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

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 univerality 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 will apply our method to calculate node polynomials for plane curves intersecting general lines in $\mathbb{P}^3$. We verify our results using 19th century geometry of Schubert.