Mori dream singular $K3$ surfaces

TL;DR

This paper investigates the conditions under which singular Mori dream K3 surfaces are classified, establishing links between their Picard lattice properties and Mori dream status, especially for rank two cases and certain singularities.

Contribution

It proves that for singular K3 surfaces, Mori dreamness of the Picard lattice implies the surface is Mori dream, and characterizes rank two cases via negative curves.

Findings

Mori dreamness of the Picard lattice implies the surface is Mori dream.

Rank two singular K3 surfaces are Mori dream if they contain two intersecting negative curves.

The study applies to K3 surfaces with A_n singularities.

Abstract

We take a first step towards the classification of singular Mori dream $K3$ surfaces. We prove that if the Picard lattice of a singular $K3$ surface is Mori dream, then the surface is Mori dream. Moreover, we show that for singular $K3$ surfaces, of Picard rank two, being Mori dream is equivalent to contain two negative curves intersecting each other, and apply this result to study Mori dreamness of $K3$ surfaces with a singular point of type $A_n$.