Asymptotic Behaviors of Moduli of One-dimensional Sheaves on Surfaces

TL;DR

This paper investigates the asymptotic properties of Betti and Picard numbers of moduli spaces of one-dimensional sheaves on surfaces, providing explicit calculations for del Pezzo surfaces and proposing a conjecture related to BPS invariants.

Contribution

It determines the intersection cohomology Betti numbers for moduli spaces on del Pezzo surfaces and introduces a new conjecture connecting BPS invariants with these geometric structures.

Findings

Explicit Betti number formulas for moduli spaces on del Pezzo surfaces

Formulation of a $P = C$ conjecture for refined BPS invariants

Insights into the asymptotic behavior of moduli space invariants

Abstract

In this paper, we study the asymptotic behaviors of the Betti numbers and Picard numbers of the moduli space $M_{\beta,\chi}$ of one-dimensional sheaves supported in a curve class $\beta$ on $S$ with Euler characteristic $\chi$. We determine the intersection cohomology Betti numbers of $M_{\beta,\chi}$ when $S$ is a del Pezzo surface and $\beta$ is sufficiently positive. As an application, we formulate a $P = C$ conjecture regarding the refined BPS invariants for local del Pezzo surfaces.