On a family of 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.