Severi Degrees in Cogenus 3

TL;DR

This paper computes the degree of the locus of 3-nodal plane curves in a linear system, expressing it as a tautological class and evaluating it using the Bott residue formula.

Contribution

It introduces a new computation method for Severi degrees in cogenus 3, linking geometric loci to tautological classes on a blow-up of the Hilbert scheme.

Findings

Degree of 3-nodal plane curves expressed as a tautological class

Evaluation of the class using Bott residue formula

Provides explicit computation for cogenus 3 Severi degrees

Abstract

In this short note, a new computation of the degree of the locus of 3-nodal plane curves in the linear system of degree d plane curves is given. The answer is expressed as a tautological class on a blow-up of the Hilbert scheme of 3 points in the plane. The class is evaluated by the Bott residue formula.