Combinatorial K3 surfaces and the Mori fan of the Dolgachev--Nikulin--Voisin family in degree 2

TL;DR

This paper introduces combinatorial K3 surfaces, a class of semistable K3 surfaces characterized by curve structures, and uses this framework to compute the Mori fan of a specific family in degree 2.

Contribution

It defines combinatorial K3 surfaces via curve structures and applies this to explicitly determine the Mori fan for the Dolgachev--Nikulin--Voisin family in degree 2.

Findings

Defined combinatorial K3 surfaces using curve structures

Described elementary modifications and their effects on Picard groups

Provided an explicit computation of the Mori fan in degree 2

Abstract

We introduce the notion of a combinatorial K3 surface. Those form a certain class of type III semistable K3 surfaces and are completely determined by combinatorial data called curve structures. Emphasis is put on degree $2$ combinatorial K3 surfaces, but the approach can be used to study higher degree as well. We describe elementary modifications both in terms of the curve structures as well as on the Picard groups. Together with a description of the nef cone in terms of curve structures, this provides an approach to explicitly computing the Mori fan of the Dolgachev--Nikulin--Voisin family in degree $2$.