An observation on the Poincar\'e polynomials of moduli spaces of one-dimensional sheaves

TL;DR

This paper observes a divisibility pattern in the Poincaré polynomials of certain moduli spaces of one-dimensional sheaves on projective plane for small degrees, revealing regularities in their topological invariants.

Contribution

It identifies a divisibility pattern and regular behavior in Poincaré polynomials of moduli spaces and related geometric objects for degrees up to six.

Findings

Poincaré polynomial of $M_{dm+1}(P^2)$ divisible by that of $P^{3d-1}$ for $0<d extless 7$

Regular behavior observed in the difference of Poincaré polynomials for specific geometric spaces at $d=4,5,6$

Patterns suggest underlying geometric or topological structures in these moduli spaces

Abstract

We notice that for $0<d\le 6$ the Poincar\'e polynomial of Simpson moduli space $M_{dm + 1}(\mathbb P_2)$ is divisible by the Poincar\'e polynomial of the projective space $\mathbb P_{3d-1}$. A somehow regular behaviour of the difference of the Poincar\'e 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 $d=4, 5, 6$.