An enriched count of nodal orbits in an invariant pencil of conics

Abstract

This work gives an equivariantly enriched count of nodal orbits in a general pencil of plane conics that is invariant under a linear action of a finite group on $\mathbb{CP}^2$. This is both inspired by and a departure from $R(G)$-valued enrichments such as Roberts's equivariant Milnor number and Damon's equivariant signature formula. Given a $G$-invariant general pencil of conics, the weighted sum of nodal orbits in the pencil is a formula in $A(G)$ in terms of the base locus considered as a $G$-set. We show this is true for all finite groups except $\mathbb{Z}/2\times \mathbb{Z}/2$, $A_4$, and $D_8$ and give counterexamples for the exceptional groups.