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2022-06

Meeting notes

Start reading:`
Start doing exercises, bring a few to discuss next time.
See LitNote-Laza-2014-Perspectives on the construction and compactification of moduli spaces-Laz14.
${\mathcal{F}}_2$ : the moduli of degree 2 K3 surfaces
Can realize ${\mathcal{F}}_2 \cong {\mathbb{D}}/\Gamma$ for ${\mathbb{D}}$ a 19-dimensional Type IV domain and $\Gamma$ an arithmetic group.
$\kappa(X) \coloneqq\limsup {\log P_n(X) \over \log(n)}$ for $P_n(X) = h^0(X; \omega_X{ {}^{ \scriptstyle\otimes_{{\mathbf{C}}}^{n} } } )$ .
- $\kappa(X) = -\infty$ :
  - Rational, so isomorphic to ${\mathbf{P}}^2$ .
  - Ruled: $X\to C$ a curve with fibers ${\mathbf{P}}^1$ .
  - 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.

How to understand $(-2){\hbox{-}}$ curves?
What is a deformation class?
How does the VHS construction for ${\mathcal{F}}_2$ 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. $K_X + {\varepsilon}R$ for $R \in {\left\lvert {nL} \right\rvert}$ .
What is a toroidal compactification?
What is the Coxeter diagram for a lattice?
- E.g. $N\coloneqq H \oplus E_8^2 \oplus A_1$
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 $G_\mathrm{Cox}$ .