1-systems of Q(6, q) and Q^-(7, q): recent results
by
Deirdre Luyckx
Ghent Universtiy
Coauthors: Joseph A. Thas

In [2], m-systems of polar spaces were introduced by Shult and Thas and a description of several classes of examples is given. However, few other examples are known until now. First, we briefly discuss the construction of a new class of 1-systems of Q(6, q), q odd, starting from a generalization of a flock of a quadratic cone in PG(3, q). Secondly, we mention a uniqueness result about 1-systems of the quadric Q^-(7, q).

An i-flock of a quadratic cone LQ(2, q) with line vertex in PG(4, q), q odd, is defined as a partition of LQ(2, q) \L in q² mutually disjoint conics such that the planes of the elements of the i-flock pairwise intersect in internal points of LQ(2, q). It can be shown that to every i-flock of LQ(2, q) a locally hermitian 1-system of Q(6, q) is associated and conversely.

Applying the theory of the i-flocks to the semi-classical non-hermitian spread S_[9] of the hexagon H(q), q odd and q \equiv 1 mod 3 (see [1]), which is locally hermitian at some line L, we find an interesting geometric construction of S_[9] starting from a rational normal cubic scroll R³ having L as directrix line. Apparently the conics on R³ determine the q² conic planes of the i-flock associated with S_[9] and hence the i-flock can be reconstructed from the cubic scroll. Surprisingly this geometric construction not only yields the 1-system S_[9]; it turns out that different cubic scrolls may give rise to non-isomorphic locally hermitian 1-systems of Q(6, q). In particular, there are (q-3)/2 orbits in the set of all non-hermitian locally hermitian 1-systems of Q(6, q) constructed from a cubic scroll, under the subgroup of PGL(7, q) fixing Q(6, q).

It can be shown that a locally hermitian non-hermitian 1-system of Q(6, q), q odd, is semi-classical if and only if it arises from a rational normal cubic scroll R³ with directrix line L subset or equal Q(6, q) and with the property that all points of R³\L are internal points of Q(6, q). As it is possible to determine all such cubic scrolls, this yields a complete characterization and determination of the locally hermitian semi-classical 1-systems of Q(6, q) for q odd.

Now, let M be a 1-system of Q^-(7, q). Then it can be shown that every line of Q^-(7, q) contains 0, 1, 2 or q+1 points on lines of M and that the latter holds if and only if the line itself belongs to M. For q odd, this implies that every plane of Q^-(7, q) either contains a line of M or an irreducible conic of points on lines of M. This observation enabled us to prove that Q^-(7, q) has a unique 1-system for q odd, up to a projectivity. This is an important result because 1-systems of Q^-(7, q) were candidates to yield new generalized quadrangles, as described in [3].

References [1] ¯ I. Bloemen, J. A. Thas and H. Van Maldeghem. Translation ovoids of generalized quadrangles and hexagons.
 Geom. Dedicata, 72(1):19-62, 1998.
[2]  E. E. Shult and J. A. Thas. m-systems of polar spaces.
 J. Combin. Theory Ser. A, 68(1):184-204, 1994.
[3]  E. E. Shult and J. A. Thas. Constructions of polygons from buildings.
 Proc. London Math. Soc. (3), 71(2):397-440, 1995

Homepage of D. Luyckx

Date received: February 21, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagg-07.