Le Potier's strange duality, quot schemes, and multiple point formulas for del Pezzo surfaces

TL;DR

This paper investigates Le Potier's strange duality on del Pezzo surfaces, employing quot schemes and multiple point formulas to analyze the structure and dimensions of moduli spaces of sheaves, providing new evidence for the duality's properties.

Contribution

It introduces novel methods using quot schemes and multiple point formulas to study the strange duality on del Pezzo surfaces, including new bounds and injectivity results.

Findings

Lower bound on the rank of the strange duality map

Evidence for injectivity when n ≤ 7

Finite and reduced quot schemes for projective plane cases

Abstract

We study Le Potier's strange duality on del Pezzo surfaces using quot schemes to construct independent sections of theta line bundles on moduli spaces of sheaves, one of which is the Hilbert scheme of n points. For n at most 7, we use multiple point formulas to count the length of the quot scheme, which agrees with the dimension of the space of sections on the Hilbert scheme. When the surface is the projective plane and n is arbitrary, we use nice resolutions of general stable sheaves to show that the quot schemes that arise are finite and reduced. Combining our results, we obtain a lower bound on the rank of the strange duality map, as well as evidence that the map is injective when n is at most 7.