Mori Dream Spaces extremal contractions of K3 surfaces

TL;DR

This paper establishes criteria for extremal contractions of K3 surfaces to produce Mori Dream Spaces, classifies Néron–Severi groups with this property, and provides explicit geometric examples, including infinitely many with Picard number 3.

Contribution

It introduces a criterion linking extremal contractions of K3 surfaces to Mori Dream Spaces and classifies relevant Néron–Severi groups with explicit examples.

Findings

Certain extremal contractions produce Mori Dream Spaces.

Classification of Néron–Severi groups with this property.

Existence of infinitely many K3 surfaces with Picard number 3 having this property.

Abstract

We will give a criterion to assure that an extremal contraction of a K3 surface which is not a Mori Dream Space produces a singular surface which is a Mori Dream Spaces. We list the possible N\'eron--Severi groups of K3 surfaces with this property and an extra geometric condition such that the Picard number is greater then or equal to 10. We give a detailed description of two geometric examples for which the Picard number of the K3 surface is 3, i.e. the minimal possible in order to have the required property. Moreover we observe that there are infinitely many examples of K3 surfaces with the required property and Picard number equal to 3.