Two New Extensions of Reider's Theorem on Algebraic Surfaces
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.
Cite
@article{arxiv.2604.21242,
title = {Two New Extensions of Reider's Theorem on Algebraic Surfaces},
author = {Aaron Bertram and Jonathon Fleck and Liebo Pan and Joseph Sullivan},
journal= {arXiv preprint arXiv:2604.21242},
year = {2026}
}