The equivariant degree and an enriched count of rational cubics

TL;DR

This paper introduces the equivariant degree for proper G-equivariant maps, enabling enriched counts of rational cubics in projective space, with applications to real and complex enumerative geometry.

Contribution

It defines the equivariant degree and local degree for G-equivariant maps, and applies these concepts to count rational plane cubics with symmetry considerations.

Findings

Provides a method to compute equivariant Euler characteristics and Euler numbers.

Enables enriched counts of rational cubics valued in representation and Burnside rings.

Recovers signed counts of real rational cubics under complex conjugation.

Abstract

We define the equivariant degree and local degree of a proper $G$-equivariant map between smooth $G$-manifolds when $G$ is a compact Lie group and prove a local to global result. We show the local degree can be used to compute the equivariant Euler characteristic of a smooth, compact $G$-manifold and the Euler number of a relatively oriented $G$-equivariant vector bundle when $G$ is finite. As an application, we give an equivariantly enriched count of rational plane cubics through a $G$-invariant set of 8 general points in $\mathbb{C}\mathbb{P}^2$, valued in the representation ring and Burnside ring of a finite group. When $\mathbb{Z}/2$ acts by pointwise complex conjugation this recovers a signed count of real rational cubics.