The KSBA compactification for the moduli space of degree two K3 pairs

TL;DR

This paper constructs an explicit geometric compactification for the moduli space of degree two K3 pairs, linking it to existing compactifications and enhancing understanding of K3 surface moduli.

Contribution

It provides the first explicit geometric compactification for degree two K3 pairs and clarifies its relation to the Baily-Borel compactification.

Findings

Constructed a compactification $ar{P_2}$ for degree two K3 pairs.

Established a natural forgetful map to the Baily-Borel compactification.

Enhanced understanding of the geometry of K3 surface moduli spaces.

Abstract

Inspired by the ideas of the minimal model program, Shepherd-Barron, Koll\'ar, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs (X,H) consisting of a degree two K3 surface X and an ample divisor H. Specifically, we construct and describe explicitly a geometric compactification $\bar{P_2}$ for the moduli of degree two K3 pairs. This compactification has a natural forgetful map to the Baily-Borel compactification of the moduli space $F_2$ of degree two K3 surfaces. Using this map and the modular meaning of $\bar{P_2}$, we obtain a better understanding of the geometry of the standard compactifications of $F_2$.