Strange duality on rational surfaces

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 ${\mathrm{\Sigma }}_e:=\mathbb{P}({\mathcal{O}}_{{\mathbb{P}}^1}\oplus {\mathcal{O}}_{{\mathbb{P}}^1}(e))$ except for $e=1$. When $e=1$, our result applies to $L=aG+bF$ with $b\geq a+[a/2]$,where$F$isthefiberclass,$G$isthesectionclasswith$G^2=-1$and$[a/2]$istheintegralpartof$a/2$.