✍ Scribed by Koen Thas
No coin nor oath required. For personal study only.
Let S be a finite generalized quadrangle (GQ) of order (s, t), s ] 1 ] t. A k-arc K is a set of k mutually non-collinear points. For any k-arc of S we have k [ st+1; if k=st+1, then K is an ovoid of S. A k-arc is complete if it is not contained in a kOE-arc with kOE > k. In S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, Pitman, Boston, 1984, it is proved that an (st -m)-arc, where -1 [ m < t/s, is always contained in a uniquely defined ovoid, hence it is a natural question to ask whether or not complete (st -t/s)-arcs exist. In this note, we prove that the classical GQ H(4, q 2 ) has no complete (q 5 -q)-arcs. We also show that a GQ S of order s with a regular point has no complete (s 2 -1)-arcs, except when s=2, i.e. S 5 Q(4, 2), and in that case there is a unique example. As a by-product there follows that no known GQ of even order s with s > 2 can have complete (s 2 -1)-arcs. Also, we prove that a GQ of order (s, s 2 ), s ] 1, cannot have complete (s 3 -s)-arcs unless s=2, i.e., S 5 Q(5, 2), in which case there is a unique example (up to isomorphism).
📜 SIMILAR VOLUMES
These year books are not only invaluable to busy practitioners of the specialties to which they relate but they also have much of interest to practitioners generally; this volume on the nervous system especially so. ## Wound Healing and Tissue Repair. Edited by W. BRADFORD PATTERSON. 9 x 52 in. P