Saturation of algebraic surfaces

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 asked whether every saturated surface is proper over its affinisation. We prove that this property holds whenever the affinisation is non-trivial. If a saturated surface has trivial affinisation, we prove that the boundary of any compactification has at most two connected components, and this bound is optimal. Furthermore, we extend the results of A. Bondal and the first named author from schemes to algebraic spaces; in particular, we prove that the saturation of a surface can be recovered from the category of reflexive sheaves.