---
date: 2022-12-25 04:07
modification date: Sunday 25th December 2022 04:07:12
title: "AG Definitions"
flashcard: "Research::Definitions"
---

# Algebraic Geometry

## Basics

- [x] What is a rational map? ✅ 2023-01-14
	- [ ] A rational map $f: V \rightarrow W$ between two varieties with $V$ irreducible is an equivalence class of pairs $\left(f_U, U\right)$ in which $f_U$ is a morphism of varieties from a non-empty open set $U \subset V$ to $W$, and two such pairs $\left(f_U, U\right)$ and $\left(f_{U^{\prime}}^{\prime}, U^{\prime}\right)$ are considered equivalent if $f_U$ and $f_{U^{\prime}}^{\prime}$ coincide on the intersection $U \cap U^{\prime}$.
- [x] What is **Kodaira dimension**? ✅ 2022-12-26
	- [ ] For$X$ normal projective and $D\in \QQ\dash\Div(X)$, $$\kappa(X, D)\da \limsup _{m \in \ZZ_{\geq 0}} \frac{\log \left(h^0\left(\mathcal{O}_X(m D)\right)\right)}{\log m}$$
	- [ ] Equivalently, the Itaka dimension of the canonical $$\kappa(X) \da \kappa(X; \omega_X)$$
	- [ ] Interpret as $P_m(X) \sim m^{\kappa(x)}$ (bounded above and below by constant multiples).
- [x] What is a general type variety? ✅ 2023-01-14
	- [ ] $\kappa(X) = \dim(X)$ is maximal, i.e. $\omega_X$ is big.
- [x] What is a big line bundle? ✅ 2023-01-14
	- [ ] $\mcl\in \Pic(X)$ is big iff $\kappa(X; \mcl) = \dim(X)$ is maximal, i.e. $\dim \im \phi_{\abs m \mcl} = \dim(X)$ for $n\gg 1$.
- [x] What is Itaka dimension? ✅ 2023-01-14
	- [ ] For $\mcl\in \Pic(X)$, define the *Itaka dimension* of $\mcl\in \Pic(X)$ as the maximal embedding dimension: $$\kappa(X; \mcl) \da \sup_{m\in \ZZ_{\geq 0}} \ts{\dim \phi_{\abs{m\mcl}}(X)},\qquad \phi_{\abs{m\mcl}}:X\to \PP^n$$
	- [ ] Always have $\kappa(X; \mcl) \in \ts{-\infty, 0, 1,\cdots, \dim(X)}$.
	- [ ] If $\mcl$ is not effective, define $\kappa(X; \mcl)= -\infty$.
	- [ ] Fact: $$\kappa(X; \mcl) + 1 = \trdeg_k \ff(R(X; L))$$ where $R(X; L)$ is the section ring.
- [x] What is the section ring? ✅ 2023-01-14
	- [ ] $$R(X, L) \da \bigoplus_{n\geq 0} H^0(X; \mcl\tensorpower{\OO_X}{n})$$
- [x] Classify curves by Kodaira dimension. ✅ 2023-01-14
	- [ ] $\kappa(X) = -\infty \iff g=0$, positive curvature.
		- [ ] $\implies X\cong \PP^1 \implies P_{\gt 0}(X) = 0$.
	- [ ] $\kappa(X) = 0 \iff g=1$, flat.
		- [ ] $\implies X\cong E$ an elliptic curve
		- [ ] $\implies K_X$ trivial.
	- [ ] $\kappa(X) = 1 \iff g\geq 2$, negative curvature.
		- [ ] $\implies$ general type.
		- [ ] $\implies K_X$ ample.
- [x] Classify surfaces by Kodaira dimension. ✅ 2023-01-14
	- [ ] For a minimal smooth projective surface $X$,
	- [ ] $\kappa(X) = -\infty \iff$rRuled, rational.
		- [ ] $\implies X\cong \PP^{2}$ or a minimal ruled surface.
		- [ ] $\implies \left|12 K_{X}\right|=\emptyset$.
	- [ ] $\kappa(X) = 0\iff$ abelian, hyperelliptic, K3s, Enriques.
		- [ ] $\implies \left|12 K_{X}\right|=\{0\}$.
	- [ ] $\kappa(X) = 1\iff$ elliptic
	- [ ] $\kappa(X) = 2 \iff$ general type.
- [x] What is **plurigenera**? ✅ 2022-12-26
	- [ ] $$P_m(X)= h^0(\omega_X^m) = h^0(\mathcal{O}_X(m K_X))$$
	- [ ] Measures the size of a canonical model of $X$, $\Proj (\bigoplus_{m\geq 0} H^0(K_X^m))$.
	- [ ] Can prove a variety is not rational by showing $P_m(X) > 0$ for some $m>0$.
- [x] What is a variety of general type? ✅ 2022-12-26
	- [ ] Maximal possible Kodaira dimension: $\kappa(X) = \dim(X)$, e.g. any variety with ample canonical.
- [x] What is **irregularity**? ✅ 2022-12-26
	- [ ] $$q(X) \da h^{0, 1} = h^1(\OO_X)$$ 
	- [ ] For a smooth complex projective surface, $q(X)=h^0(\Omega_X^1)$.
	- [ ] Measures the difference $p_g(X) - p_a(X)$ between geometric and arithmetic genus.
- [x] What is a **rational** variety? ✅ 2022-12-26
	- [ ] Normal and birational $X\birational \PP^N$ for some $N$.
- [x] What is a **unirational** variety? ✅ 2022-12-26
	- [ ] Idea: Luroth shows that if $k \subsetneq L \subseteq k(x)$ then $L = k(x)$ and asks the same for subfields $L \subseteq k(x_1,\cdots, x_n)$. If $L = k(X)$ for a variety, this induces $\PP^n \rational X$ with dense image; call such $X$ unirational.
