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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.

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