Cox rings over nonclosed fields

Abstract

We give a definition of Cox rings and Cox sheaves for varieties over nonclosed fields that is compatible with torsors under quasitori, including universal torsors. We study their existence and classification, we make the relation to torsors precise, and we present arithmetic applications.

Manin's conjecture for certain spherical threefolds

“…In order to obtain explicit descriptions of X n and X n , we use [ADHL15, Theorem 2.2.2.2, Proposition 3.3.2.9, and Construction 3.2.1.3] and [DP14], according to which the quasi-affine varieties…”

“…The goal of our project is to start the investigation of Manin's conjecture for spherical varieties by the universal torsor method. For this method, an explicit description of the universal torsors is needed; this can be obtained from the Cox rings of the underlying varieties (for details, see [DP14], for example). Cox rings of spherical varieties were determined by Brion [Bri07].…”

Imaginary quadratic points on toric varieties via universal torsors

“…We first prove (i). According to [32, p. 167], T (k) is endowed with the counting measure, and T N S (A k )/T 1 N S (A k ) is endowed with the Haar measure pullback of the usual Lebesgue measure on R r under the isomorphism induced by (22). By [32, 3.28-3.30] and [32,Theorem 4.14],…”

Cox ring of the generic fiber

“…The paper is structured as follows. In Section 1 we first recall the definition of Cox sheaf and Cox ring for a variety defined over a closed field, and then, after remembering some facts about varieties defined over a (not necessarily closed) perfect field, following [6] we construct a Cox sheaf for such varieties. In Section 2 we collect some results about the generic fiber X η of a proper surjective morphism π : X → Y , whose very general fiber is irreducible.…”