Gabriel's theorem and birational geometry

TL;DR

This paper generalizes Gabriel's theorem to birational geometry, showing that varieties are isomorphic in codimension c if their sheaf categories' quotients are equivalent, bridging reconstruction and birational equivalence.

Contribution

It extends Gabriel's theorem by characterizing isomorphism in codimension c via category quotients, connecting sheaf categories with birational geometry.

Findings

Varieties are isomorphic in codimension c if certain sheaf category quotients are equivalent.

The result interpolates between Gabriel's reconstruction theorem and birational equivalence.

Provides a categorical criterion for birational geometry.

Abstract

Extending work of Meinhardt and Partsch, we prove that two varieties are isomorphic in codimension c if and only if certain quotients of their categories of coherent sheaves are equivalent. This result interpolates between Gabriel's reconstruction theorem and the fact that two varieties are birational if and only if they have the same function field.