Thin Severi-Brauer Varieties

TL;DR

This paper explores the concept of thin Severi-Brauer varieties, which are finite sets with Galois actions related to etale algebras, and examines their embeddings and geometric analogues of algebraic constructions.

Contribution

It introduces the notion of thin Severi-Brauer varieties and studies their embeddings into classical varieties, linking algebraic and geometric perspectives.

Findings

Characterization of thin Severi-Brauer varieties as finite Galois sets

Descriptions of embeddings into classical Severi-Brauer varieties

Connections between thin quadrics and etale algebras with involution

Abstract

Severi-Brauer varieties are twisted forms of projective spaces (in the sense of Galois cohomology) and are associated in a functorial way to central simple algebras. Similarly quadrics are related to algebras with involution. Since thin projective spaces are finite sets, thin Severi-Brauer varieties are finite sets endowed with a Galois action; they are associated to etale algebras. Similarly, thin quadrics are etale algebras with involution. We discuss embeddings of thin Severi-Brauer varieties and thin quadrics in Severi-Brauer varieties and quadrics as geometric analogues of embeddings of etale algebras into central simple algebras (with or without involution), and consider the geometric counterpart of the Clifford algebra construction.