Gaeta resolutions and strange duality over rational surfaces

TL;DR

This paper establishes foundational results on Gaeta resolutions of coherent sheaves over rational surfaces and proves injectivity of the strange duality morphism in specific cases, advancing understanding of these geometric phenomena.

Contribution

It introduces new results on Gaeta resolutions over rational surfaces with exceptional sequences and proves injectivity of the strange duality map for certain classes.

Findings

Locus of non-Gaeta-resolvable semistable sheaves has codimension at least 2.

Strange duality morphism is injective for specific orthogonal classes with rank one and positive Chern class.

Quot schemes relevant to the problem are finite and reduced, enabling enumeration.

Abstract

Over the projective plane and at most two-step blowups of Hirzebruch surfaces, where there are strong full exceptional sequences of line bundles, we obtain foundational results about Gaeta resolutions of coherent sheaves by these line bundles. Under appropriate conditions, we show the locus of semistable sheaves not admitting Gaeta resolutions has codimension at least 2. We then study Le Potier's strange duality conjecture. Over these surfaces, for two orthogonal numerical classes where one has rank one and the other has sufficiently positive first Chern class, we show that the strange morphism is injective. The main step in the proof is to use Gaeta resolutions to show that certain relevant Quot schemes are finite and reduced, allowing them to be enumerated using the authors' previous paper.