	- [ ] Normal and admits a dominant rational map $\PP^N\torational X$ for some $N$, or dominated by a rational variety so that $k(\PP^n)/k(X)$ is a purely transcendental (separable) extension of finite type.
	- [ ] Idea: similar to rational, e.. $h^0(\Omega^{\geq 1}_X) = 0$, Luroth asks if unirational implies rational.
	      True for curves, false for 3-folds and higher.
- [x] What is a **uniruled** variety? ✅ 2022-12-26
	- [ ] Idea: covered by rational curves, for every $x\in X$ there is a nonconstant morphism $\PP^1\to X$ with $f(0) =x$, i.e. for every point one can find a rational curve containing it.
	- [ ] More precisely, $\exists Y$ and a dominant rational morphism $Y\times \PP^1\torational X$ which does not factor through the projection to $Y$, i.e. dominated by a ruled variety.
- [x] What is a **ruled** variety? ✅ 2022-12-27
	- [ ] Birational to $Y\times \PP^1$ for some $Y$.
- [x] What is a minimal model? ✅ 2022-12-26
	- [ ] For $\kappa(X) \geq 0$, a birational $\tilde X\to X$ with $K_{\tilde X}$ nef and $\tilde X$ terminal.
- [x] What is a terminal singularity? Canonical singularity? Log-terminal singularity? ✅ 2022-12-26
	- [ ] For a resolution $\pi: \tilde X\to X$, write $K_{\tilde X} \equiv_\QQ \pi^* K_{X} + \sum a_i E_i$ with $a_i\in \QQ$, then
		- [ ] Terminal: $a_i > 0$.
		- [ ] Canonical: $a_i \geq 0$
		- [ ] Log-terminal: $a_i > -1$.
- [x] What is Luroth's problem? ✅ 2022-12-27
	- [ ] Rational $\implies$ unirational, is the converse true? Equivalently, is every subfield of a purely transcendental field extension again purely transcendental? True in dimension 1 by Galois theory.
- [x] What is Castelnuovo's theorem on complex surfaces? ✅ 2023-01-14
	- [ ] If $q(X), P_2(X)$ vanish then $X$ is rational.
	      This implies that unirational surfaces are rational.
	      Note that $P_1(X)$ vanishing is not enough, viz. the Enriques surfaces.
- [x] What is a **rationally connected** variety? ✅ 2022-12-27
	- [ ] Any pair $p,q\in X$ can be connected by a chain of rational curves, or equivalently there is a morphism $f: \PP^1\to X$ with $f(\tv{0: 1}) = p$ and $f(\tv{1: 0}) = q$.
- [x] What is a $\QQ\dash$Gorenstein variety? ✅ 2022-12-26
	- [ ] $K_X \in \QQ\dash\CDiv(X)$.
- [x] Define the order of a nonzero meromorphic differential $\omega$ at a point $p\in C$ a curve. ✅ 2023-01-25
	- [ ] Let $X$ be a smooth integral curve over $k$, let $\omega$ be a non-zero meromorphic differential on $X$, and let $x$ be a closed point of $X$. The order or valuation of $\omega$ at $x$, denoted by $\operatorname{ord}_x \omega$, is the unique $n \in \mathbf{Z}$ such that $$\mathcal{O}_{X, x} \omega=\mathfrak{m}_{X, x}^n \Omega_{X / k, x}$$ as $\mathcal{O}_{X, x}$-submodules of $\Omega_{K(X) / k}$.
- [x] What is $\Div(\omega)$ for $\omega$ a meromorphic differential on a curve? ✅ 2023-01-25
	- [ ] $$\operatorname{div} \omega=\sum_{x \in X} \operatorname{ord}_x(\omega) x$$
- [x] What is $\div(\dt)$ for $t$ a coordinate on $\PP^1$? ✅ 2023-01-25
	- [ ] $\div(\dt) = -2[\infty]$.
- [x] Describe the canonical sheaf of a curve explicitly. ✅ 2023-01-25
	- [ ] $$\Omega_{X / k}(U)= \begin{cases}\left\{\omega \in \Omega_{K(X) / k} \mid \operatorname{ord}_x(\omega) \geq 0 \text { for all closed points } x \in U\right\} & \text { if } U \neq \emptyset, \\ 0 & \text { if } U=\emptyset\end{cases}$$
- [x] What is RR for a curve? ✅ 2023-01-25
	- [ ] Theorem $2.3$ (Riemann-Roch). Let $X$ be a smooth, projective, geometrically integral curve of genus $g$ over $k$, and let $\mathcal{K}$ be a canonical divisor on $X$. For every divisor $D$ on $X$, we have $$\operatorname{dim}_k L(X, D)-\operatorname{dim}_k L(X, \mathcal{K}-D)=1-g+\operatorname{deg} D$$
- [ ] What is the Neron-Severi group?
- [ ] What is a flip?
- [ ] Prove that $\PP^1$ is complete.
- [ ] What is the Mori cone?
	- [ ] 
- [ ] What is Krull's PID theorem?
- [ ] What are the Kodaira type fibers $I_n$?
- [ ] What is a rational polyhedral cone?
	- [ ] Generated by a finite subset of $\Lambda_\QQ$.
- [ ] What is K-stability?
	- [ ] Modern version of GIT, since GIT doesn't work well for varieties of $\dim \geq 2$.
	- [ ] For Fanos, equivalent to admitting a constant scalar curvature metric.

