Some Betti numbers of the moduli of 1-dimensional sheaves on ${\mathbb{P}}^2$

Let $M(d,\chi )$ with $(d,\chi )=1$ be the moduli space of semistable sheaves on ${\mathbb{P}}^2$ supported on curves of degree $d$ and with Euler characteristic $\chi$. The cohomology ring $H^{\ast }(M(d,\chi ),\mathbb{Z})$ of $M(d,\chi )$ is isomorphic to its Chow ring $A^{\ast }(M(d,\chi ))$ by Markman’s result. W. Pi and J. Shen have described a minimal generating set of $A^{\ast }(M(d,\chi ))$ consisting of $3d-7$ generators, which they also showed to have no relation in $A^{\ge d-2}(M(d,\chi ))$. We compute the two Betti numbers $b_{2(d-1)}$ and $b_{2d}$ of $M(d,\chi )$ and as a corollary we show that the generators given by Pi-Shen have no relations in $A^{\ge d-1}(M(d,\chi ))$ but do have three linearly independent relations in $A^d(M(d,\chi ))$.