## Abstract

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 P^{3}. We verify our results using nineteenth century geometry of Schubert.

Original language | English |
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Pages (from-to) | 4917-4959 |

Number of pages | 43 |

Journal | Selecta Mathematica, New Series |

Volume | 24 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 Nov 2018 |

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