Some non-finitely generated Cox rings

TL;DR

This paper identifies a broad class of algebraic surfaces and moduli spaces that lack finitely generated Cox rings, revealing limitations in the algebraic structure of these geometric objects.

Contribution

It introduces new examples of algebraic surfaces and moduli spaces with non-finitely generated Cox rings, expanding understanding of Cox ring properties.

Findings

Certain weighted projective planes blown up at a point lack finitely generated Cox rings

The moduli space of stable n-pointed genus zero curves is not finitely generated for n ≥ 13

Provides methods to determine Cox ring finite generation in complex geometric contexts

Abstract

We give a large family of weighted projective planes, blown up at a smooth point, that do not have finitely generated Cox rings. We then use the method of Castravet and Tevelev to prove that the moduli space of stable n-pointed genus zero curves does not have a finitely generated Cox ring if n is at least 13.