Journal article

Publication year: 1999

Pages: 139-176

Journal: Forum Mathematicum

Volume number: 11

Issue number: 2

ISSN: 0933-7741

eISSN: 1435-5337

Publisher: De Gruyter

Abstract: 

In this paper, we classify all generalized quadrangles weakly embedded in projective space of degree >2. More exactly, given a .(possibly infinite) generalized quadrangle Gamma = (P, L, I) and a map pi i from P (respectively L) to the set of points (respectively lines) of a projective space PG(d, IK), IK some skew field, d greater than or equal to 2 (but not necessarily finite), such that

(i) pi is injective on points,

(ii) if x is an element of P and L is an element of L with xIL, then x(pi) is incident with L-pi in PG(d, K),

(iii) the set of points {x(pi)\x is an element of P} generates PG (d,K),

(iv) if x, y is an element of P such that y(pi) is contained in the subspace of PG(d, K) generated by the set {z(pi)\z is collinear with x in Gamma}, then y is collinear with x in Gamma,

(v) there exists a line of PG (d, K) not in the image of pi and which meets Gamma in at least 3 points, then we show that Gamma is a Moufang quadrangle and we can explicitly describe the weak embedding of in PG(d, K). Our proof uses the recent classification of all Moufang quadrangles, as given by Tits and Wieiss [22].

Harvard Citation style: Steinbach, A. and Van Maldeghem, H. (1999) Generalized quadrangles weakly embedded of degree > 2 in projective space, Forum Mathematicum, 11(2), pp. 139-176

APA Citation style: Steinbach, A., & Van Maldeghem, H. (1999). Generalized quadrangles weakly embedded of degree > 2 in projective space. Forum Mathematicum. 11(2), 139-176.