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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 $d\rightarrow \infty$. Wealso discuss some consequence of this result for the moduli space of smoothdegree $d$ hypersurfaces in $\mathbb{P}^n$.

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