On a family of negative curves

TL;DR

This paper investigates the finite generation of Cox rings for blowups of weighted projective planes at a general point, focusing on cases with negative curves of specific classes, and provides examples of both finitely and non-finitely generated Cox rings.

Contribution

It generalizes previous examples by constructing new cases of blowups with negative curves, demonstrating when their Cox rings are finitely generated or not.

Findings

Constructed examples with finitely generated Cox rings.

Constructed examples with non-finitely generated Cox rings.

Extended understanding of Cox ring behavior in blowups with negative curves.

Abstract

Let $X$ be the blowup of a weighted projective plane at a general point. We study the problem of finite generation of the Cox ring of $X$. Generalizing examples of Srinivasan and Kurano-Nishida, we consider examples of $X$ that contain a negative curve of the class $H-mE$, where $H$ is the class of a divisor pulled back from the weighted projective plane and $E$ is the class of the exceptional curve. For any $m>0$ we construct examples where the Cox ring is finitely generated and examples where it is not.