GIT versus Baily-Borel compactification for $K3$'s which are double covers of $\mathbb P^1\times\mathbb P^1$

TL;DR

This paper proves conjectures relating GIT and Baily-Borel compactifications for moduli spaces of certain K3 surfaces, specifically double covers of smooth quadrics, using VGIT techniques.

Contribution

It provides a complete proof of the conjectural relationship between GIT and Baily-Borel compactifications for these K3 moduli spaces, advancing understanding of their birational geometry.

Findings

Confirmed conjectural decompositions of period maps into simple birational transformations.

Established geometric interpretations for these decompositions.

Applied VGIT to (2,4) complete intersection curves in this context.

Abstract

In previous work, we have introduced a program aimed at studying the birational geometry of locally symmetric varieties of Type IV associated to moduli of certain projective varieties of K3 type. In particular, a concrete goal of our program is to understand the relationship between GIT and Baily-Borel compactifications for quartic K3 surfaces, K3's which are double covers of a smooth quadric surface, and double EPW sextics. In our first paper (arXiv:1607.01324), based on arithmetic considerations, we have given conjectural decompositions into simple birational transformations of the period maps from the GIT moduli spaces mentioned above to the corresponding Baily-Borel compactifications. In our second paper (arXiv:1612.07432) we studied the case of quartic K3's; we have given geometric meaning to this decomposition and we have partially verified our conjectures. Here, we give a full proof of our conjectures for the moduli space of K3's which are double covers of a smooth quadric surface. The main new tool here is VGIT for (2,4) complete intersection curves.