Umemura quadric fibrations and maximal subgroups of $\operatorname{Cr}_{n}(\mathbb{C})$

We study the equivariant geometry of special quadric fibrations, called Umemura quadric fibrations, as well as the maximality of their automorphism groups inside \operatorname{Cr}_{n}(\mathbb{C}) . We produce infinite families of pairwise non-conjugate maximal connected algebraic subgroups of \operatorname{Cr}_{n}(\mathbb{C}) .