Affine and Unirational unique factorial domains with unmixed 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.