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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 \({\mathrm{SL}}_2({\mathbb {C}})\). The group  Open image in new window  is a semidirect product  Open image in new window . Furthermore, \(V_{18}^{{\mathrm {s}}}\) is a \({\mathrm{GL}}_2({\mathbb {C}})\)-equivariant completion of \({\mathbb {C}}^4\), and as well of \({\mathrm{GL}}_2({\mathbb {C}})\). The restriction of the \({\mathrm{GL}}_2({\mathbb {C}})\)-action on \(V_{18}^{{\mathrm {s}}}\) to  Open image in new window  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  Open image in new window  is a semidirect product  Open image in new window . For a Fano–Mukai fourfold \(V_{18}\) isomorphic neither to \(V_{18}^{{\mathrm {s}}}\), nor to \(V_{18}^{\mathrm a}\), the group  Open image in new window  is a semidirect product of \(({{\mathbb {G}}}_{\mathrm {m}})^2\) and a finite cyclic group whose order is a factor of 6. Besides, we establish that the affine cone over any polarized Fano–Mukai variety \(V_{18}\) is flexible in codimension one, and flexible if \(V_{18}=V_{18}^{{\mathrm {s}}}\).

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