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Selecta Mathematica

, Volume 24, Issue 5, pp 4917–4959 | Cite as

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

  • Ties Laarakker
Open Access
Article

Abstract

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.

Mathematics Subject Classification

Primary 14N10 Secondary 14C20 14N35 14N15 

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.UtrechtThe Netherlands

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