Cusp models of the DNV family
Gross, Hacking, Keel and Siebert have constructed a modular toroidal compactification of the moduli space of degree 2 polarized K3 surfaces. To obtain a fan structure, they construct a canonical section from the Picard group of the Dolgachev-Nikulin-Voisin family Y to the Picard group of a semi stable model \shY of Y. This section defines a slice in the Mori fan of \shY. This raises the question whether there is an intrinsic characterisation of the K3 surfaces associated to the cones in this slice. The first step towards an answer to this question is to calculate the set certain contractions of semi stable models of Y, the so called cusp models.
I will explain the motivation for these calculations in more detail and then explain our approach. This is ongoing work with Klaus Hulek and Christian Lehn.