Some necessary and sufficient condition for finite generation of symbolic Rees rings

Abstract

Consider the blow-up Y of a weighted projective plane at a point in the open orbit over a field of characteristic 0. We assume that there exists a curve C on Y such that C2<0 and C.E=1, where E is the exceptional curve.In this paper we give a (very simple) necessary and sufficient condition for finite generation of the Cox ring of Y (Theorem 1.2). It is an affirmative answer to a conjecture due to He and Kurano-Nishida.