdot 0 = 0$. - [x] What is a **rational curve**? ✅ 2022-11-26 - [ ] Birational to $\PP^1$, equivalently admits distinct points $p, q$ with $p\sim q$. - [x] Give an example of a curve that is not rational. ✅ 2022-11-26 - [ ] Take $y^2 = x^3-x$ and homogenize to $y^2z = x^3-xz^2$. - [x] What is a **reduced** divisor? An irreducible divisor? ✅ 2022-11-26 - [ ] Reduced: $n_i = 1$ for all $i$. Irreducible: $n_i = 1$ for exactly one $i$. - [x] What is a **Cartier** divisor? #syllabus ⏫✅ 2022-12-05 - [ ] A global section $s\in H^0(X; \mck\units/\OO\units)$, described by an open cover $U_i\covers X$, $f_i \in H^0(U_i; \mck\units)$, and ${f_i\over f_j}\in H^0(U_{ij}; \OO\units)$. - [ ] Equivalently, a locally principal Weil divisor. - [x] What is a **principal** Cartier divisor? ✅ 2022-11-26 - [ ] Anything in the image of $H^0(\mck\units) \to H^0(\mck\units/\OO\units)$. - [x] When do Weil and Cartier divisors coincide? ✅ 2022-11-26 - [ ] $X$ integral, separated, Noetherian, locally factorial (all local rings are UFDs). In this case, principal Weil divisors coincide with principal Cartier divisors. - [x] Give an example of a Weil divisor that is not Cartier. #redo ✅ 2022-11-26 - [ ] Take $X = V(xy-z^2) \subseteq \AA^3$ and let $Y$ be its ruling $\ts{ p\in Y\st y=z=0}$. Then $Y$ can not be locally principal in a neighborhood of the cone point. However, $2Y$ is locally principal and in fact principal. The divisor $D=V(x,z)$ is not of the form $V = (f)$ for any $f\in K(X)$ near the origin. ![](attachments/Pasted%20image%2020221126191918.png) - [x] What is an **ample divisor**? #syllabus ✅ 2022-11-26 - [ ] $\mcl$ is ample iff $\mcl^n$ is very ample for some $n$, so defines an embedding $\phi_{\abs \mcl}: X\to \PP^N$ where $\PP^N \cong \PP H^0(X; \mcl^n)\dual$. - [x] What is a **nef divisor**? ✅ 2022-11-26 - [ ] $D$ is nef iff $D\cdot C \geq 0$ for all curves iff $\OO(D)$ is nef iff $\mcl \da \OO(D)$ has positive degree over every curve. - [ ] For $D\in \QQ\dash\CDiv(X)$ and $Z\leq X$ a proper subvariety, $D$ is nef along $Z$ iff $D.C \da \deg_{\tilde C}(f^* D) \geq 0$ where $f: \tilde C\to C$ is the normalization, for all effective curves $C\subseteq Z$. - [x] What is a **big divisor**? ✅ 2022-11-26 - [ ] $D$ is big iff $h^0(\OO(mD)) \geq \bigo(m^{\dim X})$. - [x] What is a **simple normal crossings (SNC)** divisor? ✅ 2022-11-26 - [ ] Normal crossings: A Weil divisor $D = \sum D_i$ is NC iff for every $x\in X$ a local equation for $D$ is $\prod_{i=1}^r x_i$ for independent local parameters $x_i\in \OO_{X, x}$. SNC is if every $D_i$ is smooth and $r\leq n$. - [ ] Equivalently, $D$ is SNC if there exists a finite family of sections $\ts{f_i} \subseteq H^0(\OO_X)$ such that $D = \sum (f_i)$ and for each $x\in \supp D$, the restrictions $(f_i)_x \in \mfm_x$ land in the maximal ideal and form part of a regular system of parameters for $\OO_{X, x}$. - [x] Give examples and non-examples of SNC divisors. ✅ 2022-11-28 - [ ] A nodal curve is NC but not SNC (NC need not have smooth components). - [x] What is the **divisor of zeros and poles** of $f \in k(X)^{\times}$? ✅ 2022-11-26 - [ ] Writing $Z_f, P_f$ for the sets of zeros and poles, $(f) = \sum