Enumeration of Rational Cuspidal Curves via the WDVV equation

TL;DR

This paper proposes a conjectural formula for counting rational cuspidal curves in the projective plane using the WDVV equation, extending Kontsevich's recursion, and verifies it against known results, also applying it to rational quartics with E6 singularity.

Contribution

It introduces a new conjectural formula for enumerating rational cuspidal curves via the WDVV equation, extending Kontsevich's recursion to include cuspidal conditions.

Findings

The conjectural formula matches previous computations by other researchers.

The technique is extended to count rational quartics with E6 singularity.

The geometric assumption about tangent components in the closure of cuspidal curves is crucial.

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.