Strange duality on rational surfaces II: higher rank cases

We study Le Potier’s strange duality conjecture on a rational surface. We focus on the strange duality map $S{D}_{c_n^r,L}$ which involves the moduli space of rank $r$ sheaves with trivial first Chern class and second Chern class $n$, and the moduli space of 1-dimensional sheaves with determinant $L$ and Euler characteristic 0. We show there is an exact sequence relating the map $S{D}_{c_r^r,L}$ to $S{D}_{c_r^{r-1},L}$ and $S{D}_{c_r^r,L\otimes K_X}$ for all $r\ge 1$ under some conditions on $X$ and $L$ which applies to a large number of cases on ${\text{p}}^2$ or Hirzebruch surfaces . Also on ${\mathbb{P}}^2$ we show that for any $r>0$, $S{D}_{c_r^r,dH}$ is an isomorphism for $d=1,2$, injective for $d=3$ and moreover $S{D}_{c_3^3,rH}$ and $S{D}_{c_3^2,rH}$ are injective. At the end we prove that the map $S{D}_{c_n^2,L}$ ($n\ge 2$) is an isomorphism for $X={\mathbb{P}}^2$ or Fano rational ruled surfaces and $g_L=3$, and hence so is $S{D}_{c_3^3,L}$ as a corollary of our main result.