Projective quadric

A quadric or quadratic variety $Q$ in projective space $P G (𝑛, 𝕂)$ is the set of points defined by $𝑄 (𝐗) = 0$ where $𝑄$ is a quadratic form, #m/def/geo i.e.

Q = {[𝐗] : 𝐗 \in 𝕂 𝑛 + 1, 𝑄 (𝐗) = 0}

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

Canonical forms and classification

Let $Q 𝑛 \subseteq P G (𝑛, 𝑞)$ be a non-singular quadric belonging to the quadratic form $𝑄 𝑛 (𝐗)$ . Then $𝑄 𝑛$ may be transformed into one of the following forms:¹ #m/thm/geo

If $𝑛 = 2$ then $Q 𝑛$ is called a conic.
If $𝑛 > 2$ is even, $Q 𝑛$ is called parabolic quadric and has the canonical form

𝑄 𝑛 (𝐗) = 𝑋 0 𝑋 1 + 𝑋 2 𝑋 3 + \dots + 𝑋 𝑛 - 2 𝑋 𝑛 - 1

If $𝑛 > 2$ is odd, $Q 𝑛$ is called a hyperbolic quadric iff it has the canonical form

𝑄 𝑛 (𝐗) = 𝑋 0 𝑋 1 + 𝑋 2 𝑋 3 + \dots + 𝑋 𝑛 - 3 𝑋 𝑛 - 2 + 𝑋 𝑛 - 1 𝑋 𝑛

If $𝑛 > 2$ is odd, $Q 𝑛$ is called an elliptic quadric iff it has the canonical form

𝑄 𝑛 (𝐗) = 𝑋 0 𝑋 1 + 𝑋 2 𝑋 3 + \dots + 𝑋 𝑛 - 3 𝑋 𝑛 - 2 + 𝑓 (𝑋 𝑛 - 1, 𝑋 𝑛)

where $𝑓 (𝑋 𝑛 - 1, 𝑋 𝑛)$ is an irreducible polynomial and homogenous quadratic form.

Proof

#missing/proof

Properties

#state/develop | #lang/en | #SemBr

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

Projective quadric

Canonical forms and classification

Properties

Footnotes