Counting curves in a linear system with upto eight singular points

TL;DR

This paper provides explicit formulas for counting algebraic curves with up to eight singular points in a linear system, extending previous results and introducing new enumerative formulas for complex singularity configurations.

Contribution

It develops a systematic method to enumerate curves with multiple singularities, including new formulas for codimension eight cases not previously known.

Findings

Explicit formulas for curves with up to eight singular points.

Recovery of known formulas for lower codimensions.

Introduction of new enumerative results for complex singularity configurations.

Abstract

In this paper, we develop a systematic approach to enumerate curves with a certain number of nodes and one further singularity which maybe more degenerate. As a result, we obtain an explicit formula for the number of curves in a sufficiently ample linear system, passing through the right number of generic points, that have $\delta$ nodes and one singularity of codimension $k$, for all $\delta+k \leq 8$. In particular, we recover the formulas for curves with upto six nodal points obtained by Vainsencher. Moreover, all the codimension seven numbers we have obtained agree with the formulas obtained by Kazarian. Finally, in codimension eight, we recover the formula of A.Weber, M.Mikosz and P.Pragacz for curves with one singular point and we also recover the formula of Kleiman and Piene for eight nodal curves. All the other codimension eight numbers we have obtained are new.