Local BPS Invariants: Enumerative Aspects and Wall-Crossing

TL;DR

This paper investigates BPS invariants for local del Pezzo surfaces, computes Poincare polynomials for certain curve classes, and proposes a divisibility conjecture linking these polynomials to projective spaces, advancing enumerative geometry.

Contribution

It provides explicit calculations of Poincare polynomials for moduli spaces and introduces a conjecture relating these to projective space polynomials, suggesting new directions in log BPS number research.

Findings

Computed Poincare polynomials for curve classes with genus ≤ 2

Formulated a divisibility conjecture for these polynomials

Proposed a connection to log BPS numbers

Abstract

We study the BPS invariants for local del Pezzo surfaces, which can be obtained as the signed Euler characteristic of the moduli spaces of stable one-dimensional sheaves on the surface $S$. We calculate the Poincare polynomials of the moduli spaces for the curve classes $\beta$ having arithmetic genus at most 2. We formulate a conjecture that these Poincare polynomials are divisible by the Poincare polynomials of $((-K_S).\beta-1)$-dimensional projective space. This conjecture motivates upcoming work on log BPS numbers.