Categorical generic fiber

TL;DR

This paper studies the category of coherent sheaves on the generic fiber of flat proper families of algebraic varieties, establishing new results on derived equivalences and Fourier--Mukai partners through deformation techniques.

Contribution

It introduces a description of the coherent sheaves on the generic fiber as a Serre quotient and applies this to prove specialization of derived equivalences and find new Fourier--Mukai partners.

Findings

Describes the category of coherent sheaves on the generic fiber as a Serre quotient.

Proves the specialization of derived equivalence in families.

Provides new examples of Fourier--Mukai partners via deformation.

Abstract

For flat proper families of algebraic varieties with a smooth fiber, we describe the abelian category of coherent sheaves on the generic fiber as a Serre quotient. As an application, we prove specialization of derived equivalence. As another application, we provide new examples of Fourier--Mukai partners via deformations.