Affine and Unirational unique factorial domains with unmixed gradings

TL;DR

This paper characterizes a class of factorial domains with specific gradings as those defined by trinomial data, extending previous classifications of affine varieties with torus actions.

Contribution

It establishes that factorial domains with unmixed gradings and trivial invariants are exactly those described by trinomial data, generalizing earlier classifications.

Findings

Class of factorial domains matches those defined by trinomial data

Extends previous work on affine varieties with torus actions

Provides a new characterization of factorial domains with specific gradings

Abstract

This paper studies the class of unique factorial domains $B$ over an algebraically closed field $k$ which are affine or unirational over $k$ and which admit an effective unmixed $\mathbb{Z}^{d-1}$-grading with $B_0=k$, where $d$ is the dimension of $B$. Geometrically, these correspond to factorial affine $k$-varieties with an unmixed torus action of complexity one and trivial invariants. Our main result shows that this class is identical to the class of rings defined by trinomial data, thus generalizing earlier work of Mori, of Ishida, and of Hausen, Herrppich and S\"uss.