Average size of the automorphism group of smooth projective hypersurfaces over finite fields

Abstract

In this paper, we show that the average size of the automorphism group over ${\mathbb{F}}_q$ of a smooth degree $d$ hypersurface in ${\mathbb{P}}_{{\mathbb{F}}_q}^n$ is equal to $1$ as $d\to \infty$. We also discuss some consequence of this result for the moduli space of smooth degree $d$ hypersurfaces in ${\mathbb{P}}^n$.

5 pages, comments are welcome