= Projective quadric
{title2=$Q=\{[v]\in\mathbb P^n:v^TBv=0\}$}
A projective quadric is the zero set of a nonzero homogeneous <quadratic polynomial> in <projective space>. Over a <field> of <characteristic> different from two, represent it by a symmetric matrix $B$, up to a nonzero scalar. The gradient at a representative $v$ is $2Bv$, so the singular locus is the projectivization of $\ker B$. Thus the quadric is nonsingular exactly when $B$ is invertible. In the complex projective plane, ranks three, two, and one give respectively a smooth <projective conic>, two distinct <projective lines>, and a double <projective line>.
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