Strange duality on $\mathbb{P}^2$ via quiver representations

TL;DR

This paper investigates Le Potier's strange duality conjecture on the projective plane, demonstrating isomorphisms and injectivity of the duality map for specific cases using quiver representation theory.

Contribution

It applies quiver representation techniques to prove cases of the strange duality conjecture on , establishing isomorphisms for certain parameters and injectivity generally.

Findings

SD_{c^r_n,d} is an isomorphism for r=n, r=n-1, or d

SD_{c^r_n,d} is injective for all n, r, d

The approach links quiver representations to moduli spaces of sheaves on 

Abstract

We study Le Potier's strange duality conjecture on $\mathbb{P}^2$. We focus on the strange duality map $SD_{c_n^r,d}$ 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 $\mathcal{O}_{\mathbb{P}^2}(d)$ and Euler characteristic 0. By using tools in quiver representation theory, we show that $SD_{c^r_n,d}$ is an isomorphisms for $r=n$ or $r=n-1$ or $d\leq 3$, and in general $SD_{c^r_n,d}$ is injective for any $n\geq r>0$ and $d>0$.