Polyhedral Divisors, Dedekind Domains and Algebraic Function Fields

Kevin Langlois (1)

1 IF - Institut Fourier

We show that the presentation of affine $\mathbb{T}$-varieties of complexity one in terms of polyhedral divisor of Altmann-Hausen holds over an arbitrary field. We describe also 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 (non-nescessary split) torus, by using elementary facts on Galois descent. This class of affine $\mathbf{G}$-varieties are described via a new combinatorial object, which we call (Galois) invariant polyhedral divisor.

Mots clés: algebraic torus action; multigraded ring; polyhedral divisor; algebraic torus action.

Domaines: Géométrie algébrique [math.AG]

Kevin Langlois. Polyhedral Divisors, Dedekind Domains and Algebraic Function Fields. 2013. ⟨hal-00713400v3⟩