Arcs and ovals in infinite \(K\)-clan geometry (Q1281123)

Let \(F=GF(q)\) be the Galois field with \(q\) elements. A \(q\)-clan is a set \({\mathcal C}\) of \(q\), \(2\times 2\) matrices over \(F\) whose pairwise differences are anisotropic, i.e., define quadratic forms that represent 0 only trivially. Associated with a \(q\)-clan \({\mathcal C}\) is a flock \({\mathcal F}({\mathcal C})\) of a quadratic cone in \(PG(3,q)\), (and with the flock a line spread of \(PG(3,q)\) and hence translation planes of order \(q^2)\), a generalized quadrangle \(GQ({\mathcal C})\) with parameters \((q^2,q)\). When \(q\) is a power of 2, there are also ovals in \(PG(2,q)\) derived from \({\mathcal C}\), along with subquadrangles of order \(q\). \textit{F. De Clerck} and \textit{H. Van Maldeghem} [Bull. Belg. Math. Soc.-Simon Stevin 3, No. 1, 399-415 (1994; Zbl 0811.51004)] replaced \(F\) with an infinite field \(K\) and studied the problem of replacing \({\mathcal C}\) with a \(K\)-clan so as to obtain an infinite generalized quadrangle. More recently \textit{L. Bader} and the reviewer took a different approach to this problem [J. Geom. 63, No. 1-2, 1-16 (1998)]. Also, N. L. Johnson and his coauthors have studied \(K\)-clans and flocks, with some attention to the associated translation planes. See especially \textit{V. Jha} and \textit{N. L. Johnson} [J. Geom. 57, No. 1-2, 123-150 (1996; Zbl 0866.51002)] and \textit{N. L. Johnson} and the reviewer (in Lect. Notes Pure Appl. Math. 190, 51-122 (1997) and their references. The paper under review, however, seems to be the first to extend to the infinite case a study of the associated ovals in \(PG(2,K)\) when the characteristic of \(K\) is 2, along with their corresponding subquadrangles. The definition of \(K\)-clan was chosen to correspond to the situation where the natural candidate \({\mathcal F}({\mathcal C})\) for a flock really is a flock. Then a \(K\)-clan \({\mathcal C}\) is by definition a 4-gonal \(K\)-clan provided the natural candidate \(GQ({\mathcal C})\) for a generalized quadrangle really is one. De Clerck and Van Maldeghem first showed that this is the case if and only if each ``derivation'' of the flock \({\mathcal F}({\mathcal C})\) is again a flock, and algebraic conditions on the coordinatizing functions that give \({\mathcal C}\) were worked out that are equivalent to having \({\mathcal C}\) be 4-gonal. When generalizing to the infinite case, it always seems necessary to have more complicated definitions and/or special restrictions on the field. One of the main results of the present paper is the following: Let \(K\) be a perfect field of characteristic 2. Let \({\mathcal C}\) be a 4-gonal \(K\)-clan. Then each natural candidate \({\mathcal O}\) for an oval of \(PG(2,K)\) is indeed an oval, and the corresponding generalized quadrangle \(GQ({\mathcal C})\) has a subquadrangle isomorphic to the usual generalized quadrangle \(T_2({\mathcal O})\) constructed by J. Tits from the oval \({\mathcal O}\).