Elliptic quadric ($Q^{-}(2n+1,q)$)

Over a finite field ${\mathbb{F}}_q$, an elliptic quadric $Q^{-}(2n+1,q)$ is the zero set in projective space $\mathrm{PG}(2n+1,q)$ 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.^[ref]

For example, in $\mathrm{PG}(3,3)$ 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.