Some remarks on rigid sheaves, helices and exceptional vector bundles on Fano varieties over arbitrary fields

TL;DR

This paper explores the relationship between rigid sheaves and separable-exceptional objects on Fano varieties over arbitrary fields, providing criteria for decompositions and collections of such bundles.

Contribution

It introduces new criteria linking rigid sheaves to separable-exceptional bundles and their collections on Fano varieties over any field.

Findings

Criteria for a rigid vector bundle to decompose into separable-exceptional bundles.

Conditions under which a separable-exceptional bundle forms part of a full collection.

Insights into the structure of exceptional bundles on Fano varieties.

Abstract

In this paper we study the connection between rigid sheaves and separable-exceptional objects on Fano varieties over arbitrary fields. We give criteria for a rigid vector bundle on a Fano variety to be the direct sum of separable-exceptional bundles and for a separable-exceptional vector bundle to be part of a full separable-exceptional collection.