## Scheme Theory

- [x] When is $M\in \algs{k}$? ✅ 2023-01-06
	- [ ] Algebras are like "covers" of schemes: 
	- [ ] Noncommutative case: $M\in \algs{k}$ when $\exists f\in \CRing(k \to Z(M))$, so $f^*: \spec Z(M) \to \spec k$ where $Z(M)$ is the ring-theoretic center of $M$. This induces an action $\lambda . m \da f(\lambda)m$ using the ring multiplication in $M$.
	- [ ] Commutative case: $\exists f\in \CRing(k\to M)$ so $\spec M\to \spec k$.
	- [ ] In particular, $M\in \kmod$ so there should be an action by "scalars" $k\actson M$.
- [x] What is a geometric vector bundle over a scheme? ✅ 2023-01-11
	- [ ] A geometric vector bundle of rank $n$ over $X$ is a scheme $f: Z \rightarrow X$ together with an open cover $\left\{U_i\right\}$ of $X$ and isomorphisms $\psi_i: f^{-1}\left(U_i\right) \rightarrow \mathbf{A}_{U_i}^n \da U\times \AA^n_L$ such that, for any $i, j$ and any open affine $V=\operatorname{Spec}(R) \subseteq U_i \cap U_j$, the automorphism $\psi=\psi_j \circ \psi_i^{-1}$ of $\mathbf{A}_V^n$ is in $\GL_n(R)$.
- [x] What is a locally free sheaf $\mce$? ✅ 2023-01-11
	- [ ] A locally free sheaf $\mathcal{E}$ of rank $n$ is an $\mathcal{O}_X$-module $\mathcal{E}$ together with a covering $\left\{U_i\right\}$ of $X$ such that there exist $\mathcal{O}_{U_i}$-module isomorphisms $$\rho_i:\left.\mathcal{E}\right|_{U_i} \simeq \mathcal{O}_{U_i}^n .$$
- [x] What is the geometric vector bundle associated to a locally free sheaf? ✅ 2023-01-11
	- [ ] $V(\mce) \da \Spec \Sym \mce$, where $\Sym \mce$ is the sheafy symmetric algebra so $\ro{\qty{\Sym \mce}}{U_i} \cong \Sym \qty{\ro \mce {U_i}}$ over $\OO_{U_i}$, and where one takes the sheaf-theoretic spectrum.
- [x] What is $\Proj(R)$? ✅ 2023-01-12
	- [ ] Homogeneous prime ideals of $R$ **not** completely containing $R_+ \da \bigoplus_{d\geq 1} R_d$.

## Moduli

- [x] Show $\dim \mcm_g = 3g-3$. ✅ 2023-01-23
	- [ ] $\dim \mcm_g = \dim \T_{\mcm_g} = h^1(\T_C)$ where $C$ is any smooth complete curve of genus $g$.
	- [ ] Serre duality: $H^1(\T_C) = H^0(\T_C\dual\tensor \omega_C)\dual = H^0(\omega_C\tensorpower{}{2})$.
	- [ ] RR: $\chi(\omega_C\tensorpower{}{2}) = 2(g-2) + (1-g) = 3g-3$.
	- [ ] For $g\geq 2$, $H^1(\omega_C\tensorpower{}{2}) = H^0( (\omega_C\tensorpower{}{2})\dual \tensor \omega_C)=0$ since this is a line bundle of degree $4-4g + 2g-2 = 2-2g < 0$.