Strange duality on rational surfaces

TL;DR

This paper proves Le Potier's strange duality conjecture for certain rational surfaces, specifically Hirzebruch surfaces, under specific conditions on the line bundle L, expanding the cases where the conjecture holds.

Contribution

The paper establishes the truth of the strange duality conjecture for rank 2 sheaves with trivial first Chern class on rational surfaces, particularly Hirzebruch surfaces, under new conditions.

Findings

Conjecture holds for ample L on Hirzebruch surfaces with e ≠ 1.

For e=1, conjecture holds when L=aG+bF with b ≥ a + [a/2].

Results extend the validity of the conjecture to broader classes of rational surfaces.

Abstract

We study Le Potier's strange duality conjecture on a rational surface. We focus on the case involving the moduli space of rank 2 sheaves with trivial first Chern class and second Chern class 2, and the moduli space of 1-dimensional sheaves with determinant $L$ and Euler characteristic 0. We show the conjecture for this case is true under some suitable conditions on $L$, which applies to $L$ ample on any Hirzebruch surface $\Sigma_e:=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(e))$ except for $e=1$. When $e=1$, our result applies to $L=aG+bF$ with $b\geq a+[a/2]$, where $F$ is the fiber class, $G$ is the section class with $G^2=-1$ and $[a/2]$ is the integral part of $a/2$.