Fano-Mukai fourfolds of genus $10$ as compactifications of $\mathbb{C}^4$

Abstract: It is known that the moduli space of smooth Fano-Mukai fourfolds $V_{18}$ of genus $10$ has dimension one. We show that any such fourfold is a completion of $\mathbb{C}^4$ in two different ways. Up to isomorphism, there is a unique fourfold $V_{18}^{\mathrm s}$ acted upon by $\operatorname{SL}2(\mathbb{C})$. The group $\operatorname{Aut}(V{18}^{\mathrm s})$ is a semidirect product $\operatorname{GL}2(\mathbb{C})\rtimes(\mathbb{Z}/2\mathbb{Z})$. Furthermore, $V{18}^{\mathrm s}$ is a $\operatorname{GL}2(\mathbb{C})$-equivariant completion of $\mathbb{C}^4$, and as well of $\operatorname{GL}_2(\mathbb{C})$. The restriction of the $\operatorname{GL}_2(\mathbb{C})$-action on $V{18}^{\mathrm s}$ to $\mathbb{C}^4\hookrightarrow V_{18}^{\mathrm s}$ yields a faithful representation with an open orbit. There is also a unique, up to isomorphism, fourfold $V_{18}^{\mathrm a}$ such that the group $\operatorname{Aut}(V_{18}^{\mathrm a})$ is a semidirect product $({\mathbb G}{\mathrm{a}}\times{\mathbb G}{\mathrm{m}})\rtimes (\mathbb{Z}/2\mathbb{Z})$. For a Fano-Mukai fourfold $V_{18}$ neither isomorphic to $V_{18}^{\mathrm s}$, nor to $V_{18}^{\mathrm a}$, one has $\operatorname{Aut}^0 (V_{18})\cong ({\mathbb G}{\mathrm{m}})^2$, and $\operatorname{Aut}(V{18})$ is a semidirect product of $\operatorname{Aut}^0(V_{18})$ and a finite cyclic group whose order is a factor of $6$.