Gary L. Ebert

On Buekenhout-Metz unitals of odd order

Abstract The odd order Buekenhout-Metz unitals are enumerated and classified. Their inherited collineation groups are computed, they are shown to be self-dual as designs, and related designs are constructed.

Complete Systems of Lines on a Hermitian Surface over aFinite Field

The aim is to find the maximum size of a set of mutually ske lines on a nonsingular Hermitian surface in PG(3, q) for various values of q. For q = 9 such extremal sets are intricate combinatorial structures intimately connected ith hemisystems, subreguli, and commuting null polarities. It turns out they are also closely related to the classical quartic surface of Kummer. Some bounds and examples are also given in the general case.

On Buekenhout-Metz unitals of even order

Abstract The even order Buekenhout-Metz unitals are enumerated (up to projective equivalence) and their inherited collineation groups are computed. They are shown to be self-dual as designs, and certain related designs are also constructed.

Complete Spans on Hermitian Varieties

Let L be a general linear complex in PG(3, q) for any prime power q. We show that when GF(q) is extended to GF(q2), the extended lines of L cover a non-singular Hermitian surface H ≅ H(3, q2) of PG(3, q2). We prove that if Sis any symplectic spread PG(3, q), then the extended lines of this spread form a complete (q2 + 1)-span of H. Several other examples of complete spans of H for small values of q are also discussed. Finally, we discuss extensions to higher dimensions, showing in particular that a similar construction produces complete (q3 + 1)-spans of the Hermitian variety H(5, q2).

Infinite families of new semifields

We construct six new infinite families of finite semifields, all of which are two-dimensional over their left nuclei. We give constructions for both even and odd characteristics when the left nucleus has odd dimension over the center. The characteristic is odd in the one family in which the left nucleus has even dimension over the center. Spread sets of linear maps are used in all the constructions.

Mixed partitions of PG(5, q )

Abstract We prove that the projective space PG(5, q ) can be partitioned into two planes and q 3 −1 caps all of which are quadric Veroneseans. This partition is obtained by taking the orbits of a lifted Singer cycle of PG(2, q ). The possibility of getting larger caps by gluing some of these orbits together is also addressed.

Caps Embedded in Grassmannians

This paper is concerned with constructing caps embedded in line Grassmannians. In particular, we construct a cap of size q3 +2q2+1 embedded in the Klein quadric of PG(5,q) for even q, and show that any cap maximally embedded in the Klein quadric which is larger than this one must have size equal to the theoretical upper bound, namely q3+2q2+q+2. It is not known if caps achieving this upper bound exist for even q > 2.

A New Class of Translation Planes

We define a nest of reguli to be a collection P of reguli in a regular spread S of PG(3, q) such that every line of S is contained in exactly 0 or 2 reguli of P. Let U denote the lines of S contained in the reguli of some nest. If V is a partial spread of PG[3,q) covering the same points as U but having no lines in common with U , then V will be called a replacement set for U. Clearly, (S—U) U V is a spread of PG(3, q) , yielding a (potentially new) translation plane of order q 2 which is 2–dimensional over its kernel. Nests of size (q+3)/2 were first studied (under another name) by Bruen and later by many others. Whether such (q+3)/2- nests exist for q 13 and whether such nests are necessarily reversible are still open questions. In this paper we consider nests of size q. We exhibit an infinite family of ?-nests, one for each odd prime q , and show that each nest is reversible. The translation planes so obtained appear to be new, at least for q ≥ 11.

A group-theoretic characterization of classical unitals

Abstract. Let

Shult sets and translation ovoids of the Hermitian surface

\cal U