n_i Z_i - \sum m_i P_i$ where $n_i$ is the multiplicity of the zero $Z_i$ (and similarly $m_i$ for $P_i$ the order of the pole). - [x] What is the **Picard group**? #syllabus ✅ 2022-11-26 - [ ] Isomorphism classes of invertible sheaves under $\tensor$. - [ ] Equivalently, $H^1(X; \OO_X\units)$. - [x] What is the **invertible sheaf** associated to a Cartier divisor? #syllabus ✅ 2022-11-26 - [ ] Representing $D$ as $\ts{(U_i, f_i)}$, $\mcl(D) \leq \mck$ is the submodule generated by $\ts{f_i\inv}$. There is a bijection between Cartier divisors and invertible subsheaves of $\mck$. - [x] What is the relationship between the Cartier class group and $\Pic(X)$? ✅ 2022-11-26 - [ ] In general, $\Cart\Cl(X) \injects \Pic(X)$ by $D\mapsto \mcl(D)$; this is surjective when $X$ is integral. - [x] What is the relationship between $\Cl(X)$ and $\Pic(X)$? ✅ 2022-11-26 - [ ] Isomorphic when $X$ is integral, separated, locally factorial. - [x] What is an **effective** Cartier divisor? ✅ 2022-11-26 - [ ] Represented by $\ts{(U_i, f_i)}$ where each $f_i\in H^0(U_i, \OO_{U_i})$. - [x] What is the subscheme associated to a Cartier divisor? ✅ 2022-11-26 - [ ] For $D$ effective, the subscheme $Y$ of codimension 1 defined by the sheaf of ideals $\mci_Y$ which is locally generated by the $f_i$. In fact $\mci_Y \cong \mcl(-D)$ in this case. - [x] What does it mean for divisors to be **linearly equivalent**? ✅ 2022-11-26 - [ ] $D_1 - D_2 = (f)$ for some $f$. - [x] What does it mean for two divisors to be **numerically equivalent**? ✅ 2022-11-26 - [ ] $D\sim 0$ iff $D\cdot C = 0$ for every curve $C\subseteq X$, and $D_1\sim D_2 \iff D_1- D_2\sim 0$. - [x] What is the **degree** of a Weil divisor? #syllabus ✅ 2022-12-06 - [ ] $D = \sum_{Y} n_Y Y \implies \deg(D) = \sum_Y n_Y \in \ZZ$. - [x] What is the **Cartier divisor associated to a section** of an invertible sheaf? #syllabus ✅ 2022-12-06 - [ ] For $\mcl\in \Pic(X)$ and $s\in H^0(\mcl)\smz$, define $(s) = \sum_Y v_Y(s) Y$ where $v_Y$ is defined using $\sigma: \mcl_{X, V}\iso \OO_{X, V}$ and $v_Y(s) \da v_Y(\sigma(s))$ where $\mcf_V \da \mcf_{\nu_V}$, the stalk at the generic point of $V$. - [x] For $D$ a prime divisor and $Y\in \mathop{\mathrm{supp}}D$, what is the **multiplicity** of $Y$ in $D$? ✅ 2022-12-06 - [ ] $D = \sum n_{Y_i} Y_i$ and $Y= Y_j$ for some $j$ since $Y\in \supp D$, so $\mult_D(Y) = n_Y$. - [x] What is the **multiplicity of a Cartier divisor**? #syllabus ⏫ ✅ 2022-12-05 - [ ] For $D$ an effective Cartier divisor on $C$, let $f$ be a local equation at $p$ and define $\mu_p$ to be the largest $N$ such that $f\in \mfm_p^N \subseteq \OO_{X, x}$. Note $\mu_p \geq 1 \iff p\in C$ and $\mu_p = 1 \iff C$ is smooth at $p$. - [x] What is the **support of a Cartier divisor**? #syllabus ⏫ ✅ 2022-12-05 - [ ] $\supp D = \abs{D}$ is the union of poles and zeros of $D$, i.e. if $D = \ts{(U_i, f_i)}$ and $D\mapsto \sum_{Y\in \Div(X)} v_Y(f_i) Y$ where $i$ is any index with $U_i \intersect Y\neq \emptyset$, then $\supp D = \ts{Y\in\Div(X) \st v_Y(f_i) \neq 0}$. - [ ] Equivalently, define $D$ as a line bundle $\OO_X(D)$ with a rational section $s_D$, set $U \subseteq X$ to be the maximal open