Mori Dream Spaces and blow-ups of weighted projective spaces

TL;DR

The paper establishes conditions under which blow-ups of weighted projective spaces at the identity point are not Mori Dream Spaces, providing infinite examples across all dimensions three and higher.

Contribution

It introduces a sufficient condition for non-Mori Dream Space status of blow-ups and constructs infinite sequences of weights satisfying this condition in all dimensions.

Findings

Identifies a sufficient condition for blow-ups not to be Mori Dream Spaces.

Constructs infinite sequences of weights satisfying the condition in all dimensions ≥ 3.

Provides examples demonstrating the non-Mori Dream Space property in higher dimensions.

Abstract

For every $n\geq 3$, we find a sufficient condition for the blow-up of a weighted projective space $\mathbb{P}(a,b,c,d_1,\cdots,d_{n-2})$ at the identity point not to be a Mori Dream Space. We exhibit several infinite sequences of weights satisfying this condition in all dimensions $n\geq 3$.