Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local multiplicities for an equivariantly enriched non-transverse Bézout's theorem

Candace Bethea, Charanya Ravi

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 Bézout theorem, expressing an equivariant intersection number as a sum of local equivariant degrees.

Local multiplicities for an equivariantly enriched non-transverse Bézout'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 over a field and an equivariant motivic ring spectrum , we define the equivariant motivic degree and a corresponding local degree of a relatively -oriented, proper, quasi-smooth morphism of -schemes. We prove a local to global formula expressing the global degree as a sum of local contributions over -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 Bézout theorem, expressing an equivariant intersection number as a sum of local equivariant degrees.

Paper Structure

This paper contains 9 sections, 13 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Degree and Local Degree
  3. General base change
  4. Equivariant degree and local degree
  5. Local Multiplicities and Equivariant Bézout
  6. Euler number
  7. Intersection product
  8. Equivariant Euler number and Bézout's Theorem
  9. Euler number as sum of local degrees

Key Result

Theorem 1.1

Let $E_G \in \operatorname{SH}^G(k)$ be a motivic ring spectrum, and let $f\colon X\to Y$ be a proper, quasi-smooth, equivariantly smoothable $G$-equivariant map between $G$-schemes $X$ and $Y$ of the same dimension. Assume $f$ is relatively $E_G$-oriented. Let $i_y\colon \operatorname{Spec} k(y)\ho in $E^0_{G_y}(\operatorname{Spec} k(y))$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.3: Section independence
  • Theorem 1.4
  • Theorem 1.6: Equivariant non-transverse Bézout's theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Definition 2.7
  • ...and 42 more