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 $\mathbb{P}^{n}{\mathbb{F}_q}$ is equal to $1$ as $d\rightarrow \infty$. We also discuss some consequence of this result for the moduli space of smooth degree $d$ hypersurfaces in $\mathbb{P}^n$.