Fano-Mukai fourfolds of genus $10$ and their automorphism groups

TL;DR

This paper completes the classification of automorphism groups of Fano-Mukai fourfolds of genus 10, identifying specific group structures for all such fourfolds and highlighting two special cases.

Contribution

It provides a complete description of the discrete automorphism groups of Fano-Mukai fourfolds of genus 10, building on previous work on their neutral components.

Findings

Two special fourfolds have automorphism groups $GL_2(k)\rtimes\mathbb{Z}/2\mathbb{Z}$ and $(\mathbb{G}_a\times\mathbb{G}_m)\rtimes\mathbb{Z}/2\mathbb{Z}$.

Most fourfolds have automorphism group $\mathbb{G}_m^2\rtimes \mathbb{Z}/2\mathbb{Z}$.

One fourfold has automorphism group $\mathbb{G}_m^2\rtimes \mathbb{Z}/6\mathbb{Z}$.

Abstract

The automorphism groups of the Fano-Mukai fourfold of genus 10 were studied in our previous paper [arXiv:1706.04926]. In particular, we found in [arXiv:1706.04926] the neutral components of these groups. In the present paper we finish the description of the discrete parts. Up to isomorphism, there are two special Fano--Mukai fourfold of genus 10 with the automorphism groups $GL_2(k)\rtimes\mathbb{Z}/2\mathbb{Z}$ and $(\mathbb{G}_a\times\mathbb{G}m)\rtimes\mathbb{Z}/2\mathbb{Z}$, respectively. For any other Fano-Mukai fourfold $V$ of genus 10 one has $\mathrm{Aut}(V)=\mathbb{G}_m^2\rtimes \mathbb{Z}/2\mathbb{Z}$, except for exactly one of them with $\mathrm{Aut}(V)=\mathbb{G}_m^2\rtimes \mathbb{Z}/6 \mathbb{Z}$.