---
title: Algebraic Geometry
aliases:
  - Algebraic Geometry
flashcard: Reading::AG_General
created: 2023-02-05T03:50
updated: 2024-01-05T21:48
---

# Algebraic Geometry

## Basics

- [ ] What is $\Spf(R)$?
- [x] What is the Albanese variety of $Z$? Its corresponding embedding? ✅ 2023-02-05
	- [ ] $\operatorname{Alb}(Z)=H^0\left(Z, \Omega_Z^1\right)^* / H_1(Z, \mathbf{Z})$
	- [ ] $\Phi: Z\to \mathrm{Alb}(Z)$ sends a base point $p\in Z$ to the origin and $\Phi(z)(\eta) \da \int_p^z \eta$.
- [x] What is $\torsors{G}\slice X$? ✅ 2023-02-05
	- [ ] Schemes $Y\to X$ with a fiberwise $G\dash$action which etale-locally looks like $G\times X$ with $G\actson G$ by right-translation, and morphisms are $G\dash$equivariant morphisms over $X$.
- [x] What is an etale morphism in $\calg\slice\ZZ$? ✅ 2023-02-12
	- [ ] $\pi: R\to S\in \calg\slice \ZZ$ is etale iff 
		- [ ] $\pi$ is flat
		- [ ] $\del_2: S\tensorpower{R}{2}\to S$ is flat.s
		- [ ] $S$ is finitely presented over $R$
- [x] Discuss how etaleness is related to smoothness. ✅ 2023-02-12
	- [ ] Etale $\implies$ smooth.
	- [ ] Smooth and $\Omega_{S/R} = 0 \implies$ etale.
- [x] What is an ample bundle? ✅ 2023-02-14
	- [ ] Some power is very ample 
	- [ ] Very ample: bpf/gg and $\phi_{\abs L}: X\to \PP^{h^0(L) - 1}$ is a closed immersion.
	- [ ] Tensoring high powers with a coherent sheaf gives many global sections.
- [x] When is $\OO_{\PP^n}(d)$ ample? Very ample? ✅ 2023-02-14
	- [ ] BPF iff $d\geq 0$
	- [ ] Very ample iff ample iff $d\geq 1$.
- [x] What is a big bundle? ✅ 2023-02-14
	- [ ] Sections grow maximally: $h^0(L^k)\geq \bigo( k^{\dim X} )$.
- [x] What is a nef bundle? ✅ 2023-02-14
	- [ ] $L$ has non-negative degree on every irreducible curve $C$
	- [ ] Degree: $\deg \div(s)$ for $s$ any rational section of $L$.
- [x] What is a big and nef bundle? ✅ 2023-02-14
	- [ ] $L^2 > 0$ and $L$ is nef.
	      
## Definitions 

- [ ] What is the period domain for a Hodge structure on $V\in \cmod$?
- [ ] What is unipotent monodromy?
- [ ] What is the residue associated to a connection?
- [x] What is a large complex structure limit? ✅ 2023-02-12
	- [ ] Idea: line bundle $L$ over the total space of a family which specifies $c_1(L)$ and thus a symplectic class.
	- [ ] A polarized algebraic family of $n$-dim CY manifolds $X \rightarrow S \backslash\{0\}$ over a punctured algebraic curve, whose 'essential skeleton' (simplicial subcomplex of dual complex) has dimension $n$.
	- [ ] Also called "maximal degeneration"
	- [ ] Semistable SNC model.
- [ ] What is $\Dmod$?
- [ ] What is the Gauss-Manin connection?
- [ ] What is the Picard-Fuchs operator?
- [ ] What is a theta divisor?
- [ ] What are the 1st and 2nd Voronoi fans?
- [x] Describe a toric Fano variety and its singularities. ✅ 2023-02-12
	- [ ] Primitive generators form a convex polytope in $N$
	- [ ] $-K$ is ample
	- [ ] If integral points on boundary: canonical singularities
	- [ ] If all integral points outside: terminal
	- [ ] ![](attachments/2023-02-08-fano-termin.png)
- [ ] What is a canonical model?
- [ ] What are flips and flops?
- [x] What is a stable curve? ✅ 2023-02-08
	- [ ] Nodal with $\size \Aut(X) < \infty$ (nodes: $xy=0$).
- [x] What is the scheme-theoretic definition of an elliptic curve over $R$? ✅ 2023-02-12
	- [ ] A smooth proper morphism $\pi: E\to \spec R$ with a section $0: \spec R\to E$ where the geometric fibers of $\pi$ are connected genus 1 curves.
	- [ ] Implies $h^0(E; \Omega_{E/R}) =1$.
- [x] What is a hyperplane section? Give an example. ✅ 2023-02-12
	- [ ] $Y \subseteq X$ of the form $Y = X\intersect H$ for $X\embeds \PP^N$ and $H\subseteq \PP^N$ a hyperplane.
	- [ ] Any hypersurface is an example.
- [x] What is the Lefschetz decomposition? ✅ 2023-02-12
	- [ ] $$H^k(X; \CC) = \bigoplus_{2r\leq k}L^r H^{k-2r}_\prim(X)$$

