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Let = 2, = GF(). Let and be arbitrary 2 × 2 matrices over .

Since in this section we depend so strongly on the computational setup, we review the notation one more time.

Recall that denotes the full group of collineations of , and that denotes the subgroup of fixing the points ((0, 0), (0, 0), 0) and (∞).

By a we mean one for which there is some modulo + 1 for which the automorphism of G⊗ given explicitly by Eq. (3.21) is a collineation of . (See Theorem 3.8.1.) In [COP03] the authors gave a unified construction that included three previously known cyclic families plus a new one. We have modified their presentation to obtain what we call the version. (See [Pa02a] for the connection between the original construction, which we do not need, and the one given here.) Moreover, we go on to show that the unified construction really does give cyclic GQ (see [CP03]).

To obtain the canonical form of the classical -clan put = = 1 in Eqs. (4.7) and (4.8). A simple computation shows that if , so = 1, then the classical -clan in 1/2-normalized form is given by

algebraic plane curve of degree PG(2, ) = = {() ∈ PG(2, ): f() = 0}, .

We now pick up right where we left off at the end of Section 4.8, except that = 2e with even, and (mod + 1), . The unique linear map known that stabilizes the oval is the involution given by The fixed points of this involution are the points of the line = 0, i.e., the points (0, ). But clearly the unique oval point on this line is the point (0, , 1), hence this line is a tangent line. The generator of the known stabilizer is , which acts on the points of this line as (0, ) ↦ (0, y2/δ, , from which it follows that exactly three points on this line are fixed: the oval point (0, , 1) and two others: (0, 1, 0) and (0, 0, 1). But the secant line through and passes through the point (0, 1, 0), implying that the nucleus must be (0, 0, 1).

Suppose that is a -clan for which each of the functions , , is a monomial function. In a rather remarkable paper, T. Penttila and L. Storme [PS98] show that up to the usual equivalence of -clans, the three known examples are the only ones. Since the two non-classical families exist only for odd, we assume throughout this chapter that is odd. Then the three known families have the following appearance. There is some positive integer for which

Let be a GQ with parameters (), 1, 1. A of is a set of lines that partition the points of . Dually, an of is a set of points of such that each line of is incident with a unique point of . It is easy to see that a spread must have 1 + lines and an ovoid must have 1 + points. For example, if a GQ of order is contained as a subquadrangle in a GQ with order , then each point of a line exterior to is on a unique line of . Hence the +1 lines of that meet form a spread of said to be subtended by . Spreads and ovoids of GQ have been studied a great deal and have a wide variety of connections with other geometric objects. For a general reference see J. A. Thas and S. E. Payne [TP94]. For = 2e see especially [BOPPR1] and [BOPPR2]. In this section we give a very brief introduction to the material contained in these latter two papers.