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 ${\mathrm{SL}}_2(\mathbb{C})$. The group $\mathrm{Aut}(V_{18}^{\mathrm{s}})$ is a semidirect product ${\mathrm{GL}}_2(\mathbb{C})⋊(\mathbb{Z}/2\mathbb{Z})$. 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 ${\mathbb{C}}^4\hookrightarrow V_{18}^{\mathrm{s}}$ 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 $\mathrm{Aut}(V_{18}^{\mathrm{a}})$ is a semidirect product $({\mathbb{G}}_{\mathrm{a}}\times {\mathbb{G}}_{\mathrm{m}})⋊(\mathbb{Z}/2\mathbb{Z})$. For a Fano-Mukai fourfold $V_{18}$ neither isomorphic to $V_{18}^{\mathrm{s}}$, nor to $V_{18}^{\mathrm{a}}$, one has ${\mathrm{Aut}}^0(V_{18})\cong ({\mathbb{G}}_{\mathrm{m}}{)}^2$, and $\mathrm{Aut}(V_{18})$ is a semidirect product of ${\mathrm{Aut}}^0(V_{18})$ and a finite cyclic group whose order is a factor of $6$.

59p.; revised: some typos fixed, Theorem 13.5 extended, formulations of Proposition 14.2 and Theorem 14.3 corrected, acknowledgments added

References in corpus (2)

• Hilbert schemes of lines and conics and automorphism groups of Fano threefolds
• Isotrivial VMRT-structures of complete intersection type

Cited by in corpus (1)

• Fano threefolds as equivariant compactifications of the vector group