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

TL;DR

This paper classifies absolutely split vector bundles on proper schemes over a field and explores their connection to Brauer--Severi varieties, providing a detailed correspondence with Picard scheme points.

Contribution

It establishes a one-to-one correspondence between Picard scheme points and indecomposable absolutely split vector bundles, advancing the understanding of their structure and applications.

Findings

Closed points of the Picard scheme correspond to indecomposable absolutely split vector bundles.

Results provide insights into the geometry of Brauer--Severi varieties.

Classification aids in understanding vector bundle decompositions on proper schemes.

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.