Enumeration of curves with one singular point
Somnath Basu, Ritwik Mukherjee
TL;DR
The paper develops a topological enumeration framework for complex degree $d$ plane curves with a single singularity of type $\mathfrak{X}_k$ ($k\le 7$) through $\delta_d-(k+n)$ generic points, using Euler classes of carefully constructed vector-bundle data over stratified moduli spaces. By lifting to augmented spaces $\mathcal{P}\mathfrak{X}_k$ with marked directions and formulating counts $\mathcal{N}(\mathfrak{X}_k,n)$ and $\mathcal{N}(\mathcal{P}\mathfrak{X}_k,n,m)$, the authors derive a recursive scheme controlled by boundary contributions and transversality conditions valid for sufficiently large $d$. They provide explicit formulas and algorithmic procedures (implemented in Mathematica) to compute these numbers, generalizing prior algebraic-geometry results and enabling extensions to other complex surfaces and multi-singularity configurations. The approach hinges on counting zeros of generic holomorphic sections via Euler classes, with meticulous handling of closures, boundary strata, and the local normal forms of singularities. Overall, the work offers a concrete, topological method to compute refined enumerative invariants for plane curves and sets the stage for broader applications in complex-surface geometry.
Abstract
In this paper we obtain an explicit formula for the number of degree d curves in two dimensional complex projective space, passing through (d(d+3)/2 -k) generic points and having a codimension k singularity, where k is at most 7. In the past, many of these numbers were computed using techniques from algebraic geometry. In this paper we use purely topological methods to count curves. Our main tool is a classical fact from differential topology: the number of zeros of a generic smooth section of a vector bundle V over M, counted with a sign, is the Euler class of V evaluated on the fundamental class of M.