Counting rational curves with an $m$-fold point

TL;DR

This paper derives a recursive formula for counting rational degree d curves in the complex projective plane with an m-fold singular point, extending previous work on triple points using a family version of Kontsevich's recursion.

Contribution

It introduces a new recursive approach based on a family version of Kontsevich's formula to count rational curves with m-fold singularities, generalizing prior results.

Findings

Derived recursive formula for m-fold singular points

Explicit calculations for low degree cases

Extended counting methods beyond triple points

Abstract

We obtain a recursive formula for the number of rational degree $d$ curves in $\mathbb{CP}^2$ that pass through $3d+1-m$ generic points and that have an $m$-fold singular point. The special case of counting curves with a triple point was solved earlier by other authors. We obtain the formula by considering a family version of Kontsevich's recursion formula, in contrast to the excess intersection theoretic approach of others. A large number of low degree cases have been worked out explicitly.