Local multiplicities for an equivariantly enriched non-transverse Bézout's theorem
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arXiv
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| Format: | Preprint |
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2026
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| author | Bethea, Candace Ravi, Charanya |
| author_facet | Bethea, Candace Ravi, Charanya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00289 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local multiplicities for an equivariantly enriched non-transverse Bézout's theorem Bethea, Candace Ravi, Charanya Algebraic Geometry 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. |
| title | Local multiplicities for an equivariantly enriched non-transverse Bézout's theorem |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2604.00289 |