Quadratic Euler characteristic of symmetric powers of curves.

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 that the power structure on the Grothendieck-Witt ring introduced by Pajwani-Pál computes the compactly supported $A^1$ -Euler characteristic of symmetric powers for all curves.