### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 059, 23 pages      arXiv:2202.11611      https://doi.org/10.3842/SIGMA.2022.059
Contribution to the Special Issue on Enumerative and Gauge-Theoretic Invariants in honor of Lothar Göttsche on the occasion of his 60th birthday

### Node Polynomials for Curves on Surfaces

Steven Kleiman a and Ragni Piene b
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

Received February 24, 2022, in final form July 28, 2022; Published online August 02, 2022

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.

Key words: enumerative geometry; nodal curves; nodal polynomials; Bell polynomials; Enriques diagrams; Hilbert schemes.

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