Absolutely split locally free sheaves on proper $k$-schemes and Brauer--Severi varieties

Abstract

We classify absolutely split vector bundles on proper $k$-schemes. More precise, we prove that the closed points of the Picard scheme are in one-to-one correspondence with indecomposable absolutely split vector bundles. Furthermore, we apply the obtained results to study the geometry of (generalized) Brauer--Severi varieties.

17 pages, revised version with corrected and improved result, comments are welcome!

References in corpus (1)

• Severi-Brauer varieties; a geometric treatment

Cited by in corpus (3)

• Tilting objects on twisted forms of some relative flag varieties
• No full exceptional collections on non-split Brauer--Severi varieties of dimension $\le 3$
• Some remarks on rigid sheaves, helices and exceptional vector bundles on Fano varieties over arbitrary fields