Monomial Flocks and Herds Containing a Monomial Oval

✍ Scribed by Tim Penttila; L Storme

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 367 KB
• Volume: 83
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1997.2855

✦ Synopsis

Let F be a flock of the quadratic cone Q: X 2 2 =X 1 X 3 , in PG(3, q), q even, and let 6 t : X 0 = x t X 1 + t 1Â2 X 2 + z t X 3 , t # F q , be the q planes defining the flock F. A flock is equivalent to a herd of ovals in PG(2, q), q even, and to a flock generalized quadrangle of order (q 2 , q). We show that if the herd contains a monomial oval, this oval is the Segre oval. This implies a result on the existence of subquadrangles T 2 (O) in the corresponding flock generalized quadrangle. To obtain this result, we prove that if x t and z t both are monomial functions of t, then the flock is either the linear, FTWKB-, or Payne P 1 flock. This latter result implies, in the even case, the classification of regular partial conical flocks, as introduced by Johnson.