Lynn Margaret Batten

Rank functions of closure spaces of finite rank

We characterize those functions that are the rank functions of closure spaces of finite rank. In case such a function is defined on a finite set, we are able to improve this characterization.

Translation triple systems

We prove the following result. Let S be a Steiner triple system embedded in the projective plane П of order n, such that r=n+1, and such that there exists a line l of Π exterior to S. Let G be a collineation group of Π fixing S, fixing l and transitive on the blocks of S. Then n=3 and S=Π∖l=AG(2, 3), and G contains the group of translations of S with respect to l.

Quadriques partielles d'indice deux

RésuméLorigine de ce travail réside dans lobservation que le groupe de Higman-Sims possède une géométrie très proche de celle dune quadrique dindice de Witt deux, constituée de 100 points, de droites de 2 points et de cercles de 6 points. Notre but est de décrire un système daxiomes qui caractérise simultanément la géométrie des droites et des cercles des quadriques finies dindice deux et la “quadrique” de Higman-Sims.

Minimally projectively embeddable Steiner systems

We study Steiner systems which embed “in a minimal way” in projective planes, and consider connections between the automorphism group of the Steiner systems and corresponding planes. Under certain conditions we are able to show (see Theorem 2) that such Steiner systems are either blocking sets or maximal arcs.

Embedding pseudo-complements of quadratics in PG ( n, q ), n ≥ 9T3, q ≥ 9T2

Abstract Let S be a finite, planar, linear space of dimension n⩾3 such that (1) each line has q − 1, q, or q + 1 points; (2) in any subspace R, the number of lines on any point of R is (q dim R − 1) (q − 1) , where q⩾2. We prove that S embeds in a unique way in PG(n, q). If in addition S has at most qn points, it follows, using a result of Tallini, that S is the complement in PG(n, q) of a parabolic or hyperbolic quadric, a parabolic quadric plus a subspace of its nucleus space, a cone projecting from a PG(n − 3, q) a plane (q + 1)-arc plus a subspace of the PG(n − 2, q) joining the knot of the arc with the PG(n − 3, q), or a hyperplane along with a subspace of PG(n, q).