Characterisations of elementary pseudo-caps and good eggs

In this note, we use the theory of Desarguesian spreads to investigate good eggs. Thas showed that an egg in $\mathrm{PG}(4n-1, q)$, $q$ odd, with two good elements is elementary. By a short combinatorial argument, we show that a similar statement holds for large pseudo-caps, in odd and even characteristic. As a corollary, this improves and extends the result of Thas, Thas and Van Maldeghem (2006) where one needs at least 4 good elements of an egg in even characteristic to obtain the same conclusion. We rephrase this corollary to obtain a characterisation of the generalised quadrangle $T_3(\mathcal{O})$ of Tits. Lavrauw (2005) characterises elementary eggs in odd characteristic as those good eggs containing a space that contains at least 5 elements of the egg, but not the good element. We provide an adaptation of this characterisation for weak eggs in odd and even characteristic. As a corollary, we obtain a direct geometric proof for the theorem of Lavrauw

Switching of generalized quadrangles of order s and applications

Peer Reviewe

Generalized quadrangles of orrder (s, s2), I

AbstractIn this paper generalized quadrangles of order (s, s2), s > 1, satisfying property (G) at a line, at a pair of points, or at a flag, are studied. Property (G) was introduced by S. E. Payne (Geom. Dedicata32 (1989), 93–118) and is weaker than 3-regularity (see S. E. Payne and J. A. Thas, “Finite Generalized Quadrangles,” Pitman, London, 1984). It was shown by Payne that each generalized quadrangle of order (s2, s), s > 1, arising from a flock of a quadratic cone, has property (G) at its point (∞). In particular translation generalized quadrangles satisfying property (G) are considered here. As an application it is proved that the Roman generalized quadrangles of Payne contain at least s3 + s2 classical subquadrangles Q(4, s). Also, as a by-product, several classes of ovoids of Q(4, s), s odd, are obtained; one of these classes is new. The goal of Part II is the classification of all translation generalized quadrangles satisfying property (G) at some flag ((∞), L)

Finite semifields and nonsingular tensors

In this article, we give an overview of the classification results in the theory of finite semifields (note that this is not intended as a survey of finite semifields including a complete state of the art (see also Remark 1.10)) and elaborate on the approach using nonsingular tensors based on Liebler (Geom Dedicata 11(4):455-464, 1981)

On generalized quadrangles with some concurrent axes of symmetry

Let S be a finite Generalized Quadrangle (GQ) of order (s, t), s not equal 1 not equal t, and suppose L is a line of S. A symmetry about L is an automorphism of S which fixes every line concurrent with L. A line L is an axis of symmetry if there is a full group of size s of symmetries about L. A point of a generalized quadrangle is a translation point if every line through it is an axis of symmetry. If there is a point p in a GQ S = (P, B, 1) for which there is a group G of automorphisms of the GQ which fixes p linewise, and such that G acts regularly on the points of P \ p(perpendicular to), then S is called an elation generalized quadrangle, and instead of S, often the notations (S-(P), G) or S(P) are used. If G is abelian, then (S-(P), G) is a translation generalized quadrangle (TGQ), and a GQ is a TGQ S-(P) if and only if p is a translation point, see [9]. We study the following two problems. (1) Suppose S is a G Q of order (8, t) s not equal 1 not equal t. How many distinct axes of symmetry through the same point p are needed to conclude that every line through p is an axis of symmetry, and hence that S-(P) is a TGQ? (2) Given a TGQ (S-(P), G), what is the minimum number of distinct lines through p such that G is generated by the symmetries about these lines

Flocks, ovoids and generalized quadrangles

In this paper we will discuss some of the connections between flocks of quadratic cones, ovoids of PG(3, q) and generalized quadrangles