Normal ovoids and affine planes associated with generalized quadrangles of order (s, s-2)
by
Mark A. Miller
Marietta College

Generalized quadrangles (GQ) of order (s, t) with |s-t| <= 2 have been studied extensively. In the 1980s and 1990s progress was made in the area of characterizing these GQ by, among others, De Soete, Kantor, Miller, Payne, Thas, van Maldeghem, and most recently in a paper by De Bruyn and Payne (to appear in Bull. ICA). During this same period results regarding extended generalized quadrangles whose residual GQ had these parameters were studied in part by Del Fra, Kasikova, Pasechnik, Pasini, Shult, Thas, and Yoshiara.

In this note we consider the affine planes associated with normal ovoids in GQ(s, s-2). Every known GQ with these parameters arises from a q-arc in a plane embedded in PG(3, q) where q=s-1. When the q-arc extends to a translation hyperoval the points of the GQ can be partitioned into a fan of normal ovoids, and the affine planes associated with these ovoids are all desarguesian. We provide sufficient conditions for having isomorphic affine planes arise from normal ovoids in fans of arbitrary GQ(s, s-2).

Date received: February 19, 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-04.