The KSBA compactification for the moduli space of degree two $K$3 pairs

Abstract

Inspired by the ideas of the minimal model program, Shepherd-Barron, Kollár, 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 ${\overline{\mathcal{P}}}_2$ for the moduli of degree two $K$3 pairs. This compactification has a natural forgetful map to the Baily–Borel compactification of the moduli space ${\mathcal{F}}_2$ of degree two $K$3 surfaces. Using this map and the modular meaning of ${\overline{\mathcal{P}}}_2$, we obtain a better understanding of the geometry of the standard compactifications of ${\mathcal{F}}_2$.