Moduli spaces of 1-dimensional semi-stable sheaves and Strange duality on $\mathbb{P}^2$

TL;DR

This paper proves Le Potier's strange duality conjecture for specific pairs of moduli spaces of sheaves on the projective plane, advancing understanding of dualities in algebraic geometry.

Contribution

It establishes the validity of the strange duality conjecture for pairs involving rank 2 and 1-dimensional sheaves on b^2, a case previously unresolved.

Findings

Strange duality holds for the pair (W(2,0,2), M(d,0)) with d > 0.

The proof confirms the conjecture for a new class of sheaf moduli spaces.

Results contribute to the broader understanding of dualities in algebraic geometry.

Abstract

We study Le Potier's strange duality conjecture on $\mathbb{P}^2$. We show the conjecture is true for the pair ($W(2,0,2),~M(d,0)$) with $d>0$, where $W(2,0,2)$ is the moduli space of semistable sheaves of rank 2, zero first Chern class and second Chern class 2, and $M(d,0)$ is the moduli space of 1-dimensional semistable sheaves of first Chern class $dH$ and Euler characteristic 0.