Local multiplicities for an equivariantly enriched non-transverse B\'ezout's theorem

Abstract

We introduce the degree and local degree in equivariant motivic homotopy theory for the purpose of studying equivariant enumerative problems over general fields. Given a finite, tame group scheme $G$ over a field $k$ and an equivariant motivic ring spectrum $E_G$, we define the equivariant motivic degree and a corresponding local degree of a relatively $E_G$-oriented, proper, quasi-smooth morphism of $G$-schemes. We prove a local to global formula expressing the global degree as a sum of local contributions over $G$-orbits. Using these constructions, we define the Euler number of an oriented vector bundle on a quasi-smooth, proper derived stack and show that the Euler number is independent of the choice of section under appropriate hypotheses. In the presence of a finite group action, the equivariant Euler number can be computed as a sum of local equivariant degrees. As an application, we obtain an equivariantly enriched local multiplicity formula for an equivariant non-transverse B\'ezout theorem, expressing an equivariant intersection number as a sum of local equivariant degrees.