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

Abstract

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 ${\mathbb{P}}^4$.

Errors identified in several places. A corrected version may be drafted in the future, but the timeline is uncertain for now

References in corpus (1)

• The class of Eisenbud--Khimshiashvili--Levine is the local A1-Brouwer degree