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

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 $d \to \infty$. 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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This material is based upon work supported by the National Science Foundation under Grant Number DMS 1641020.

Received 24 October 2019

Accepted 2 June 2020

Published 22 November 2021