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

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

10 pages. Comments are welcome

References in corpus (1)

• Tautological integrals on curvilinear Hilbert schemes

Cited by in corpus (3)

• Rational Cuspidal Curves in a moving family of ${\mathbb{P}}^2$
• Counting planar curves in ${\mathbb{P}}^3$ with degenerate singularities
• Node Polynomials for Curves on Surfaces