The Mori fan of the mirror family of K3 surfaces in genus 2

To construct a geometrically meaningful compactification of the moduli space of polarized K3 surfaces is a notoriously difficult problem. Gross, Hacking, Keel und Siebert have started a program to construct such a compactification. Their starting point is the mirror family of polarized $K3$ surfaces of given genus (degree). This is a $1$-dimensional family of lattice-polarized $K3$ surfaces. One of the main ingredients of the GHKS program is the Mori fan of this mirror family. In this talk we discuss this fan of in genus $2$. This is joint work with Carsten Liese.