Enumeration of rational curves in a moving family of $\mathbb{P}^2$
Ritwik Mukherjee, Anantadulal Paul, Rahul Kumar Singh
TL;DR
This work develops a family version of enumerating rational degree $d$ curves in $\mathbb{P}^3$ whose images lie in a moving plane $\mathbb{P}^2_\eta$, encapsulated in the invariant $N_d^{\mathbb{P}^3,\mathrm{Planar}}(r,s)$. A fibre-bundle moduli framework over the dual projective space $\hat{\mathbb{P}}^3$ is used to define intersection-theoretic counts with constraints for passing through lines and points, leading to a recursive formula that computes these numbers for all $d\ge 2$. The recursion hinges on a WDVV-type decomposition into one- and two-component contributions, with an auxiliary class $B_{d_1,d_2}$ expressed through products of planar counts of smaller degree; explicit base cases $d=1,2$ are provided. Low-degree checks up to $d=6$ align with T. Laarakker’s results on $\delta$-nodal planar curves in $\mathbb{P}^3$, supporting the conjectured enumerativity threshold $d \ge 1+\big\lfloor \frac{\delta}{2} \big\rfloor$ (analogous to Göttsche’s threshold in $\mathbb{P}^2$). These findings reinforce the expectation that the planar invariants genuinely count geometric objects, with degenerations (e.g., double lines) appearing beyond the threshold (e.g., at $d=7$).
Abstract
We obtain a recursive formula for the number of rational degree $d$ curves in $\mathbb{P}^3$, whose image lies in a $\mathbb{P}^2$, passing through $r$ lines and $s$ points, where $r + 2s = 3d+2$. This can be viewed as a family version of the classical question of counting rational curves in $\mathbb{P}^2$. We verify that our numbers are consistent with those obtained by T. Laarakker, where he studies the parallel question of counting $δ$-nodal degree $d$ curves in $\mathbb{P}^3$ whose image lies inside a $\mathbb{P}^2$. Our numbers give evidence to support the conjecture, that the polynomials obtained by T. Laarakker are enumerative when $d \geq 1 + [\fracδ{2}]$, which is analogous to the {G}öttsche threshold for counting nodal curves in $\mathbb{P}^2$.