CHAPTER III: CONICS AND QUADRICS - OCW UPM

AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
Observations: Let Q be a projective quadric generated by a quadratic form
ω, with polar form f and associated matrix A.
1. Let sign(Q) be the set of singular points of Q; this is,
sign(Q) = {X ∈ P 3 | f(X, Y ) = 0, for every Y ∈ P 3 }
= {X ∈ P 3 | AX = 0}.
We have
dim(sign(Q)) = 3 − rank(A).
2. If X ∈ P 3 is a singular point, then X ∈ Q.
Proof. We have to check that ω(X) = 0. We have ω(X) = f(X, X) = 0
as X is conjugated with any point, in particular with itself.