Polyhedral divisors and torus actions of complexity one over arbitrary fields

Abstract

We show that the presentation of affine $\mathbb{T}$-varieties of complexity one in terms of polyhedral divisors holds over an arbitrary field. We also describe a class of multigraded algebras over Dedekind domains. We study how the algebra associated to a polyhedral divisor changes when we extend the scalars. As another application, we provide a combinatorial description of affine $\mathbf{G}$-varieties of complexity one over a field, where $\mathbf{G}$ is a (not-nescessary split) torus, by using elementary facts on Galois descent. This class of affine $\mathbf{G}$-varieties is described via a new combinatorial object, which we call (Galois) invariant polyhedral divisor.

31 pages, published version and erratum. Journal of Pure and Applied Algebra 219 (2015), no. 6, 2015-2045

References in corpus (1)

• Affine T-varieties of complexity one and locally nilpotent derivations