An observation on the Poincaré polynomials of moduli spaces of one-dimensional sheaves

Abstract

We notice that for $0<d\le 6$ the Poincaré polynomial of Simpson moduli space $M_{dm+1}({\mathbb{P}}_2)$ is divisible by the Poincaré polynomial of the projective space ${\mathbb{P}}_{3d-1}$. A somehow regular behaviour of the difference of the Poincaré polynomials of the Hilbert scheme of $\frac{(d-2)(d-1)}{2}$ points on ${\mathbb{P}}_2$ and the moduli space of Kronecker modules $N(3;d-2,d-1)$ is noticed for $d=4,5,6$.