Saturation of algebraic surfaces

TL;DR

This paper extends the concept of saturation of algebraic surfaces from schemes to algebraic spaces, exploring properties related to properness over affinisation and providing characterizations based on affinisation dimension.

Contribution

It generalizes the theory of saturation from schemes to algebraic spaces and addresses properness over affinisation for saturated surfaces.

Findings

Saturation of surfaces can be recovered from reflexive sheaves.

Passing to algebraic spaces ensures properness over affinisation when non-trivial.

Characterizations depend on the dimension of the affinisation.

Abstract

The saturation of an algebraic surface is the maximal open embedding with complement of dimension zero. For schemes, it was introduced by the first named author and A. Bondal, who proved that the saturation of a surface X can be recovered from the category of reflexive sheaves on X. In this article, we extend these results to algebraic spaces. Furthermore, we address the question of A. Bondal whether every saturated surface X is proper over its affinisation. We prove that passing from schemes to algebraic spaces guarantees that this property holds whenever the affinisation is non-trivial. Finally, we give some characterisation of saturated surfaces depending on the dimension of their affinisation.