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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


Jiuya Wang’s, UGA NT seminar talk

#web/quick-notes #personal/idle-thoughts