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

TL;DR

This paper computes specific Betti numbers of the moduli space of 1-dimensional sheaves on the projective plane, revealing the structure of its cohomology and Chow rings, and clarifying relations among generators.

Contribution

It provides explicit calculations of Betti numbers for the moduli space, advancing understanding of its cohomological and algebraic structure.

Findings

Computed Betti numbers $b_{2(d-1)}$ and $b_{2d}$ for the moduli space.

Showed generators have no relations in $A^{ geq d-1}$ but three relations in $A^d$.

Connected Betti number calculations to relations among Chow ring generators.

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 $A^d(M(d,\chi))$.