Translation Generalized Quadrangles for Which the Translation Dual Arises From a Flock

Abstract

Abstract. It is shown that each finite translation generalized quadrangle (TGQ) S, which is the translation dual of the point-line dual of a flock generalized quadrangle, has a line [∞] each point of which is a translation point. This leads to the fact that the full group of automorphisms of S acts 2-transitively on the points of [∞], and the observation applies to the point-line duals of the Kantor flock generalized quadrangles, the Roman generalized quadrangles and the recently discovered Penttila-Williams generalized quadrangle. Moreover, by previous work of the author, the non-classical generalized quadrangles (GQ's) which have two distinct translation points, are precisely the TGQ's of which the translation dual is the point-line dual of a non-classical flock GQ.We emphasize that, for a long time, it has been thought that every non-classical TGQ which is the translation dual of the point-line dual of a flock GQ has only one translation point. There are important consequences for the theory of generalized ovoids (or eggs) in PG(4n − 1, q), the study of span-symmetric generalized quadrangles, derivation of flocks of the quadratic cone in PG(3, q), subtended ovoids in generalized quadrangles, and the understanding of automorphism groups of certain generalized quadrangles. Several problems on these topics will be solved completely.2000 Mathematics Subject Classification. 51E12.

Determination of generalized quadrangles with distinct elation points

“…In [38], the converse of (iii) was (unexpectedly) obtained: (a) they have some line L each point of which is an elation point (in fact, each of these points is a translation point); (b) each line M of the GQ which meets L is a regular line (including L).…”

“…As there are s + 1 distinct -orbits in S\ , it immediately follows that L is contained in at least (and then precisely, by straightforward counting) s 3 + s 2 distinct subGQ's of S of order s, all isomorphic to Q(4, s). By [38] (see also, implicitly, [30]), it follows that each point incident with L is a translation point, and so by [37], S is the translation dual of the point-line dual of a flock GQ S(F). Now suppose that does not act semiregularly on S\ .…”

Semifield flocks, eggs, and ovoids of Q (4,q)

“…r Remark 4.3. After the writing of this paper was completed, in [12] it was shown that there exists a symmetry about a line not on the point ðyÞ (of the form calculated in [5, 5.4]), from which it then follows that every point of the line EðyÞ is a translation point of TðEÞ, and hence that for a fixed subGQ S and a fixed point of type (i) z of TðEÞnS, there exists an automorphism of TðEÞ fixing S and mapping the point z to a point of type (ii). This implies that the ovoid of S subtended from z is isomorphic to an ovoid subtended from a point of type (ii).…”

“…This implies that the ovoid of S subtended from z is isomorphic to an ovoid subtended from a point of type (ii). In this way the solution of the isomorphism problem was completed in [12].…”

Subquadrangles of order s of generalized quadrangles of order (s,s2), Part II

“…With the BLT-set of W ðsÞ there corresponds a BLT-set on Qð4; sÞ: Let wAE and project from w onto a PGð3; sÞCPGð4; sÞ with Qð4; sÞCPGð4; sÞ and wePGð3; sÞ: With Qð4; sÞ there corresponds a GQ T 2 ðCÞ in PGð3; sÞ; C a conic, and with E\fwg there corresponds a set E 0 of size s in PGð3; sÞ\g; with g the plane of C: If e 1 ; e 2 AE 0 ; e 1 ae 2 ; then /e 1 ; e 2 S-E 0 ¼ J has size s 0 and J,ð/e 1 ; e 2 S-gÞ is a subline over GFðs 0 Þ of /e 1 ; e 2 S: It follows that any flock corresponding to the BLTset is a semifield flock, that is, it follows that the point-line dual # S of S is a TGQ with any line of S incident with ðNÞ as base point; see [6,10,11,19]. Now, by Theorem 44 of Thas [24], the flocks corresponding to the BLT-set are Kantor semifield flocks, that is, S is a Kantor semifield flock GQ. Hence, by Theorem 4.1, S 0 DW ðsÞ and either SDHð3; s 2 Þ or S 0 is one of the s 3 þ s 2 subquadrangles of order s containing the point ðNÞ: & Proof.…”