Constructing non-Mori Dream Spaces from negative curves

TL;DR

This paper develops a geometric method to classify when blowups of certain toric surfaces are Mori Dream Spaces or not, expanding the understanding of their structure through negative curves.

Contribution

It introduces a new geometric approach for constructing and classifying Mori Dream Spaces among blowups of weighted projective planes and toric surfaces.

Findings

Classified examples from two families of negative curves.

Provided a general method applicable to various cases.

Extended positive characteristic techniques to geometric classification.

Abstract

We study blowups of weighted projective planes at a general point, and more generally blowups of toric surfaces of Picard number one. Based on the positive characteristic methods of Kurano and Nishida, we give a general method for constructing examples of Mori Dream Spaces and non-Mori Dream Spaces among such blowups. Compared to previous constructions, this method uses the geometric properties of the varieties and applies to a number of cases. We use it to fully classify the examples coming from two families of negative curves.