A projective quadric is the zero locus of a nonzero homogeneous quadratic polynomial , with symmetric over a field of characteristic different from two. A projective point on it is singular when all first partial derivatives vanish. Since , the quadric is nonsingular exactly when is invertible.
Over the complex numbers, a nonzero quadratic form in three variables can be diagonalized by an invertible change of basis. Its rank classifies a projective conic:
- Rank three gives a nonsingular conic.
- Rank two gives two distinct projective lines, since the reduced form factors as . Their intersection is the singular point.
- Rank one gives a double projective line, defined by a squared linear form. The gradient vanishes along that line.
The zero polynomial is excluded from the definition of a conic. Thus these last two cases are precisely the singular conics.
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