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

TL;DR

This paper proves that the average size of automorphism groups of smooth hypersurfaces over finite fields approaches one as the degree increases, impacting the understanding of their moduli spaces.

Contribution

It establishes the asymptotic behavior of automorphism group sizes for smooth hypersurfaces over finite fields as degree grows large.

Findings

Average automorphism group size tends to 1 as degree increases

Implications for the structure of moduli spaces of hypersurfaces

Provides new insights into symmetry properties of hypersurfaces

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 $\mathbb{P}^n$.