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 $\mathbb{P}^n$.
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