Poincar\'e polynomials of moduli spaces of one-dimensional sheaves on the projective plane

Let $M_{\beta}$ denote the moduli space of stable one-dimensional sheaves on a del Pezzo surface $S$, supported on curves of class $\beta$ with Euler characteristic one. We show that the divisibility property of the Poincar\'e polynomial of $M_{\beta}$, proposed by Choi-van Garrel-Katz-Takahashi follows from Bousseau's conjectural refined sheaves/Gromov-Witten correspondence. Since this correspondence is known for $S=\mathbb{P}^2$, our result proves Choi-van Garrel-Katz-Takahashi's conjecture in this case. For $S=\mathbb{P}^2$, our proof also introduces a novel approach to computing the Poincar\'e polynomials using Gromov-Witten invariants of local $\mathbb{P}^2$ and a local elliptic curve. Specifically, we compute the Poincar\'e polynomials of $M_{d}$ with degrees $d\leq 16$ and derive a closed formula for the leading Betti numbers $b_i(M_d)$ with $d\geq 6$ and $i\leq 4d-22$. We also propose a conjectural formula for the leading Betti numbers $b_i(M_d)$ with $d\geq 4$ and $i\leq 6d-20$. In the Appendix (by M. Moreira), a more general conjecture concerning the higher range Betti numbers of $M_{d}$ is presented, along with another conjecture that involves refinements from the perverse/Chern filtration.