The Kleiman–Piene conjecture and node polynomials for plane curves in $$\mathbb {P}^3$$ P 3

For a relative effective divisor $$\mathcal {C}$$ on a smooth projective family of surfaces $$q:\mathcal {S}\rightarrow B$$ , we consider the locus in B over which the fibres of $$\mathcal {C}$$ are $$\delta $$ -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 $$\gamma (\mathcal {C})\in A^*(B)$$ 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 $$q_*(c_1(\mathcal {O}(\mathcal {C}))^a c_1(T_{\mathcal {S}/B})^b c_2(T_{\mathcal {S}/B})^c)$$ . Under an ampleness assumption, we show that $$\gamma (\mathcal {C})\cap [B]$$ is the class of a natural effective cycle with support equal to the closure of the locus of $$\delta $$ -nodal curves. Finally, we apply our method to calculate node polynomials for plane curves intersecting general lines in $$\mathbb {P}^3$$ . We verify our results using nineteenth century geometry of Schubert.