Reider-type theorems on normal surfaces via Bridgeland stability

TL;DR

This paper extends Reider-type theorems to normal surfaces using Bridgeland stability, covering cases with singularities and positive characteristic, and confirms optimal bounds for complex surfaces with rational double points.

Contribution

It generalizes Reider-type theorems to normal surfaces via Bridgeland stability, including singular and positive characteristic cases, and aligns with Fujita's conjecture for certain complex surfaces.

Findings

Reider-type theorems are valid for normal surfaces using Bridgeland stability.

Results hold in positive characteristic and when $ ext{ω}_X ensor L$ is not a line bundle.

Optimal bounds for global generation and very ampleness are confirmed for surfaces with rational double points.

Abstract

Using Langer's construction of Bridgeland stability conditions on normal surfaces, we prove Reider-type theorems generalizing the work done by Arcara-Bertram in the smooth case. Our results still hold in positive characteristic or when $\omega_X \otimes L$ is not necessarily a line bundle. They also hold when the dualizing sheaf is replaced by a variant arising from the theory of Du Bois complexes. For complex surfaces with at most rational double point singularities, we recover the optimal bounds for global generation and very ampleness as predicted by Fujita's conjecture.