Representations in Strange Duality: Hilbert Schemes paired with higher rank spaces

TL;DR

This paper investigates the strange duality conjecture for pairs of moduli spaces of sheaves on P^2, including the Hilbert scheme of points and higher rank spaces, by computing sections of theta bundles and applying the finite Quot scheme method.

Contribution

It constructs higher rank moduli spaces using exceptional bundle resolutions and verifies the strange duality conjecture through explicit section computations and the finite Quot scheme approach.

Findings

Computed sections of theta bundles as SL(X) representations.

Constructed higher rank moduli spaces with exceptional bundle resolutions.

Applied the finite Quot scheme method to verify duality.

Abstract

We examine a sequence of examples of pairs of moduli spaces of sheaves on $\mathbb P^2$ where Le Potier's strange duality is expected to hold. One of the moduli spaces in these pairs is the Hilbert scheme of two points. We compute the sections of the relevant theta bundle as a representation of $SL(X)$, where $\mathbb P^2=\mathbb P(X)$. For the higher rank space, we construct a moduli space using the resolution of exceptional bundles from Coskun, Huizenga, and Woolf. We compute a subspace of the sections of the theta bundle which is dual to the sections on the Hilbert scheme. In the second part, we use a result from Goller and Lin to rigorously apply the "finte Quot scheme method", introduced in Marian and Oprea, and Bertram, Goller, and Johnson. This requires us to prove that the kernels in appearing in the Quot scheme have the appropriate resolution by exceptional bundles.