Strange duality on 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 $a/2$.