Characteristic numbers of rational curves with cusp or prescribed triple contact

Abstract

This note pursues the techniques of [Graber-Kock-Pandharipande] to give concise solutions to the characteristic number problem of rational curves in P 2 or P 1 × P 1 with a cusp or a prescribed triple contact. The classes of such loci are computed in terms of modified psi classes, diagonal classes, and certain codimension-2 boundary classes. Via topological recursions the generating functions for the numbers can then be expressed in terms of the usual characteristic number potentials.

Rational cuspidal curves on del-Pezzo surfaces

“…Theorem 1.2 also happens to be true when X := P 1 × P 1 . In [4], Koch obtains a formula for C β using an algebro-geometric method, which is consistent with (1.1). In order to keep the exposition here more streamlined, we decided to omit working out this case separately.…”

Counting curves in a linear system with upto eight singular points

“…Let us define B 0 as before; it denotes the component of the boundary where all the points are distinct. The section Ψ PD 4 does not vanish on A δ 1 • PA 4 ; hence (19), the section does not vanish on B 0 .…”

“…The question of enumerating genus zero curves with higher singularities is a much harder question. Pandahripande and later Kock computed the number of rational cuspidal curves in P 2 and P 1 × P 1 using algebro geometric methods (cf [29] and [19]). Later on, Zinger used methods from Symplectic Geometry to approach the question of enumerating rational curves with cusps and higher singularities (cf [49], [52], [50] and [51]).…”

Genus two enumerative invariants in del-Pezzo surfaces with a fixed complex structure

“…When X is P 1 × P 1 , we use the formula obtained by Kock [7] for the number of rational cuspidal curves on P 1 ×P 1 through δ β −1 generic points. There are no assumptions on β required to use Kock's formula.…”