# 2021-10-27

Tags: #web/quick-notes

Refs: ?

## 15:17

Kristin DeVleming, UGA AG seminar talk on moduli of quartic Unsorted/K3 surfaces.

• See K-stability, log Fano pairs, Fano varieties, Hassett-Keel program
• There’s a way to take the volume of the anticanonical divisor $$-K_X$$, see the delta invariant.
• Defines a moduli space with a natural wall crossing framework.
• See du Val singularities and ADE singularities.
• A polarized K3 is a pair $$(S, L)$$ with $$S$$ a K3 and $$L$$ an ample line bundle.
• From a Hodge theoretic perspective, there is a natural period domain.
• See GIT moduli spaces, Hodge bundle, Heegner divisor
• There is a map $$\mkern 1.5mu\overline{\mkern-1.5mu{\mathcal{M}}\mkern-1.5mu}\mkern 1.5mu^{\mathrm{GIT}} \to {\mathcal{F}}_4^*$$ where the LHS are quartics in $${\mathbb{P}}^1$$, and the RHS has two nontrivial divisors $$H_k$$ parameterizing hyperellptic K3s and $$H_u$$ parameterizing unigonal K3s.
• See weighted projective space, here $${\mathbb{P}}^1 \times {\mathbb{P}}^1$$ is a smooth quadric in $${\mathbb{P}}^3$$ while a singular one is $${\mathbb{P}}(1, 1, 2)$$:

• Can reduce moduli of K3s to studying moduli of curves plus stability conditions. Studying unigonal K3s reduces to studying elliptic fibrations, i.e. maps $$S\to C \subseteq {\mathbb{P}}^3$$ a twisted cubic whose fibers are elliptic.
• By Leza-O’Grady, there is a nice VGIT wall crossing framework.
• Theorem: can interpolate between $${\mathcal{M}}_4^{\GIT}$$ and $${\mathcal{F}}_4^*$$ via a sequence of explicit $$K{\hbox{-}}$$moduli wall crossings

## 16:24

Jiuya Wang’s, UGA NT seminar talk

• See ramified primes, inertia group, class group, discriminant
• Malle’s conjecture implies the inverse Galois problem.
• Unsorted/Kronecker-Weber theorem in class field theory.
#web/quick-notes #personal/idle-thoughts