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Aart Blokhuis is a mathematician working in finite geometry and combinatorics. With G. Eric Moorhouse, he calculated characteristic-$p$ rank formulas for the point-hyperplane incidence matrix of a set system associated with finite quadrics, and applied them to bounds and nonexistence results for ovoids.^[ref]

The coefficient field is essential to this method. For an elementary illustration, the integer matrix $\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$ has rank two over $\mathbb{Q}$, but rank one over ${\mathbb{F}}_2$. Relations revealed by reduction modulo a prime number can therefore constrain configurations that a characteristic-zero calculation misses. His paper on few-distance sets provides another example of turning geometric restrictions into algebraic counting bounds.^[ref]

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.

G. Eric Moorhouse is a mathematician whose work includes finite geometry, incidence structures and coding theory. His author homepage identifies him as Eric Moorhouse and records a University of Wyoming faculty appointment beginning in 1989.^[ref]

With Aart Blokhuis, he computed characteristic-$p$ rank formulas for the point-hyperplane incidence matrix of a set system in finite orthogonal geometries. Their calculations yield bounds on ovoids and rule out their existence in several families.^[ref] The link is that an incidence matrix converts containment statements into vectors over a finite field. A proposed geometric configuration must satisfy the resulting linear constraints, so linear algebra can prove that the configuration cannot exist. This makes incidence ranks useful structural information, rather than merely a numerical summary of a matrix.

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.

A Suzuki ovoid is a special set of $q^2+1$ points in the three-dimensional projective space over ${\mathbb{F}}_q$, for $q=2^{2a+1}$. It is an ovoid: no three of its points are collinear. Its projective symmetries include the Suzuki group of Lie type $Sz(q)$, acting two-transitively on these points.

For $q\ge 8$, fixing one point leaves a stabilizer of order $q^2(q-1)$, with a subgroup of order $q^2$ acting regularly on the other $q^2$ points. Thus the geometry realizes the natural permutation action of the group, rather than only its four-dimensional matrix action. For $q=8$, there are $65$ ovoid points and $|Sz(8)|=29120$.

The absence of collinear triples means that choosing two ovoid points determines a secant containing exactly those two ovoid points. There are therefore $\left(\genfrac{}{}{0ex}{}{q^2+1}{2}\right)$ such secants. This turns part of the geometry into an explicit pair-counting problem.^[ref]