$\mathbb{A}^{1}$Local Degree via Stacks
Abstract
We extend results of KassWickelgren to define an Euler class for a nonorientable (or nonrelatively orientable) vector bundle on a smooth scheme, valued in the GrothendieckWitt 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 KassWickelgren'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 $\mathbb{P}^4$.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.05955
 Bibcode:
 2019arXiv191105955K
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Algebraic Topology;
 Mathematics  KTheory and Homology
 EPrint:
 22 pages, submitted version, global root stack \mathcal{X} replaced with local root stacks \mathcal{U}, statements updated to reflect this change, Lem. 5.8 added