set on which $s_D$ is defined and nonzero, and define $\abs{D} \da X\sm U$. - [x] For $D$ a divisor, what is ${\mathcal{O}}_X(D)$? ✅ 2022-11-26 - [ ] $\ts{f\in \OO_X \st (f) + D \geq 0}$. - [x] What is a **linear system**? ✅ 2022-11-26 - [ ] Any subspace of a complete linear system. - [x] What is a **complete linear system**? ✅ 2022-11-26 - [ ] $\abs D = \ts{ D' \st D'\sim D}$ - [x] What is the linear system associated to the canonical divisor $K_X$? ✅ 2022-11-26 - [ ] $\abs{K_X} = \PP H^0(X; \omega_X)$. - [x] What does it mean for a linear system to be **base point free**? ✅ 2022-11-26 - [ ] A point $p\in X$ is a base point of a linear system $S$ iff $p$ is contained in the support of every divisor $D\in S$. $S$ is bpf iff it contains no basepoints. - [ ] Equivalently, the corresponding line bundle $\mcl$ is generated by global sections, where $S$ corresponds to $V \subseteq H^0(X; \mcl)$ for some $\mcl\in \Pic(X)$, since $p$ is a base point iff $s_p\in \mfm_p \mcl_p$ for all $s\in V$. - [x] When is the morphism defined by a linear system a closed immersion? ✅ 2022-11-26 - [ ] Separates points and separates tangents: $\forall p, q\in X$ there exists $D\in S$ with $p\in D$ and $q\not\in D$; given $p\in X, t\in \T_p X$, there is a $D\in S$ with $p\in D$ but $t\not \in \T_p(D) \da \qty{\mfm_{P, D}/\mfm_{P, D} }\dual$ - [x] What is the **base locus** of a linear system? ✅ 2022-11-26 - [ ] The union of all base points in $S$. - [x] What is the **plurigenus** of a smooth projective variety? ✅ 2022-11-26 - [ ] $P_d(X) \da h^0( (\det \Omega_{X/k})^d ) \da h^0(K_X^d)$. This is a birational invariant, and e.g. $X$ is not rational if $P_d(X) > 0$. - [x] What is the **irregularity** of a smooth projective variety? ✅ 2022-11-26 - [ ] The hodge number $q\da h^{0, 1} = h^1(\OO_X)$. - [x] What is the Kodaira dimension? ✅ 2022-11-26 - [ ] $\kappa(X) \da -\infty$ if $P_d(X) = 0$ for all $d>0$, otherwise $\kappa(X) = \min \ts{k \in \ZZ \st {P_d\over d^k} \leq \bigo(1) }$ (i.e. the smallest $k$ for which this is bounded, or the minimal $k$ so that $P_d(X) \in \bigo(d^k)$). Generally $\kappa(X) \in \ts{-\infty}\union [0, \dim X]$. - [x] What is the **arithemetic genus**? The **geometric genus**? ✅ 2022-11-26 - [ ] $p_a(X) \da (-1)^r \qty{\chi(\OO_X) - 1}$ and $p_g(X) = h^{n, 0} = h^0(\Omega_{X/k}^n)$. - [x] What is a **hyperplane section**? ✅ 2022-11-26 - [ ] For $X \subseteq \PP^n$ projective, the intersection $X \intersect H$ for some hyperplane $X = V(f)$. - [x] What is a **polarization**? What is a principal polarization? ✅ 2022-11-27 - [ ] A choice of $\mcl\in \NS(X)^\mathrm{amp}$, where $\NS(X) \da \Pic(X)/\Pic^0(X)$. Principal if $\deg \mcl = 1$. - [x] What is the **ramification divisor** of a finite morphism $f:X\to Y$ of smooth projective curves? ✅ 2022-11-26 - [ ] $R \da \sum_{p\in X} \length(\Omega_{X/Y})_p \cdot p$. - [x] What is the **ramification formula** for a finite morphism of curves? ✅ 2022-11-26 - [ ] $K_X \sim f^* K_Y + R$. - [x] Prove the ramification formula for canonical divisors. ✅ 2022-11-27 - [ ] For $f: X\to Y$ ake the cotangent exact sequence $$f^* \Omega_{Y/k} \injects \Omega_{X/k} \surjects \Omega_{X/Y}$$ tensor by $\Omega_{X/k}\dual$ to get $$f^* \Omega_{Y/k}\tensor \Omega_{X/k}\dual \injects \OO_X \surjects \Omega_{X/Y} \tensor \Omega_{X/k}\dual$$ Use that $\Omega_{X/Y}$ has finite support, so $\Omega_{X/Y} \cong \Omega_{X/Y} \tensor \Omega_{X/k}\dual$ and $$f^* \Omega_{Y/k}\tensor \Omega_{X/k}\dual \injects \OO_X \surjects \OO_R$$ but $\ker(\OO_X \surjects \OO_R) \cong \mcl(-R)$, so $$f^* \Omega_{Y/k} \tensor \Omega_{X/k}\dual \cong \mcl(-R)$$ and taking associated divisors yields $f^*K_Y, -K_X, -R$ respectively, so $f^* K_Y - K_X \sim -R \implies K_X \sim f^* K_Y + R$. - [x] What is the Riemann-Hurwitz formula for $f:X\to Y$? ✅ 2022-11-26 - [ ] $\chi(X) = \deg(f)\cdot \chi(Y) + \deg R \implies 2g_X - 2 = \deg(f)\cdot(2g_Y - 2) + \sum_{p\in X} (e_p - 1)$. - [x] Can $\PP^1$ admit an unramified cover by a curve? ✅ 2022-11-26 - [ ] No: if $f: X\to \PP^1$ is a finite etale cover of degree $n \geq 1$ then $\deg R = 0$, so by Riemann-Hurwitz $\chi(X) = n\chi(\PP^1) \implies 2g_X - 2 = n(-2) \leq -2 \implies g_X \leq 0$, so this forces $g_X = 0$ and $n=1$, so $f$ must be the identity. - [x] What is the **ramification index**? ✅ 2022-11-26 - [ ] For $f:X\to Y$ a morphism of curves and $p\in X$, letting $q\da f(p)$ and $t\in \OO_{Y, q}$ be a local coordinate, define $e_p \da v_p (f^\sharp(t))$ where $f^\sharp: \OO_{Y, q} \to \OO_{X, p}$. - [x] What is a **branch point**? A **ramification point**? ✅ 2022-11-26 - [ ] $e_p = 1\implies$ unramified, $e_p > 1 \implies p$ is ramified and $q\da f(p)$ is a branch point. - [x] What is an **etale morphism**? ✅ 2022-11-26 - [ ] For varieties: locally finite and unramified. - [ ] Flat and unramified - [ ] Formally etale and of finite presentation, where $f\in \CRing(A, B)$ is formally etale iff $\LL_{B/A} \homotopic 0$. - [x] What is the **branch locus**? ✅ 2022-11-26 - [ ] For $f:X\to Y$, the set of critical values in $Y$. Compare to the ramification locus in $X$, which is its preimage. - [x] What is the result of taking determinants in an exact sequence? ✅ 2022-11-27 - [ ] $A\to B \to C$ a SES in $\mods{\OO_X}$ yields $\det(A) \tensor_{\OO_X} \det(C) \cong \det(B)$. - [x] Discuss ampleness of divisors on curves. ✅ 2022-11-26 - [ ] If $\deg D \geq 2g$, $D$ is bpf, and if $\deg D \geq 2g+1$ then $D$ is very ample. - [x] What is a **canonical divisor**? ✅ 2022-11-27 - [ ] A divisor $K_X$ defined by any meromorphic section of $\omega_{X/k}$. - [x] What is the statement of **Riemann-Roch**? ✅ 2022-11-27 - [ ] $h^0(\mcl(D)) - h^0(\mcl(K-D)) = \deg(D) + (1-g)$. - [x] What is $\deg(K_X)$ for $X$ a curve? ✅ 2022-11-27 - [ ] $\deg K_X = 2g-2$; Take $D\da K_X$ in RR and use that $h^0(\mcl(K)) = g$ and $h^0(\mcl(0)) = 1$ to get $g-1$ on the LHS and $\deg(K_X) + (1-g)$. - [x] What is a **very ample sheaf**? ✅ 2022-11-28 - [ ] $\mcl\in \mods{\OO_X}$ invertible is very ample relative to $f:X\to Y$ if $\exists \mce \in \QCoh(Y)$ and an immersion $\iota: X\to \PP(\mce)$ such that $\mcl \cong \iota^* \OO_{\PP(\mce)}(1)$. - [x] What is an **ample sheaf**? ✅ 2022-11-28 - [ ] For any $\mcf\in \QCoh(X)^\ft$, the twist $\mcf(n) \da \mcf \tensor \mcl\tensorpower{}{n}$ is globally generated for all $n> N_0 \gg 0$.