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# Meeting notes

• Start doing exercises, bring a few to discuss next time.

• $${\mathcal{F}}_2$$: 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.

## Questions

• 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 $$G_\mathrm{Cox}$$.