Gaeta Resolutions and Strange Duality Over Rational Surfaces

Abstract Over 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. 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 Quot schemes are finite and reduced, allowing them to be enumerated using the authors’ previous paper.