Counting rational curves with an $m$-fold point

Indranil Biswas, Chitrabhanu Chaudhuri, Apratim Choudhury, Ritwik Mukherjee, Anantadulal Paul

Published 2022-12-03Version 1

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.