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

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.