Projective space

Projective quadric

A quadric or quadratic variety in projective space is the set of points defined by where is a quadratic form, 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 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 and homogenous quadratic form.

Proof

proof

Properties

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

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develop | en | sembr

Footnotes

1. 2020. Finite geometries, ¶4.47–4.48, pp. 99–103 ↩