Reconstruction of a surface from the category of reflexive sheaves

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 Xˆ together with the canonical open embedding X→Xˆ.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.