On blowing up the weighted projective plane

TL;DR

This paper studies the blow-up of weighted projective planes, providing criteria and algorithms to determine when the resulting surface is a Mori dream space, along with Cox ring computations and applications to moduli spaces.

Contribution

It introduces new criteria and algorithms for identifying Mori dream surfaces after blow-ups and computes Cox rings in specific cases, with applications to moduli space analysis.

Findings

Criteria for Mori dream surface after blow-up

Algorithms for testing Mori dream property

Computed Cox rings in several cases

Abstract

We investigate the blow-up of a weighted projective plane at a general point. We provide criteria and algorithms for testing if the result is a Mori dream surface and we compute the Cox ring in several cases. Moreover applications to the study of $\overline{M}_{0,n}$ are discussed.