 Ovoid

In finite projective space $\mathrm{PG}(3,q)$ with $q>2$, an ovoid is a set of $q^2+1$ points with no three collinear. At each of its points there is a unique tangent plane meeting the set only at that point. It is a finite-geometric analogue of a strictly convex surface, although a definition using real convexity does not make sense over a finite field.^[ref]

The elliptic quadric supplies examples. Over ${\mathbb{F}}_3$, the equation $x_0^2+x_1^2+x_2^2=x_3^2$ defines ten projective points. In the chart $x_3=1$, exactly one of the first three coordinates is nonzero, giving six points. At infinity all three are nonzero, giving four more after quotienting by simultaneous sign. Thus the count is $6+4=3^2+1$. This example links the incidence condition to a concrete homogeneous polynomial equation.