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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 \(G_\mathrm{Cox}\).