Projective space

Projective quadric

A quadric or quadratic variety in projective space is the set of points defined by where is a quadratic form, #m/def/geo i.e.

and is called the quadric belonging to . A quadric is said to be singular iff by change of coördinates can be made to contain fewer variables.

Canonical forms and classification

Let be a non-singular quadric belonging to the quadratic form . Then may be transformed into one of the following forms:¹ #m/thm/geo

1. If then is called a conic.
2. If is even, is called parabolic quadric and has the canonical form

3. If is odd, is called a hyperbolic quadric iff it has the canonical form

3. If is odd, is called an elliptic quadric iff it has the canonical form

where is an irreducible polynomial and homogenous quadratic form.

Properties

• A quadric is singular iff its matrix is singular away from 2
• Orthogonality by a quadric

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