GIT versus Baily-Borel compactification for quartic K3 surfaces

TL;DR

This paper explores the relationship between GIT and Baily-Borel compactifications for quartic K3 surfaces, providing evidence for a conjectural decomposition of the period map into elementary birational modifications.

Contribution

It proposes a matching between arithmetic strata in the period space and GIT moduli space strata, supporting a conjectural framework for quartic surface moduli.

Findings

Proposed a correspondence between period space strata and GIT moduli strata.

Partially verified the proposed matching.

Supports the Hassett-Keel-Looijenga program for quartic surfaces.

Abstract

Looijenga has introduced new compactifications of locally symmetric varieties that give a complete understanding of the period map from the GIT moduli space of plane sextics to the Baily-Borel compactification of the moduli space polarized K3's of degree 2, and also of the period map of cubic fourfolds. On the other hand, the period map of the GIT moduli space of quartic surfaces is significantly more subtle. In our previous paper, we introduced a Hassett-Keel-Looijenga program for certain locally symmetric varieties of Type IV. As a consequence, we gave a complete conjectural decomposition into a product of elementary birational modifications of the period map for the GIT moduli spaces of quartic surfaces. The purpose of this note is to provide compelling evidence in favor of our program. Specifically, we propose a matching between the arithmetic strata in the period space and suitable strata of the GIT moduli spaces of quartic surfaces. We then partially verify that the proposed matching actually holds.