Rational Cuspidal Curves in a moving family of $\mathbb{P}^2$
Ritwik Mukherjee, Rahul Kumar Singh
TL;DR
The paper advances the classical problem of counting rational cuspidal curves by extending it to a moving family of planes: it counts genus-zero, degree $d$ planar cuspidal curves in $\mathbb{P}^3$ meeting specified incidences. It expresses the count as the Euler class of the bundle $\mathbb{L}^* \otimes \mathrm{ev}^*W$ over the planar moduli space and accounts for a degenerate boundary contribution from ghost bubbles via dynamic intersection methods. A recursive formula for $C_d^{\mathbb{P}^3,\mathrm{Planar}}(r,s)$ is derived in terms of tautological intersections $\Phi_d(i,j,r,s,\theta)$ and known characteristic numbers of planar curves, with explicit level-by-level (0,1,2) expressions. The authors validate their framework through low-degree checks against established nodal-count data and provide a Mathematica implementation, thereby connecting moving-family characteristic numbers to classical planar counts and extending prior results of Ran, Pandharipande, Zinger, and others.
Abstract
In this paper we obtain a formula for the number of rational degree d curves in $\mathbb{P}^3$ having a cusp, whose image lies in a $\mathbb{P}^2$ and that passes through $r$ lines and $s$ points (where $r + 2s = 3d + 1$). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in $\mathbb{P}^2$, which has been studied earlier by Z. Ran, R. Pandharipande and A. Zinger. We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger and I. Biswas, S. D'Mello, R. Mukherjee and V. Pingali. We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author, where they compute the characteristic number of $δ$-nodal planar curves in $\mathbb{P}^3$ with one cusp (for $δ\leq 2$).