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

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 $\overline{P_2}$ 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 $\overline{P_2}$, we obtain a better understanding of the geometry of the standard compactifications of $F_2$.