Semiorthogonal decomposition for algebraic varieties

TL;DR

This paper introduces a criterion for functors between derived categories to be full and faithful, constructs semiorthogonal decompositions for intersections of quadrics, and explores how derived categories behave under birational transformations, also proving a reconstruction theorem.

Contribution

It provides new criteria for functor fullness and faithfulness, develops semiorthogonal decompositions for specific varieties, and advances understanding of derived categories' behavior under birational changes.

Findings

Established a criterion for full and faithful functors between derived categories.

Constructed semiorthogonal decompositions for intersections of quadrics.

Proved a theorem on reconstructing varieties from their derived categories.

Abstract

A criterion for a functor between derived categories of coherent sheaves to be full and faithful is given. A semiorthogonal decomposition for the derived category of coherent sheaves on the intersection of two even dimensional quadrics is obtained. The behaviour of derived categories with respect to birational transformations is investigated. A theorem about reconstruction of a variety from the derived category of coherent sheaves is proved.