Quadratic Euler Characteristic of Symmetric Powers of Curves

TL;DR

This paper calculates the quadratic Euler characteristic of symmetric powers of smooth projective curves over fields not of characteristic two, utilizing motivic techniques, and applies this to relate power structures on Grothendieck-Witt rings to $A^1$-Euler characteristics.

Contribution

It introduces a method to compute quadratic Euler characteristics of symmetric powers of curves using the Motivic Gauss-Bonnet Theorem and connects this to $A^1$-Euler characteristics via power structures.

Findings

Quadratic Euler characteristic of symmetric powers computed for curves.

Power structure on Grothendieck-Witt ring relates to $A^1$-Euler characteristic.

Results hold over fields of characteristic zero, excluding characteristic two.

Abstract

We compute the quadratic Euler characteristic of the symmetric powers of a smooth, projective curve over any field $k$ that is not of characteristic two, using the Motivic Gauss-Bonnet Theorem of Levine-Raksit. As an application, we show over a field of characteristic zero that the power structure on the Grothendieck-Witt ring introduced by Pajwani-P\'al computes the compactly supported $\mathbb{A}^1$-Euler characteristic of symmetric powers for all curves.