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Abstract Liese

The KSBA compactification of the moduli space of degree 2 K3 pairs: a toroidal interpretation

Work of Gross,Hacking,Keel and Siebert shows that the Gross-Siebert reconstruction algorithm provides a partial toroidal  compactification of the moduli space of polarized K3 surfaces for any genus. The construction comes with a family $\mathfrak{X}\to \mathbb{P}^g$ over a  subset of the Kollar-Shepherd-Barron moduli space of stable K3 pairs $M_{SP}$. A conjecture of Keel says that $\mathfrak{X}\to \mathbb{P}^g$ to the main component $\bar{\mathbb{P}^g}$ of $M_{SP}$. In the genus $2$ case, $\bar{\mathbb{P}^g}$ is known by work of Laza. We show that all degenerate  type III $K3$ surfaces appear as fibres of $\mathfrak{X}\to \mathbb{P}^g$.