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Symmetry, Integrability and Geometry: Methods and Applications, 2022, Volume 18, 059, 23 pp.
DOI: https://doi.org/10.3842/SIGMA.2022.059 (Mi sigma1855) 		 
Node Polynomials for Curves on Surfaces

Steven Kleimana, Ragni Pieneb

a Room 2-172, Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
b Department of Mathematics, University of Oslo, PO Box 1053, Blindern, NO-0316 Oslo, Norway
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DOI: https://doi.org/10.3842/SIGMA.2022.059
Abstract: We complete the proof of a theorem we announced and partly proved in [Math. Nachr. 271 (2004), 69–90, math.AG/0111299]. The theorem concerns a family of curves on a family of surfaces. It has two parts. The first was proved in that paper. It describes a natural cycle that enumerates the curves in the family with precisely $r$ ordinary nodes. The second part is proved here. It asserts that, for $r\le 8$, the class of this cycle is given by a computable universal polynomial in the pushdowns to the parameter space of products of the Chern classes of the family.
Keywords: enumerative geometry, nodal curves, nodal polynomials, Bell polynomials, Enriques diagrams, Hilbert schemes.
Received: February 24, 2022; in final form July 28, 2022; Published online August 2, 2022
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Document Type: Article
MSC: 14N10, 14C20, 14H40, 14K05
Language: English
Citation: Steven Kleiman, Ragni Piene, “Node Polynomials for Curves on Surfaces”, SIGMA, 18 (2022), 059, 23 pp.
Citation in format AMSBIB
\Bibitem{KlePie22}
\by Steven~Kleiman, Ragni~Piene
\paper Node Polynomials for Curves on Surfaces
\jour SIGMA
\yr 2022
\vol 18
\papernumber 059
\totalpages 23
\mathnet{http://mi.mathnet.ru/sigma1855}
\crossref{https://doi.org/10.3842/SIGMA.2022.059}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4459536}
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