*Shult Sets and Translation Ovoids of H(3, q^{2})*

Gary L. Ebert (Department of Mathematical Sciences - University of Delaware)

(Joint work with the group in Potenza)

A Hermitian surface, denoted by H = H(3, q^{2}), is defined to be
the set of all isotropic points of a nondegenerate unitary polarity
defined on PG(3, q^{2}). The generators of H are the lines of
PG(3, q^{2}) lying totally on the surface, and the points of H
together with its generators form a finite polar space (as well as one of
the classical generalized quadrangles). The dual polar space is the
five-dimensional elliptic quadric Q^{-}(5, q) of order q. An ovoid
on H is a set of points having exactly one point in common with every
generator of the polar space. In recent years many inequivalent ovoids of
the Hermitian surface have been found, but very few "translation ovoids"
are known. Here a translation ovoid is an ovoid admitting a collineation
group which fixes one of its points, fixes all the generators incident
with that point, and acts regularly on the remaining points of the ovoid.

In this talk we describe newly discovered infinite families of translation
ovoids on the Hermitian surface, and various connections with locally
Hermitian 1-spreads of Q^{-}(5, q) and semifield spreads of PG(3,
q). Our technique is to start with carefully chosen sets of points in the
Desarguesian affine plane AG(2, q^{2}), an idea first formulated
by Ernie Shult.

Two previously known examples of translation ovoids on the Hermitian
surface are the so-called classical and semiclassical examples, the first
being simply a non-tangent planar section of the Hermitian surface, and
the second being obtained by taking all the chords passing through some
point of an embedded Baer elliptic quadric Q^{-}(3, q) which is
permutable with the Hermitian surface. We discuss how these examples arise
from Shult sets in AG(2, q^{2}), and mention some characterization
results in the classical and semiclassical cases. The Klein correspondence
will be used heavily throughout the talk.