Finite generation of Cox rings

TL;DR

This paper explores Cox rings, a class of graded algebras linked to algebraic varieties, focusing on conditions for their finite generation and implications for variety presentations.

Contribution

It provides a geometric perspective on the finite generation problem of Cox rings and presents examples illustrating both finitely and non-finitely generated cases.

Findings

Finite generation of Cox rings leads to quotient presentations of varieties.

Examples demonstrate cases of both finite and infinite Cox ring generation.

Finite generation relates to the geometric structure of the underlying variety.

Abstract

In this expository note we discuss a class of graded algebras named Cox rings, which are naturally associated to algebraic varieties generalizing the homogeneous coordinate rings of projective spaces. Whenever the Cox ring is finitely generated, the variety admits a quotient presentation by a quasitorus, which resembles the quotient construction of the projective space. We discuss the problem of the finite generation of Cox rings from a geometric perspective and provide examples of both the finitely and non-finitely generated cases.