  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.
