In this paper, we study the asymptotic behaviors of the Betti numbers and Picard numbers of the moduli space Mβ,χ of one-dimensional sheaves supported in a curve class β on S with Euler characteristic χ. We determine the intersection cohomology Betti numbers of Mβ,χ when S is a del Pezzo surface and β is sufficiently positive. As an application, we formulate a $P = C$ conjecture regarding the refined BPS invariants for local del Pezzo surfaces.
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Si et al. (2024) studied this question.
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