On the enumeration of complex plane curves with two singular points

TL;DR

This paper extends previous methods to enumerate complex plane curves with two prescribed singular points, providing solutions for ordinary multiple points and reducing more complex cases to simpler ones with detailed examples.

Contribution

It generalizes the enumeration method for plane curves with two singular points, including arbitrary linear types, and introduces a reduction technique for complex singularities.

Findings

Solved enumeration for two ordinary multiple points

Reduced complex singularity enumeration to simpler cases

Provided numerous examples and numerical results

Abstract

We study equi-singular strata of plane curves with two singular points of prescribed types. The method of the previous work [Kerner06] is generalized to this case. In particular we consider the enumerative problem for plane curves with two singular points of linear singularity types. First the problem for two ordinary multiple points of fixed multiplicities is solved. Then the enumeration for arbitrary linear types is reduced to the case of ordinary multiple points and to the understanding of "merging" of singular points. Many examples and numerical answers are given.