Moduli spaces of semistable sheaves of dimension 1 on ${\mathbb{P}}^2$

Let $M(d,\chi )$ be the moduli space of semistable sheaves of rank 0, Euler characteristic $\chi$ and first Chern class $dH(d>0)$, with $H$ the hyperplane class in ${\mathbb{P}}^2$. We give a description of $M(d,\chi )$, viewing each sheaf as a class of matrices with entries in ${⨁}_{i\ge 0}{H}^0({\mathcal{O}}_{{\mathbb{P}}^2}(i))$. We show that there is a big open subset of $M(d,1)$ isomorphic to a projective bundle over an open subset of a Hilbert scheme of points on ${\mathbb{P}}^2\mathrm{.}$ Finally we compute the classes of M(4,1), M(5,1) and M(5,2) in the Grothendieck group of varieties, especially we conclude that M(5,1) and M(5,2) are of the same class.