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

Abstract: 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 $H^*(M(d,\chi),\mathbb{Z})$ of $M(d,\chi)$ is isomorphic to its Chow ring $A^*(M(d,\chi))$ by Markman's result. W. Pi and J. Shen have described a minimal generating set of $A^*(M(d,\chi))$ consisting of $3d-7$ generators, which they also showed to have no relation in $A^{\geq d-2}(M(d,\chi))$. 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 $A^{\geq d-1}(M(d,\chi))$ but do have three linearly independent relations in $A^d(M(d,\chi))$.