Peterson's Deformations of Higher Dimensional Quadrics

We provide the first explicit examples of deformations of higher dimensional quadrics: a straightforward generalization of Peterson's explicit 1-dimensional family of deformations in $\mathbb C^3$ of 2-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere $\mathbb S^2\subset\mathbb C^3$ to an explicit $(n-1)$-dimensional family of deformations in $\mathbb C^{2n-1}$ of $n$-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere $\mathbb S^n\subset\mathbb C^{n+1}$ and non-degenerate joined second fundamental forms. It is then proven that this family is maximal.