Enumeration of rational curves in a moving family of $\mathbb{P}^2$

TL;DR

This paper develops a recursive formula to count rational degree d curves in P^3 lying in a P^2, extending classical enumerative geometry questions and supporting conjectures about polynomial enumerativity thresholds.

Contribution

It introduces a recursive method for enumerating rational curves in a moving family within P^3, generalizing classical fixed-family counts and connecting to nodal curve conjectures.

Findings

Recursive formula for counting rational curves in P^3 within P^2.

Consistency with previous results on nodal curve counts by T. Laarakker.

Evidence supporting the conjecture on polynomial enumerativity thresholds.

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 $\delta$-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 $d \geq 1 + [\frac{\delta}{2}]$, which is analogous to the {G}\"ottsche threshold for counting nodal curves in $\mathbb{P}^2$.