$\mathbb{A}^{1}$-Local Degree via Stacks

Abstract

We extend results of Kass-Wickelgren to define an Euler class for a non-orientable (or nonrelatively 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 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 P 4 .