Examples of non-finitely generated Cox rings

TL;DR

This paper provides examples of higher-dimensional toric varieties, including weighted projective 3-spaces, that have non-finitely generated Cox rings, extending previous results from lower-dimensional cases.

Contribution

It introduces new examples of toric varieties with non-finitely generated Cox rings in higher dimensions and Picard numbers, broadening the understanding of Cox ring properties.

Findings

Examples of non-finitely generated Cox rings in higher dimensions

Generalization of previous surface results to 3-spaces

Weighted projective 3-spaces blown up at a point

Abstract

We bring examples of toric varieties blown up at a point in the torus that do not have finitely generated Cox rings. These examples are generalizations of previous work where toric surfaces of Picard number 1 were studied. In this article we consider toric varieties of higher Picard number and higher dimension. In particular, we bring examples of weighted projective 3-spaces blown up at a point that do not have finitely generated Cox rings.