Automorphisms of symmetric powers and motivic zeta functions

TL;DR

This paper proves that for certain smooth projective varieties, automorphisms of the variety correspond exactly to automorphisms of their symmetric powers, and explores motivic zeta functions of Severi-Brauer surfaces.

Contribution

It establishes an isomorphism between automorphisms of a variety and its symmetric powers under specific conditions and computes motivic zeta functions for Severi-Brauer surfaces.

Findings

Automorphisms of the variety and symmetric powers are isomorphic.

Partial computation of motivic zeta functions for Severi-Brauer surfaces.

Relations between classes of Severi-Brauer varieties in the Grothendieck ring.

Abstract

We prove that if $X$ is a smooth projective variety of dimension greater than 1 over a field $K$ of characteristic zero such that $\operatorname{Pic}(X_{\bar{K}}) = \mathbb{Z}$ and $X_{\bar{K}}$ is simply connected, then the natural map $\rho: \operatorname{Aut}(X) \to \operatorname{Aut}(\operatorname{Sym}^d(X))$ is an isomorphism for every $d > 0$. We also partially compute the motivic zeta function of a Severi-Brauer surface and explain some relations between the classes of Severi-Brauer varieties in the Grothendieck ring of varieties.