= Elliptic quadric
{title2=$Q^-(2n+1,q)$}
Over a <finite field> $\mathbb F_q$, an <elliptic quadric> $Q^-(2n+1,q)$ is the zero set in <projective space> $\operatorname{PG}(2n+1,q)$ of a nonsingular <quadratic form> on a $(2n+2)$-dimensional <vector space> whose <Witt index> is $n$. Thus the largest vector subspaces on which the form vanishes identically have dimension $n$, one less than for a hyperbolic form in the same dimension. The word elliptic describes this type of quadratic form.
The especially important case $Q^-(3,q)$ contains $q^2+1$ points and no projective line. A line therefore meets it in at most two points, making it an example of an <ovoid>. These properties make elliptic quadrics useful configurations in <finite geometry> and in the study of incidence ranks.<Some p-ranks Related to Orthogonal Spaces by Aart Blokhuis and G. Eric Moorhouse>[section 2, orthogonal spaces and elliptic type]{ref}
For example, in $\operatorname{PG}(3,3)$ the equation $x_0^2+x_1^2+x_2^2=x_3^2$ gives an elliptic quadric. With $x_3=1$ there are six solutions, since exactly one of the first three coordinates is nonzero. With $x_3=0$ all three must be nonzero; the eight vectors give four projective points after identifying scalar multiples. There are ten points altogether, as $3^2+1$ predicts.
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