Compact moduli of K3 surfaces

Abstract

Let $F$ be a moduli space of lattice-polarized K3 surfaces. Suppose that one has chosen a canonical effective ample divisor $R$ on a general K3 in $F$. We call this divisor "recognizable" if its flat limit on Kulikov surfaces is well defined. We prove that the normalization of the stable pair compactification $\overline{F}^R$ for a recognizable divisor is a Looijenga semitoroidal compactification. For polarized K3 surfaces $(X,L)$ of degree $2d$, we show that the sum of rational curves in the linear system $|L|$ is a recognizable divisor, giving a modular semitoroidal compactification of $F_{2d}$ for all $d$.