Two New Extensions of Reider's Theorem on Algebraic Surfaces

TL;DR

This paper extends Reider's Theorem to new contexts, providing nefness criteria for linear series on blow-ups and estimates for the ample cone of Hilbert schemes, using Bridgeland stability.

Contribution

It introduces two new extensions of Reider's Theorem, connecting stability conditions with nefness and the geometry of Hilbert schemes of surfaces.

Findings

Reider-type inequalities imply nefness of certain linear series on blow-ups.

Sharp estimates for the ample cone of Hilbert schemes are obtained.

The proofs utilize Bridgeland stability and determinant line bundles.

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.