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

Let $M(d,χ)$ be the moduli space of semistable sheaves of rank 0, Euler characteristic $χ$ and first Chern class $dH (d>0)$, with $H$ the hyperplane class in $\mathbb{P}^2$. We give a description of $M(d,χ)$, viewing each sheaf as a class of matrices with entries in $\bigoplus_{i\geq0}H^0(\mathcal{O}_{\mathbb{P}^2}(i))$. 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 $\mathbb{P}^2.$ 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.

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