$\mathbb{A}^{1}$-Local Degree via Stacks
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866916359612399616 |
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| author | Kobin, Andrew Taylor, Libby |
| author_facet | Kobin, Andrew Taylor, Libby |
| contents | We extend results of Kass--Wickelgren to define an Euler class for a non-orientable (or non-relatively orientable) vector bundle on a smooth scheme, valued in the Grothendieck--Witt group of the ground field. We use a root stack construction to produce this Euler class and discuss its relation to other versions of an Euler class in $\mathbb{A}^{1}$-homotopy theory. This allows one to apply Kass--Wickelgren's technique for arithmetic enrichments of enumerative geometry to a larger class of problems; as an example, we use our construction to give an arithmetic count of the number of lines meeting $6$ planes in $\mathbb{P}^4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1911_05955 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | $\mathbb{A}^{1}$-Local Degree via Stacks Kobin, Andrew Taylor, Libby Algebraic Geometry Algebraic Topology K-Theory and Homology We extend results of Kass--Wickelgren to define an Euler class for a non-orientable (or non-relatively orientable) vector bundle on a smooth scheme, valued in the Grothendieck--Witt group of the ground field. We use a root stack construction to produce this Euler class and discuss its relation to other versions of an Euler class in $\mathbb{A}^{1}$-homotopy theory. This allows one to apply Kass--Wickelgren's technique for arithmetic enrichments of enumerative geometry to a larger class of problems; as an example, we use our construction to give an arithmetic count of the number of lines meeting $6$ planes in $\mathbb{P}^4$. |
| title | $\mathbb{A}^{1}$-Local Degree via Stacks |
| topic | Algebraic Geometry Algebraic Topology K-Theory and Homology |
| url | https://arxiv.org/abs/1911.05955 |