The geography of negative curves

TL;DR

This paper investigates the role of negative curves in the birational geometry of blowups of weighted projective planes and toric surfaces, revealing their influence on the Mori Dream Space property and cataloging known and new negative curves.

Contribution

It catalogs all known negative curves in the parameter space of rational triangles and introduces two new families, including the first infinite family of special negative curves.

Findings

Negative curves significantly influence the MDS property.

Identification of two new families of negative curves.

Negative curves often determine the MDS property in these varieties.

Abstract

We study the Mori Dream Space (MDS) property for blowups of weighted projective planes at a general point and, more generally, blowups of toric surfaces defined by a rational plane triangle. The birational geometry of these varieties is largely governed by the existence of a negative curve in them, different from the exceptional curve of the blowup. We consider a parameter space of all rational triangles, and within this space we study how the negative curves and the MDS property vary. One goal of the article is to catalogue all known negative curves and show their location in the parameter space. In addition to the previously known examples we construct two new families of negative curves. One of them is, to our knowledge, the first infinite family of special negative curves. The second goal of the article is to show that the knowledge of negative curves in the parameter space often determines the MDS property. We show that in many cases this is the only underlying mechanism responsible for the MDS property.