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.
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(ALEXANDER DUNCAN) The author was partially supported by National Science Foundation RTG grants DMS 0838697 and DMS 0943832.
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DUNCAN, A. TWISTED FORMS OF TORIC VARIETIES. Transformation Groups 21, 763–802 (2016). https://doi.org/10.1007/s00031-016-9394-5
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DOI: https://doi.org/10.1007/s00031-016-9394-5