Local BPS Invariants: Enumerative Aspects and Wall-Crossing

Abstract

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 Poincaré polynomials of the moduli spaces for the curve classes $\beta $ having arithmetic genus at most 2. We formulate a conjecture that these Poincaré polynomials are divisible by the Poincaré polynomials of $((-K_S).\beta -1)$-dimensional projective space. This conjecture motivates the upcoming work on log BPS numbers [8].