Monomial flocks of monomial cones in even characteristic
This article generalizes the concept of a flock of a quadratic cone and of an \(\alpha\)-flock of a translation oval cone to more general types of cones. A flock of a quadratic cone of \(PG(3,q)\) is a set of \(q\) planes not passing through the vertex which do not intersect each other at a point of this quadratic cone. One of the most interesting features of flocks is that they are related to a large variety of other structures. They give rise to certain generalized quadrangles of order \((q^2,q)\) and can be used to construct translation planes. In \textit{W. Cherowitzo, T. Penttila, I. Pinneri} and \textit{G. Royle} [Geom. Dedicata 60, No. 1, 17-37 (1996; Zbl 0855.51008)], a new connection was found. Flocks of quadratic cones give rise to a set of \(q+1\) ovals in a projective plane \(PG(2,q)\), \(q\) even; called a herd of ovals. This link, found earlier by Payne, led to the discovery of the Payne ovals and to the discovery of the Subiaco ovals. So, flocks of quadratic cones give rise to ovals, but not every oval is related to a flock of a quadratic cone. This led Cherowitzo in: [\textit{W. Cherowitzo}, Geom. Dedicata 72, No. 3, 221-246 (1998)] to define flocks of cones \(\Sigma_\alpha\) with base a translation oval. And this, with the goal of linking them to ovals and with the goal of finding new ovals. This approach was successful; these flocks of the quadratic cones \(\Sigma_\alpha\), also called \(\alpha\)-flocks, are indeed linked to an oval, and this approach made it possible to prove the existence of the infinite class of Cherowitzo ovals. But still there remain ovals which are not linked to such \(\alpha\)-flocks. So, with the goal of finding even more links with ovals, an extension of the definition of \(\alpha\)-flocks is presented in this article. Now, flocks of monomial cones are studied. A monomial cone in \(PG(3,q)\), \(q=2^h\), is a set of points \(\Sigma_\beta=\{(x,y,z,w)||y^\beta=xz^{\beta-1} \}\), where \((\beta,q-1)=(\beta-1,q-1)=1\). The author gives the conditions which a set of \(q\) planes must satisfy in order to form a flock of \(\Sigma_\beta\), and gives some examples of monomial flocks. These are flocks which are equivalent to a flock having as planes \(\pi_t: t^a x+t^by+\kappa t^cz+w=0\), \(t\in GF(q)\). A geometric construction of some of these flocks is also given. The article ends with a number of open problems. The most important among these questions is whether this idea of flocks of monomial cones can lead to the discovery of new ovals in \(PG(2,q)\), \(q\) even.
scientific article; zbMATH DE number 1266811