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 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.