Saturation of algebraic surfaces
Agnieszka Bodzenta, Tomasz Pełka, Dario Weißmann
TL;DR
This work extends Bondal’s saturation theory from schemes to algebraic spaces by introducing a universal saturation $X^{\mathrm{sat}}$ for normal surfaces and proving its existence. It shows that the saturation is the adjunction unit for inverting all morphisms in the category of surfaces with big open embeddings, and that $X^{\mathrm{sat}}$ can be reconstructed from the category of reflexive sheaves on $X$; moreover, saturation is functorial under proper morphisms. The paper then connects saturation to affinisation, proving that a saturated surface is proper over its non-trivial affinisation, and provides a detailed boundary-divisor criterion that classifies saturated open subspaces according to the dimension of $X^{\mathrm{aff}}$. It also clarifies when the affinisation map is surjective in the schematic vs algebraic-space setting and offers a variety of examples distinguishing the new notions from the classical scheme-saturation, illustrating the richness of saturation phenomena beyond schemes.
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.