Node polynomials for curves on surfaces

We complete the proof of a theorem we announced and partly proved in [16]. The theorem concerns a family of curves on a family of surfaces. It has two parts. The first was proved in [16]. It describes a natural cycle that enumerates the curves in the family with precisely r ordinary nodes. The second part is proved here. It asserts that, for r ≤ 8, the class of this cycle is given by a computable universal polynomial in the pushdowns to the parameter space of products of the Chern classes of the family.