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Node polynomials for curves on surfaces

Thomas Dedieu

TL;DR

This work develops explicit node polynomials for counting δ-nodal curves on smooth surfaces in a fixed linear system, expressing counts as N_δ = (1/δ!) P_δ(a_1,...,a_δ) where the a_i are universal polynomials in the Chern numbers (d = L^2, k = L·K_S, s = K_S^2, x = c_2(S)). The core method builds an inductive framework via jet bundles and blow-ups, introducing induced pairs (F_i/X_i, D_i) and a recursive relation that reduces δ-nodal counts to lower-order data, ultimately yielding closed forms in terms of Bell polynomials. The text also connects these node polynomials to universality conjectures (Göttsche), presents explicit computations for curves on fixed surfaces and plane curves in solids, and places the results in the broader contexts of equisingularity, Enriques diagrams, and Kazarian’s Thom polynomials. Practical impact includes explicit, computable formulas for nodal counts across broad geometries, enabling precise enumerations in algebraic geometry and informing related enumerative theories such as those arising in Calabi–Yau and Fano contexts.

Abstract

This text is a presentation of a set of formulae, first found by Vainsencher (for $δ\leq 6$) and shortly after improved by Kleiman and Piene, counting $δ$-nodal curves in a complete linear system on a smooth surface, if $δ\leq 8$ and the corresponding line bundle is sufficiently positive. We also discuss a complement by Qviller, and related results due to Kazarian, Ohmoto, and others.

Node polynomials for curves on surfaces

TL;DR

This work develops explicit node polynomials for counting δ-nodal curves on smooth surfaces in a fixed linear system, expressing counts as N_δ = (1/δ!) P_δ(a_1,...,a_δ) where the a_i are universal polynomials in the Chern numbers (d = L^2, k = L·K_S, s = K_S^2, x = c_2(S)). The core method builds an inductive framework via jet bundles and blow-ups, introducing induced pairs (F_i/X_i, D_i) and a recursive relation that reduces δ-nodal counts to lower-order data, ultimately yielding closed forms in terms of Bell polynomials. The text also connects these node polynomials to universality conjectures (Göttsche), presents explicit computations for curves on fixed surfaces and plane curves in solids, and places the results in the broader contexts of equisingularity, Enriques diagrams, and Kazarian’s Thom polynomials. Practical impact includes explicit, computable formulas for nodal counts across broad geometries, enabling precise enumerations in algebraic geometry and informing related enumerative theories such as those arising in Calabi–Yau and Fano contexts.

Abstract

This text is a presentation of a set of formulae, first found by Vainsencher (for ) and shortly after improved by Kleiman and Piene, counting -nodal curves in a complete linear system on a smooth surface, if and the corresponding line bundle is sufficiently positive. We also discuss a complement by Qviller, and related results due to Kazarian, Ohmoto, and others.

Paper Structure

This paper contains 53 sections, 35 theorems, 212 equations, 2 tables.

Table of Contents

  1. Node polynomials for curves on surfaces
  2. Introduction and statement of results
  3. The inductive procedure
  4. The induced pairs
  5. The recursive formula
  6. The computation --- Proof of the main theorem
  7. Example \ref{['pedestre']} continued
  8. Proof of Theorem \ref{['t:KP']} when $\delta \leqslant 7$
  9. Discussion of Theorem \ref{['t:KP']} when $\delta = 8$
  10. Comparison with Vainsencher's computation
  11. Applications
  12. Curves in a linear system on a fixed surface
  13. Plane curves in a solid in $\mathbf{P}^4$
  14. Computations for Formula \ref{['f:solids']}
  15. Equisingularity classes
  16. ...and 38 more sections

Key Result

Theorem (1.3)

Let $\delta\leqslant 8$ be an integer, and assume that $L$ may be written as $L = mA+B$ for an integer $m\geqslant 3\delta$ and line bundles $A$ and $B$, such that $A$ is very ample and $B$ is globally generated.if $\delta=8$, one needs to add the technical assumption that among curves in $|L|$ with

Theorems & Definitions (75)

  • Theorem (1.3): Vainsencher, Kleiman--Piene
  • Theorem (1.6): Vainsencher, Kleiman--Piene
  • Theorem (1.7): Vainsencher, Kleiman--Piene
  • Lemma (2.16)
  • proof
  • Definition (2.17)
  • Proposition (2.19)
  • proof
  • Proposition (2.20)
  • proof
  • ...and 65 more