On cubic torsors, biextensions and Severi-Brauer varieties over Abelian varieties

TL;DR

This paper explores the classification of Severi-Brauer varieties over Abelian varieties through the lens of cubical structures and biextensions, connecting classical results with modern torsor theory.

Contribution

It provides a new interpretation of Severi-Brauer varieties classification using cubical structures and biextensions, linking previous classifications to torsor theory.

Findings

Classification of Severi-Brauer varieties over Abelian varieties clarified

Connection established between classical results and modern torsor theory

Interpretation within the framework of cubical structures and biextensions

Abstract

We study the homogeneous irreducible Severi-Brauer varieties over an Abelian variety $A$. Such objects were classified by Brion, \cite{Bri}. Here we interpret that result within the context of cubical structures and biextensions for certain $\G_m$-torsors over finite subgroups of $A$. Our results can be seen as an instance of the theory developed by Breen, \cite{Breen:1983}, and Moret-Bailly, \cite{Moret-Bailly}.