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 $\bigoplus_{i\geq0}H^0(\mathcal{O}_{\mathbb{P}^2}(i))$. 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. 
Yao Yuan
math.AG