Reconstruction of a surface from the category of reflexive sheaves

TL;DR

This paper introduces the concept of codim-2-saturated models for normal surfaces, characterizes them via reflexive sheaves, and develops categorical tools to reconstruct these models from sheaf categories.

Contribution

It defines codim-2-saturation for normal surfaces, provides a reconstruction method from reflexive sheaves, and introduces weakly localising Serre subcategories for categorical analysis.

Findings

Every normal surface has a codim-2-saturated model.

The category of reflexive sheaves is quasi-abelian.

A criterion for codim-2-saturation based on Nagata compactification.

Abstract

We define a normal surface $X$ to be codim-2-saturated if any open embedding of $X$ into a normal surface with the complement of codimension 2 is an isomorphism. We show that any normal surface $X$ allows a codim-2-saturated model $\widehat{X}$ together with the canonical open embedding $X\to \widehat{X}$. Any normal surface which is proper over its affinisation is codim-2-saturated, but the converse does not hold. We give a criterion for a surface to be codim-2-saturated in terms of its Nagata compactification and the boundary divisor. We reconstruct the codim-2-saturated model of a normal surface $X$ from the additive category of reflexive sheaves on $X$. We show that the category of reflexive sheaves on $X$ is quasi-abelian and we use its canonical exact structure for the reconstruction. In order to deal with categorical issues, we introduce a class of weakly localising Serre subcategories in abelian categories. These are Serre subcategories whose categories of closed objects are quasi-abelian. This general technique might be of independent interest.