K-moduli of curves on a quadric surface and K3 surfaces

TL;DR

This paper establishes a connection between K-moduli spaces of certain log Fano pairs and VGIT quotients of complete intersection curves, revealing a natural interpolation between different moduli spaces of curves and K3 surfaces.

Contribution

It demonstrates that K-moduli spaces of specific log Fano pairs coincide with VGIT quotients, linking moduli of curves on a quadric surface to K3 surface moduli.

Findings

K-moduli spaces match VGIT quotients of (2,4) complete intersection curves

These spaces interpolate between GIT moduli of (4,4)-curves and K3 surface compactifications

Wall crossings in K-moduli correspond to VGIT wall crossings

Abstract

We show that the K-moduli spaces of log Fano pairs $(\mathbb{P}^1\times\mathbb{P}^1, cC)$ where $C$ is a $(4,4)$-curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ complete intersection curves in $\mathbb{P}^3$. This, together with recent results by Laza-O'Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$-curves on $\mathbb{P}^1\times\mathbb{P}^1$ and the Baily-Borel compactification of moduli of quartic hyperelliptic K3 surfaces.