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arXiv · 2604.21242

Two New Extensions of Reider's Theorem on Algebraic Surfaces

Aaron Bertram·Jonathon Fleck·Liebo Pan·Joseph Sullivan
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Abstract

Reider's Theorem on the very ampleness of adjoint linear series on a complex projective algebraic surface is extended in two new directions. First, Reider-type inequalities are shown to imply nefness of linear series of the form dH - E on the blow-up of projective space along the embedded surface. This can be thought of as a weak analogy of Saint-Donat's Theorem on the generators of the ideal of a curve embedded by an adjoint linear series. Next, Reider-type inequalities give a sharp estimate for the ample cone of the Hilbert schemes of length d subschemes of the surface. The proofs consist of (a) finding a natural family of objects parametrized by the base (either the blow-up along the surface or the Hilbert scheme) and (b) finding the largest chamber in the stability manifold of the surface where the objects in the family are all Bridgeland semistable. A Theorem of Bayer-Macri then gives nefness of the determinant line bundle on the base of the family.

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Aaron Bertram, Jonathon Fleck, Liebo Pan, Joseph Sullivan. 2026-04-23. Two New Extensions of Reider's Theorem on Algebraic Surfaces. https://arxiv.org/abs/2604.21242

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