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,qn) and a certain set of points in PG(r-1,qn) \ PG(r-1,q), called indicator set. If the indicator set is a subspace of (r-1,qn), 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,qn) as the set of vectors with coordinates in GF(q) with respect to a fixed basis of V(n,qn). 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 qn. 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 { < a1t,a2t,...,akt > | t in GF(qn)*}, 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.