Birational geometry of the moduli space of quartic K3 surfaces

TL;DR

This paper investigates the birational geometry of the moduli space of degree 4 K3 surfaces, refining existing methods to understand the relationship between different compactifications and proposing a conjectural interpolation between them.

Contribution

It extends Looijenga's work to handle the complex cases of degree 4 K3 surfaces and double EPW sextics, introducing a conjectural interpolation between GIT and Baily-Borel compactifications.

Findings

Conjectural one-parameter interpolation between compactifications for degree 4 K3 surfaces.

Refinement of Looijenga's work to address complex arithmetic issues.

Verification of predictions for hyperelliptic cases via VGIT.

Abstract

By work of Looijenga and others, one has a good understanding of the relationship between GIT and Baily-Borel compactifications for the moduli spaces of degree 2 K3 surfaces, cubic fourfolds, and a few other related examples. The similar-looking cases of degree 4 K3 surfaces and double EPW sextics turn out to be much more complicated for arithmetic reasons. In this paper, we refine work of Looijenga to allow us to handle these cases. Specifically, in analogy with the so-called Hassett-Keel program for the moduli space of curves, we study the variation of log canonical models for locally symmetric varieties of Type IV associated to D lattices. In particular, for the dimension 19 case, we conjecturally obtain a continuous one-parameter interpolation (via a directed MMP) between the GIT and Baily-Borel compactifications for the moduli of degree 4 K3 surfaces. The predictions for the similar 18 dimensional case, corresponding geometrically to hyperelliptic degree 4 K3 surfaces, can be verified by means of VGIT -- this will be discussed elsewhere.