Local BPS Invariants: Enumerative Aspects and Wall-Crossing

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 the upcoming work on log BPS numbers [8].