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

TL;DR

This paper develops a new Euler class for vector bundles on smooth schemes using stack constructions, extending $A^1$-homotopy theory tools to broader enumerative geometry problems with arithmetic counts.

Contribution

It introduces an Euler class for non-orientable bundles via stacks, broadening the applicability of $A^1$-homotopy techniques in arithmetic enumerative geometry.

Findings

Defined an Euler class valued in the Grothendieck--Witt group.

Connected the new Euler class to existing $A^1$-homotopy Euler classes.

Provided an arithmetic count of lines meeting six planes in projective 4-space.

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 $\mathbb{P}^4$.