The Group of Units on an Affine Variety

TL;DR

This paper investigates the structure of the group of units in the coordinate ring of certain affine varieties, revealing conditions under which it is trivial or has a specific algebraic form, using Galois cohomology methods.

Contribution

It applies Galois cohomology to describe the units in coordinate rings of cyclic covers and hyperelliptic curves, providing new criteria for their structure.

Findings

O*(X) = k* for cyclic covers with prime degree and irreducible ramification divisor.

O*(X) = k* for sufficiently general affine hyperelliptic curves.

O*(X)/k* is isomorphic to Z^(r-1) under certain conditions on the divisor at infinity.

Abstract

The object of study is the group of units O^\ast(X) in the coordinate ring of a normal affine variety X over an algebraically closed field k. Methods of Galois cohomology are applied to those varieties that can be presented as a finite cyclic cover of a rational variety. On a cyclic cover X \rightarrow A^m of affine m-space over k such that the ramification divisor is irreducible and the degree is prime, it is shown that O^\ast(X) is equal to k^\ast, the nonzero scalars. The same conclusion holds, if X is a sufficiently general affine hyperelliptic curve. If X has a projective completion such that the divisor at infinity has r components, then sufficient conditions are given for O^\ast(X)/k^\ast to be isomorphic to Z^(r-1).