The Kleiman–Piene conjecture and node polynomials for plane curves in P³

For a relative effective divisor C on a smooth projective family of surfaces q: S→ B, we consider the locus in B over which the fibres of C are δ-nodal curves. We prove a conjecture by Kleiman and Piene on the universality of an enumerating cycle on this locus. We propose a bivariant class γ(C) ∈ A^∗(B) motivated by the BPS calculus of Pandharipande and Thomas, and show that it can be expressed universally as a polynomial in classes of the form q∗(c1(O(C))ac1(TS/B)bc2(TS/B)c). Under an ampleness assumption, we show that γ(C) ∩ [B] is the class of a natural effective cycle with support equal to the closure of the locus of δ-nodal curves. Finally, we apply our method to calculate node polynomials for plane curves intersecting general lines in P³. We verify our results using nineteenth century geometry of Schubert.