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It is known that the moduli space of smooth Fano-Mukai fourfolds $V_{18}$ ofgenus $10$ 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 uniquefourfold $V_{18}^{\mathrm s}$ 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 $\operatorname{GL}_2(\mathbb{C})$-equivariantcompletion 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 $\mathbb{C}^4\hookrightarrow V_{18}^{\mathrm s}$ yieldsa faithful representation with an open orbit. There is also a unique, up toisomorphism, fourfold $V_{18}^{\mathrm a}$ such that the group$\operatorname{Aut}(V_{18}^{\mathrm a})$ is a semidirect product $({\mathbbG}_{\mathrm{a}}\times{\mathbb G}_{\mathrm{m}})\rtimes(\mathbb{Z}/2\mathbb{Z})$. For a Fano-Mukai fourfold $V_{18}$ neitherisomorphic 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 afactor of $6$.

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