Twisted Forms of Toric Varieties

Abstract

We consider the set of forms of a toric variety over an arbitrary field: those varieties which become isomorphic to a toric variety after base field extension. In contrast to most previous work, we also consider arbitrary isomorphisms rather than just those that respect a torus action. We define an injective map from the set of forms of a toric variety to a non-abelian second cohomology set, which generalizes the usual Brauer class of a Severi-Brauer variety. Additionally, we define a map from the set of forms of a toric variety to the set of forms of a separable algebra along similar lines to a construction of A. Merkurjev and I. Panin. This generalizes both a result of M. Blunk for del Pezzo surfaces of degree 6, and the standard bijection between Severi-Brauer varieties and central simple algebras.

Galois actions on analytifications and tropicalizations

“…We begin by providing a brief review the theory of toric varieties twisted by a Galois‐action, as developed by Elizondo, Lima‐Filho, Sottile and Teitler in , and by Duncan in . This theory has a long gestation period that precedes and . A thorough account of this development appears in [, § 1].…”

Derived categories of centrally‐symmetric smooth toric Fano varieties

“…The existence of such collections was settled affirmatively in [27,28] (see also [10]). Moving beyond to general fields and twisted forms of toric varieties (also called arithmetic toric varieties [21,23,32]), one has the opportunity to further advance our grasp of the general situation. Indeed, this presents a non-trivial challenge: it is not known whether all smooth projective arithmetic toric varieties admit full exceptional collections.…”

Separable algebras and coflasque resolutions

“…Both Severi-Brauer varieties and del Pezzo surfaces are examples of arithmetic toric varieties: normal varieties which admit a faithful action of a torus (Definition 2.6) with dense open orbit. In [Dun16], it is shown that one can distinguish isomorphism classes of k-forms of an arithmetic toric variety X by separable k-algebras whenever forms of X with a rational point are retract rational. In all these cases, the separable algebras are the direct sums of endomorphism algebras of certain indecomposable vector bundles on the variety X.…”