- [x] Draw a picture of 3-dimensional vanishing cycles. ✅ 2023-02-12
	- [ ] ![](attachments/2023-02-12-vanishing.png)

## Theorems

- [ ] What is the "base change and cohomology" theorem?
- [x] What is Bertini's theorem? Give a consequence. ✅ 2023-02-12
	- [ ] A generic hyperplane section is smooth.
- [ ] What is Chow's theorem?
- [x] What is the Torelli theorem for K3s? ✅ 2023-01-07
	- [ ] Isomorphisms $\sigma: X^{\prime} \rightarrow X$ are in bijection with the isometries $\sigma^*: H^2(X, \mathbb{Z}) \rightarrow H^2\left(X^{\prime}, \mathbb{Z}\right)$ satisfyings $\sigma^*\left(H^{2,0}(X)\right)=H^{2,0}\left(X^{\prime}\right)$ and $\sigma^*\left(\mathcal{K}_X\right)=\mathcal{K}_{X^{\prime}}$.
- [x] What is the hard Lefschetz theorem? ✅ 2023-02-07
	- [ ] For $X$ compact Kahler, for $k\leq n\da \dim X$, 
	- [ ] $L^{n-k}: H^k(X; \RR)\iso H^{2n-k}(X; \RR)$
	- [ ] $H^k(X; \RR) = \bigoplus_{i\geq 0} L^i H^{k-2i}(X; \RR)_\prim$
	- [ ] $H^k(X; \RR)_\prim \tensor_\RR \CC =\bigoplus_{p+q=k} H^{p, q}(X)_\prim$.
- [x] What is the Hodge index theorem? ✅ 2023-02-07
	- [ ] For $X$ a compact Kahler surface, the intersection form satisfies $$\sgn H^2(X; \ZZ) = (2h^{2, 0} + 1,\, h^{1,1} -1), \qquad \sgn H^{1,1}(X) = (1,\, h^{1,1}-1)$$
	- [ ] Equivalently, if $L\in \NS(X)^\amp$ (or even $L^2 \gt 0$) then the intersection form is negative definite on $L^\perp$.
- [x] What is the Lefschetz $(1, 1)\dash$theorem? ✅ 2023-02-07
	- [ ] For $X$ compact Kahler, $\Pic(X) \surjects H^{1, 1}(X)$.
	      
## Conceptual

- [x] What is nonabelian Hodge theory? ✅ 2023-02-05
	- [ ] $\mcm_B$ is the moduli of irreducible reps of $\pi_1(X)$, $\mcm_{\mathrm{Dol}}$ of stable Higgs bundles, and $\mcm_{\dR}$ of stable flat connections: ![](attachments/2023-02-05-nonab.png)
- [x] What is $\Bun_G$? ✅ 2023-02-05
	- [ ] $\Bun_G(k)$ is the maximal subgroupoid of $\torsors{G}\slice X$.
	- [ ] Forms an algebraic stack.
	- [ ] Observation due to Weil: $\dcoset{G(F)}{G(\AA_F)}{\prod_{x\in \abs X}' G(\OO_x)} \iso \Bun_G$ where $F= k(X)$, $F_x$ is the completion at $x$, and $\OO_x$ is its valuation ring. Supposed to resemble $$\dcoset{G(F)}{G(\AA_F)}{K }$$
- [x] What is a rigid representation? ✅ 2023-02-05
	- [ ] Only trivial deformations, isolated point in character variety.
	- [ ] Conjecture: rigid representations are "motivic", of geometric origin.
	- [ ] E.g. irreducible rigid flat connections are subquotients of the Gauss-Manin connection on a family of smooth projective varieties over an open dense subvvariety of $X$.
	- [ ] The underlying local system is a summand of $\RR^if_* \ul{\CC}$ for $f:Y\to U$ a smooth projective morphism to a dense subvariety.
- [x] What are Higgs bundles? Why care? ✅ 2023-02-05
	- [ ] $(E, \theta)$ where $E$ is a holomorphic bundle and $\theta: E\to E\tensor \Omega^1_{X}$ is an endomorphism-valued 1-form satisfying $\theta\wedge \theta =0$.
	- [ ] Stable Higgs bundles with $c_1(E) = 0$ biject with irreducible representations of $\pi_1(X)$.
- [ ] What is a Hitchin fibration?
- [ ] What is a Hodge bundle?
- [x] What is Batyrev-Borisov's mirror symmetry? ✅ 2023-02-12
	- [ ] For e.g. complete intersections in Fanos, $H^1(X; \T\dual X) \cong H^1(X\dual; \T X\dual)$.
	- [ ] LHS: variations of $\omega_X$ a symplectic form (A-model)
	- [ ] RHS: variations of complex structure (B-model)
- [ ] What is a limiting MHS? How is this related to zeta values?
- [ ] What is Bridgeland stability?
- [ ] What is Gieseker stability?