Projective Quadrics

Abstract

A (Projective) quadric in P(E) is a non-zero element α of PQ(E), i.e. a quadrativ from q, over E, considered up to a non-zero scalar. Such a from q representing the class α is called an equation of α.

In this chapter E denotes a vector space of dimension n + 1 over a commutative field K of characteristic different from 2; the associated projective space is denoted by P(E), and p: E \ 0 → P(E) is the canonical projection. We denote by Q(E) the vector space of quadratic forms over E, and by PQ(E) the associated projective space P(Q(E)); the polar form of q ∈ Q(E) is denoted by P. We will always have n ≥ 1.