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created: 2023-04-17T12:22
updated: 2023-04-17T12:22
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date: 2022-06-02 20:21
modification date: Thursday 2nd June 2022 20:21:13
title: "2022-06-02_20-21"
aliases: [2022-06-02_20-21]
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Last modified date: NaN

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- Tags: 
	- #web/quick-notes #research/notes
- Refs:

- Links:
	- #todo/create-links 

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# 2022-06

# Meeting notes

- [ ] Start reading:`
	- [ ] [LitNote-Cox et al.-2011-Toric varieties-CLS11](Zotnotes/LitNote-Cox%20et%20al.-2011-Toric%20varieties-CLS11.md) ⏫ 
	- [ ] [LitNote-Fulton-1993-Introduction to toric varieties-Ful93](Zotnotes/LitNote-Fulton-1993-Introduction%20to%20toric%20varieties-Ful93.md) ⏫ 
	- [ ] [LitNote-Huybrechts-2016-Lectures on K3 Surfaces-Huy16](Zotnotes/LitNote-Huybrechts-2016-Lectures%20on%20K3%20Surfaces-Huy16.md) ⏫ 
	- [ ] [LitNote-Harder and Thompson-2015-The Geometry and Moduli of K3 Surfaces-HT15](Zotnotes/LitNote-Harder%20and%20Thompson-2015-The%20Geometry%20and%20Moduli%20of%20K3%20Surfaces-HT15.md) ⏫ 
- Start doing exercises, bring a few to discuss next time.
- [ ] See [LitNote-Laza-2014-Perspectives on the construction and compactification of moduli spaces-Laz14](Zotnotes/LitNote-Laza-2014-Perspectives%20on%20the%20construction%20and%20compactification%20of%20moduli%20spaces-Laz14.md). ⏫ 

- [ ] Todo: process these into their own pages. 🔼 
- $\mcf_2$: the moduli of degree 2 K3 surfaces
- Can realize $\mcf_2 \cong \DD/\Gamma$ for $\DD$ a 19-dimensional Type IV domain and $\Gamma$ an arithmetic group.
- $\kappa(X) \da \limsup {\log P_n(X) \over \log(n)}$ for $P_n(X) = h^0(X; \omega_X\tensorpower{\CC}{n} )$.
	- $\kappa(X) = -\infty$:
		- Rational, so isomorphic to $\PP^2$.
		- Ruled: $X\to C$ a curve with fibers $\PP^1$.
		- Type 7.
	- $\kappa(X) = 0$:
		- Enriques surfaces
		- Hyperelliptic
		- K3s
		- Toric or abelian surfaces
		- Kodaira surfaces
	- $\kappa(X) = 1$:
		- Proper, quasi-elliptic
	- $\kappa(X) = 2$:
		- General type.

## Questions

- [ ] How to understand $(-2)\dash$curves?
- [ ] What is a deformation class?
- [ ] How does the VHS construction for $\mcf_2$ work?
	- [ ] Is the KSBA construction related?
- [ ] What are rational polyhedral decompositions?
	- [ ] See maybe Looijenga 03.
- [ ] What is type $\mathrm{IV}$?
- [ ] What are slc singularities?
- [ ] What are log canonical classes?
	- [ ] E.g. $K_X + \eps R$ for $R \in \abs{nL}$.
- [ ] What is a toroidal compactification?
- [ ] What is the Coxeter diagram for a lattice?
	- [ ] E.g. $N\da H \oplus E_8^2 \oplus A_1$
- [ ] What is the toric descriptions of degenerations of PPAVs?
- [ ] What is $\mathrm{Vor}(B), \mathrm{Del}(B)$?
- [ ] What are Kulikov models?
- [ ] How do degenerations of K3s relate to $G_\mathrm{Cox}$.