On a fibre bundle version of the Caporaso-Harris formula

TL;DR

This paper extends the Caporaso-Harris recursive enumeration method to fiber bundle settings, enabling counting of cuspidal and nodal curves in higher-dimensional projective spaces.

Contribution

It introduces two generalizations of the Caporaso-Harris formula, including a fiber bundle version and applications to counting specific singular curves.

Findings

Extended enumeration to cuspidal curves on P^2.

Developed fiber bundle version of the recursive formula.

Counted characteristic numbers of nodal cubics in P^3.

Abstract

The Caporaso-Harris formula gives a recursive algorithm to enumerate delta nodal degree d curves in P^2. The recursion is obtained in terms of curves of lower degree that are tangent to a given divisor. This paper presents two generalizations of this method. The first result is on enumeration of one cuspidal curves on P^2, and the second result is an extension to the fiber bundle setting. We solve the question of counting the characteristic number planar nodal cubics in P^3 by extending the idea of Caporaso-Harris.