Polyhedral divisors and torus actions of complexity one over arbitrary fields

Abstract

We show that the presentation of affine 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 G-varieties of complexity one over a field, where G is a (not necessarily split) torus, by using elementary facts on Galois descent. This class of affine G-varieties is described via a new combinatorial object, which we call (Galois) invariant polyhedral divisor.

Flexible affine cones and flexible coverings

“…Here, the case of a complexityone torus is the most widely studied one, with contributions by many different authors [2,16,21,24,35,37]. In the following we restrict ourselves to the case of rational T -varieties of complexity one.…”

On intersection cohomology with torus actions of complexity one

“…Note that contraction-free T-varieties of complexity one were studied by Mumford in [KKMS73,Chapter IV]. These combinatorial descriptions admit a generalization to the setting of T-varieties (see [AH06,AHS08,Tim08,Lan14]). The description in [AH06] of an affine T-variety is in term of a divisor on a normal variety where its coefficients are polyhedra in N Q .…”

“…Preliminaries on T-varieties. In this section, we recall some basic notions on algebraic torus actions of complexity one (see [AH06,AHS08,Tim08,Lan14] for details).…”

Smooth varieties with torus actions

“…Hence, our results are also valid in positive characteristic provided the A-H description is valid too. In particular, see [Al10] for Luna's Slice Theorem in positive characteristic and [La15] for a different proof of the A-H description in complexity one over arbitrary fields.…”