---
date: 2021-10-27
tags:
  - web/quick-notes
created: 2021-10-27T19:35
updated: 2024-04-19T16:19
---

# 2021-10-27

Tags:
#web/quick-notes 

Refs:
?

## 15:17

> Kristin DeVleming, UGA AG seminar talk on moduli of quartic [K3 surfaces](K3%20surfaces.md).

- See [K-stability](K-stability), [log Fano pairs](log%20Fano%20pairs), [Fano varieties](Fano%20varieties), [Hassett-Keel program](Hassett-Keel%20program.md)
  - There's a way to take the volume of the anticanonical divisor $-K_X$, see the [delta invariant](delta%20invariant).
  - Defines a moduli space with a natural [wall crossing](wall%20crossing) framework.
  - See [du Val singularities](du%20Val%20singularities) and [ADE singularities](ADE%20singularities.md).
  - A [polarized](polarized) K3 is a pair $(S, L)$ with $S$ a [K3](K3) and $L$ an [ample](ample.md) line bundle.
  - From a [Hodge theoretic](Unsorted/Hodge%20theory%20MOC.md) perspective, there is a natural [period domain](period%20domain).
  - See [GIT](GIT) [moduli spaces](moduli%20spaces), [Hodge bundle](Hodge%20bundle.md), [Heegner divisor](Heegner%20divisor)
  - There is a map $\bar{\mcm}^{\mathrm{GIT}} \to \mcf_4^*$ where the LHS are quartics in $\PP^1$, and the RHS has two nontrivial divisors $H_k$ parameterizing [hyperellptic](hyperellptic.md) K3s and $H_u$ parameterizing [unigonal](unigonal.md) K3s.
  - See [weighted projective space](weighted%20projective%20space.md), here $\PP^1 \times \PP^1$ is a smooth quadric in $\PP^3$ while a singular one is $\PP(1, 1, 2)$:

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- Can reduce moduli of K3s to studying moduli of curves plus stability conditions.
  Studying unigonal K3s reduces to studying [elliptic fibrations](elliptic%20fibrations), i.e. maps $S\to C \subseteq \PP^3$ a [twisted cubic](twisted%20cubic.md) whose fibers are elliptic.
- By Leza-O'Grady, there is a nice VGIT wall crossing framework.
- Theorem: can interpolate between $\mcm_4^{\GIT}$ and $\mcf_4^*$ via a sequence of explicit $K\dash$moduli [wall crossings](wall%20crossings)
  - #personal/idle-thoughts Sequences of wall crossings look like [correspondences](correspondences.md) or [spans](spans)

## 16:24

> Jiuya Wang's, UGA NT seminar talk

- See [ramified primes](ramified%20primes), [inertia group](inertia%20group), [class group](class%20group.md), [discriminants](Unsorted/discriminants.md)
- [Malle's conjecture](Malle's%20conjecture) implies the [inverse Galois problem](inverse%20Galois%20problem.md).
- [Unsorted/Kronecker-Weber theorem](Unsorted/Kronecker-Weber%20theorem.md) in [class field theory](class%20field%20theory.md).