Nonexistence of Complete (st−t/s)-Arcs in Generalized Quadrangles of Order (s, t), I

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 k′-arc with k′>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, q2) has no complete (q5−q)-arcs. We also show that a GQ S of order s with a regular point has no complete (s2−1)-arcs, except when s=2, i.e. S≅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 (s2−1)-arcs. Also, we prove that a GQ of order (s, s2), s≠1, cannot have complete (s3−s)-arcs unless s=2, i.e., S≅Q(5, 2), in which case there is a unique example (up to isomorphism).