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Severi-Brauer varieties; a geometric treatment

János Kollár

Abstract

These notes present a geometric treatment of Severi-Brauer varieties, without using any results from the theory of central simple algebras or from Galois cohomology. 2026 version: major revisions

Severi-Brauer varieties; a geometric treatment

Abstract

These notes present a geometric treatment of Severi-Brauer varieties, without using any results from the theory of central simple algebras or from Galois cohomology. 2026 version: major revisions

Paper Structure

This paper contains 10 sections, 38 theorems, 100 equations.

Table of Contents

  1. Geometrically split vector bundles
  2. First examples of Severi--Brauer varieties
  3. Twisted linear subvarieties
  4. Splitting fields
  5. Non-minimal Severi--Brauer varieties
  6. Twisted linear systems
  7. The Brauer group
  8. Index of a Severi--Brauer variety
  9. Norm forms and Severi--Brauer varieties
  10. Severi--Brauer schemes

Key Result

Lemma 8

Notation and assumptions are as in Definition . Then all morphisms in $T( {\mathcal{L}})$ split and there is a unique vector bundle $E( {\mathcal{L}})\in T({\mathcal{L}})$ such that every other member of $T( {\mathcal{L}})$ is a sum of copies of $E( {\mathcal{L}})$.

Theorems & Definitions (60)

  • Definition 2
  • Definition 4: Severi--Brauer variety
  • Lemma 8
  • Remark 9
  • Corollary 10
  • Corollary 12
  • Corollary 13
  • Corollary 14: Châtelet's theorem
  • Corollary 15: Wedderburn's theorem
  • Theorem 17
  • ...and 50 more