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

TL;DR

This paper provides a simple necessary and sufficient condition for the finite generation of the Cox ring of a blow-up of a weighted projective plane at a point, confirming a conjecture by He and Kurano-Nishida.

Contribution

It establishes a clear criterion for finite generation of symbolic Rees rings in a specific geometric setting, resolving a conjecture in the field.

Findings

Established a simple criterion for finite generation of Cox rings

Confirmed the conjecture by He and Kurano-Nishida

Provided insights into the structure of blow-ups of weighted projective planes

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 C^2<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.