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

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 $G{L}_2(k)⋊\mathbb{Z}/2\mathbb{Z}$ and $({\mathbb{G}}_a\times \mathbb{G}m)⋊\mathbb{Z}/2\mathbb{Z}$, respectively. For any other Fano-Mukai fourfold $V$ of genus 10 one has $\mathrm{Aut}(V)={\mathbb{G}}_m^2⋊\mathbb{Z}/2\mathbb{Z}$, except for exactly one of them with $\mathrm{Aut}(V)={\mathbb{G}}_m^2⋊\mathbb{Z}/6\mathbb{Z}$.

arXiv commentThe paper has grown up from an earlier version of the preprint [arXiv:2005.12092], where the major results were already present. This is the final version to appear in European J. Math. A new reference added; minor corrections are done due to referees remarks