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 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.

💡 Research Summary

The paper develops a comprehensive theory of “saturation” for normal algebraic surfaces, extending the notion previously defined only for schemes to the broader category of algebraic spaces. A surface X is called saturated if every big open embedding X ↪ Y —an open immersion whose complement has codimension at least two (i.e. dimension 0)—is necessarily an isomorphism. In other words, a saturated surface cannot be enlarged by adding isolated closed points.

Main Results

1. Existence and Categorical Description (Theorem A).
   – Every normal surface admits a saturation η_X : X ↪ X_sat.
   – The saturation map is identified as the unit of an adjunction s∗ ⊣ s, where s localizes the category D_alg of surfaces and big open embeddings at all morphisms, and s∗ is its left adjoint. Consequently, saturation is a universal construction that makes all morphisms invertible.
   – Saturation is stable under proper surjective morphisms: if π : X → Y is proper and surjective, then X is saturated iff Y is, and there is a unique proper morphism π_sat : X_sat → Y_sat making the natural diagram commute.

2. Reconstruction from Reflexive Sheaves (Theorem B).
   The saturated surface X_sat can be recovered purely from the abelian category Ref(X) of reflexive coherent sheaves on X. This extends the earlier result for schemes and shows that the reflexive sheaf category encodes all the missing isolated points.

3. Relation to Affinisation (Theorem C).
   Let X_aff = Spec H⁰(X, O_X) be the affinisation of X. If dim X_aff > 0, then X is saturated if and only if the canonical morphism X → X_aff is proper. The forward direction uses the fact that a saturated surface cannot have a non‑proper affinisation when the target has positive dimension; the converse follows from a simple properness criterion. This answers a question of Bondal: while the converse fails for schemes, it holds for algebraic spaces as soon as the affinisation is non‑trivial. Moreover, even when the affinisation is proper but not an isomorphism, the map is always surjective for saturated surfaces.

4. Numerical Characterisation via Boundary Divisors (Theorem D).
   Choose a normal compactification \bar X of X and set D = \bar X \ X, a Weil divisor on \bar X. Then:
   – X is saturated iff D contains no connected component that is negative definite.
   – If X is saturated but not proper over its affinisation, the dimension of X_aff is determined by the sign of D:
   * dim X_aff = 2 ⇔ D is not negative semi‑definite.
   * dim X_aff = 1 ⇔ there exists a proper morphism X → C to a smooth curve such that D is a union of some fibres (so D behaves like a fibration).
   * dim X_aff = 0 ⇔ D is negative semi‑definite and X contains only finitely many proper irreducible curves; equivalently, at most one proper reduced Weil divisor on \bar X fails to be negative definite.

   The theorem refines classical contractibility criteria and shows that the geometry of the boundary divisor completely controls the saturation and the behaviour of the affinisation.

Technical Tools

• Pushouts in D_alg. Lemma 2.4 proves that big open embeddings admit pushouts in the category of separated algebraic spaces, and the pushout remains a surface. This is essential for constructing the saturation as a colimit of a stabilising chain of embeddings (Proposition 3.5).
• Galois Covers. Lemmas 2.1–2.3 show that any normal surface can be expressed as a coarse quotient of a normal scheme by a finite Galois group. This reduction allows the authors to work with schemes when necessary, while preserving the structure of big open embeddings.
• Reflexive Sheaf Theory. The reconstruction theorem uses the fact that reflexive sheaves on a normal surface are determined by their restrictions to any big open subset, together with results from