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 $q:\mathcal{S}\to B$, 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 $\gamma (\mathcal{C})\in A^{\ast }(B)$ 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}$/B})^c)$\mathrm{.}Underanamplenessassumption,weshowthat$\gamma($\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.