On the automorphism groups of smooth Fano threefolds

TL;DR

This paper establishes numerical restrictions on the size of automorphism groups of smooth Fano threefolds with Picard rank 1 and genus up to 10, excluding certain special cases, using classification and topological methods.

Contribution

It provides explicit divisibility conditions on automorphism group orders for a broad class of Fano threefolds, extending previous classifications and topological analyses.

Findings

Automorphism group order divides a specific explicit number depending on genus.

Restrictions apply to Fano threefolds with genus ≤ 10, excluding Gushel-Mukai types.

Uses classification via complete intersections and topology of regular sections.

Abstract

Let $\mathcal{X}$ be a smooth Fano threefold over the complex numbers of Picard rank $1$ with finite automorphism group. We give numerical restrictions on the order of the automorphism group $\mathrm{Aut}(\mathcal{X})$ provided the genus $g(\mathcal{X})\leq 10$ and $\mathcal{X}$ is not an ordinary smooth Gushel-Mukai threefold. More precisely, we show that the order $|\mathrm{Aut}(\mathcal{X})|$ divides a certain explicit number depending on the genus of $\mathcal{X}$. We use a classification of Fano threefolds in terms of complete intersections in homogeneous varieties and the previous paper of A. Gorinov and the author regarding the topology of spaces of regular sections.