Compact Moduli Spaces of K3 Surfaces coming from Mirror Symmetry

In this talk, I will present the results of joint work, partially still in progress, with Hulek-Liese and Dutour Sikirić - Hulek about a certain compactification of the moduli space of polarized K3 surfaces of degree 2d. The construction of the compactification is due to Gross-Hacking-Keel-Siebert and uses the birational geometry of the Dolgachev mirror family. It is a toroidal compactification in the sense of Mumford if 2d=2, or rather semi-toroidal in the sense of Looijenga for higher d. For 2d=2, we obtain very explicit information by counting the maximal cones, respectively the rays of the toric fan in question. These results are obtained by counting so-called curve structures, which are combinatorial objects associated to the various birational models of the mirror family.