Local BPS Invariants: Enumerative Aspects and Wall-Crossing

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 β having arithmetic genus at most 2. We formulate a conjecture that these Poincaré polynomials are divisible by the Poincaré polynomials of ((−KS).β−1)-dimensional projective space. This conjecture motivates the upcoming work on log BPS numbers [8].