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Enumeration of Rational Cuspidal Curves via the WDVV equation

Indranil Biswas, Apratim Choudhury, Ritwik Mukherjee, Anantadulal Paul

TL;DR

The paper develops a conjectural WDVV-based framework to enumerate rational plane curves with cusps by extending Kontsevich’s recursion. It defines and interrelates $\mathsf{N}^d_m$, $\mathsf{C}^d_m(n)$, and $\mathsf{T}^{d_1,d_2}_{m_1,m_2}(n)$, deriving a detailed cusp-recursion formula that couples cusp counts to ordinary rational counts and two-component tangency data, with a key conjectural tangency degeneration. It then provides algorithms to compute the tangency numbers via tautological intersections, applies the method to compute quartics with an $E_6$ singularity, and verifies the results against known data from Ran, Pandharipande, Zinger, and Ernström–Kennedy. The work also offers explicit branched-tangency cases and low-degree checks, along with a Mathematica code repository, supporting the validity and utility of the approach. Overall, the paper offers a coherent framework to enumerate cuspidal and related singular plane curves through WDVV, while highlighting conjectural aspects and paving the way for generalizations to other surfaces and singularities.

Abstract

We give a conjectural formula for the characteristic number of rational cuspidal curves in the projective plane by extending the idea of Kontsevich's recursion formula (namely, pulling back the equality of two divisors in the four pointed moduli space). The key geometric input that is needed here is that in the closure of rational cuspidal curves, there are two component rational curves which are tangent to each other at the nodal point. While this fact is geometrically quite believable, we haven't as yet proved it; hence our formula is for the moment conjectural. The answers that we obtain agree with what has been computed earlier Ran, Pandharipande, Zinger and Ernstrom and Kennedy. We extend this technique (modulo another conjecture) to obtain the characteristic number of rational quartics with an E6 singularity.

Enumeration of Rational Cuspidal Curves via the WDVV equation

TL;DR

The paper develops a conjectural WDVV-based framework to enumerate rational plane curves with cusps by extending Kontsevich’s recursion. It defines and interrelates , , and , deriving a detailed cusp-recursion formula that couples cusp counts to ordinary rational counts and two-component tangency data, with a key conjectural tangency degeneration. It then provides algorithms to compute the tangency numbers via tautological intersections, applies the method to compute quartics with an singularity, and verifies the results against known data from Ran, Pandharipande, Zinger, and Ernström–Kennedy. The work also offers explicit branched-tangency cases and low-degree checks, along with a Mathematica code repository, supporting the validity and utility of the approach. Overall, the paper offers a coherent framework to enumerate cuspidal and related singular plane curves through WDVV, while highlighting conjectural aspects and paving the way for generalizations to other surfaces and singularities.

Abstract

We give a conjectural formula for the characteristic number of rational cuspidal curves in the projective plane by extending the idea of Kontsevich's recursion formula (namely, pulling back the equality of two divisors in the four pointed moduli space). The key geometric input that is needed here is that in the closure of rational cuspidal curves, there are two component rational curves which are tangent to each other at the nodal point. While this fact is geometrically quite believable, we haven't as yet proved it; hence our formula is for the moment conjectural. The answers that we obtain agree with what has been computed earlier Ran, Pandharipande, Zinger and Ernstrom and Kennedy. We extend this technique (modulo another conjecture) to obtain the characteristic number of rational quartics with an E6 singularity.
Paper Structure (13 sections, 1 theorem, 71 equations, 9 figures)

This paper contains 13 sections, 1 theorem, 71 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Recursive formula for cuspidal curves
  3. Kontsevich's recursion formula
  4. Derivation of recursive formula for cuspidal curves
  5. Intersection of Tautological Classes
  6. Computation of two component tangential curves
  7. Rational plane quartic with $E_6$ singularity via WDVV
  8. Rational cuspidal curves with branched tangency
  9. Low degree checks
  10. Verification with tangencies
  11. Verification with the result by Ernström and Kennedy
  12. Future directions
  13. Acknowledgement

Key Result

Lemma 5.1

The intersection numbers $\Phi_{d}(1, \alpha, m)$ are given by the recursive formula

Figures (9)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 4 more figures

Theorems & Definitions (2)

  • Lemma 5.1
  • proof