*Desarguesian spreads in subgeometries and an application to
the theory of eggs in finite projective spaces*

Michel Lavrauw, Università degli studi di Napoli "Federico II"

An (n-1)-spread in PG(r-1,q) is a set of (n-1)-subspaces partitioning the
pointset of PG(r-1,q). One can construct a spread using the method of
Segre by embedding PG(r-1,q) as a subgeometry in PG(r-1,q^{n}) and
a certain
set of points in PG(r-1,q^{n}) \ PG(r-1,q), called *indicator
set*. If the indicator set is a subspace of (r-1,q^{n}), then
the spread is called *Desarguesian*. Let V(n,q) be the n-dimensional
vectorspace over the finite field of order GF(q), and consider V(n,q)
embedded in V(n,q^{n}) as the set of vectors with coordinates in
GF(q) with respect to a fixed basis of V(n,q^{n}). It is well
known that one can define a multiplication in V(n,q) such that V(n,q) is
isomorphic to the finite field of order q^{n}. We give a
construction of such an isomorphism, construct an indicator set for a
Desarguesian (n-1)-spread of PG(kn-1,q) of which the elements are of the
form { < a_{1}t,a_{2}t,...,a_{k}t > | t in
GF(q^{n})^{*}}, and illustrate its use by giving a new
proof of a theorem by J. A. Thas on the relation between Veronese
varieties and good eggs in PG(4n-1,q), q odd.

This research has been supported by a Marie Curie Fellowship of the
European Community programme "Improving the Human Research Potential and
the Socio-Economic knowledge Base" under the contract number
HMPF-CT-2001-01386.