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 ${\mathbf{P}}^1$, and the RHS has two nontrivial divisors $H_k$ parameterizing hyperellptic K3s and $H_u$ parameterizing unigonal K3s.
  • See weighted projective space, here ${\mathbf{P}}^1 \times {\mathbf{P}}^1$ is a smooth quadric in ${\mathbf{P}}^3$ while a singular one is ${\mathbf{P}}(1, 1, 2)$:

• Can reduce moduli of K3s to studying moduli of curves plus stability conditions. Studying unigonal K3s reduces to studying elliptic fibrations, i.e. maps $S\to C \subseteq {\mathbf{P}}^3$ 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
  • #personal/idle-thoughts Sequences of wall crossings look like correspondences or spans

16:24

Jiuya Wang’s, UGA NT seminar talk

• See ramified primes, inertia group, class group, discriminants
• Malle’s conjecture implies the inverse Galois problem.
• Unsorted/Kronecker-Weber theorem in class field theory.