Cox rings, semigroups and automorphisms of affine algebraic varieties

TL;DR

This paper explores Cox rings of affine varieties, providing explicit descriptions, universal properties, and automorphism lifting results, revealing infinite-dimensional automorphism groups and constructing wild automorphisms in specific cases.

Contribution

It introduces explicit Cox realization descriptions, proves a universal property, and demonstrates automorphism lifting, leading to new insights into automorphism groups of affine varieties.

Findings

Automorphisms of affine varieties can be lifted to Cox rings.

Automorphism groups of certain affine varieties are infinite-dimensional.

Constructed a wild automorphism of the 3D quadratic cone.

Abstract

We study the Cox realization of an affine variety, i.e., a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi quasitorus. The realization is described explicitly for the quotient space of a linear action of a finite group. A universal property of this realization is proved, and some results on the divisor theory of an abstract semigroup emerging in this context are given. We show that each automorphism of an affine variety can be lifted to an automorphism of the Cox ring normalizing the grading. It follows that the automorphism group of a non-degenerate affine toric variety of dimension $\geq 2$ has infinite dimension. We obtain a wild automorphism of the three-dimensional quadratic cone that rises to Anick's automorphism of the polynomial algebra in four variables.