Peterson's Deformations of Higher Dimensional Quadrics

TL;DR

This paper introduces explicit higher-dimensional deformations of quadrics, generalizing Peterson's 1D family, and proves the maximality of this deformation family in complex Euclidean spaces.

Contribution

It provides the first explicit higher-dimensional deformation examples of quadrics and establishes their maximality, extending Peterson's classical 1D results.

Findings

Explicit construction of higher-dimensional quadric deformations

Generalization of Peterson's 1D deformation family

Proof of maximality of the deformation family

Abstract

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.