# Strange duality on rational surfaces II: higher rank cases

## TL;DR

This paper advances the understanding of Le Potier's strange duality conjecture on rational surfaces, establishing new isomorphisms and injectivity results for higher rank sheaves, especially on $P^2$ and Hirzebruch surfaces.

## Contribution

It introduces an exact sequence relating different strange duality maps and proves several cases of the conjecture as isomorphisms or injections, extending previous results to higher ranks.

## Key findings

- Established exact sequences linking duality maps for all ranks.
- Proved $SD_{c^r_r,dH}$ is an isomorphism for $d=1,2$ on $P^2$.
- Showed $SD_{c_n^2,L}$ is an isomorphism for certain rational surfaces.

## Abstract

We study Le Potier's strange duality conjecture on a rational surface. We focus on the strange duality map $SD_{c_n^r,L}$ 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 $SD_{c_r^r,L}$ to $SD_{c^{r-1}_{r},L}$ and $SD_{c_r^r,L\otimes K_X}$ for all $r\geq1$ under some conditions on $X$ and $L$ which applies to a large number of cases on $\p^2$ or Hirzebruch surfaces . Also on $\mathbb{P}^2$ we show that for any $r>0$, $SD_{c^r_r,dH}$ is an isomorphism for $d=1,2$, injective for $d=3$ and moreover $SD_{c_3^3,rH}$ and $SD_{c_3^2,rH}$ are injective. At the end we prove that the map $SD_{c_n^2,L}$ ($n\geq2$) is an isomorphism for $X=\mathbb{P}^2$ or Fano rational ruled surfaces and $g_L=3$, and hence so is $SD_{c_3^3,L}$ as a corollary of our main result.