ISSN:
1572-9168
Schlagwort(e):
51E15
;
51E25
Quelle:
Springer Online Journal Archives 1860-2000
Thema:
Mathematik
Notizen:
Abstract This paper is a survey on the existence and non-existence of ovoids and spreads in the known finite generalized quadrangles. It also contains the following new results. We prove that translation generalized quadrangles of order (s,s 2), satisfying certain properties, have a spread. This applies to three known infinite classes of translation generalized quadrangles. Further a new class of ovoids in the classical generalized quadranglesQ(4, 3 e ),e≥3, is constructed. Then, by the duality betweenQ(4, 3 e ) and the classical generalized quadrangleW (3 e ), we get line spreads of PG(3, 3 e ) and hence translation planes of order 32e . These planes appear to be new. Note also that only a few classes of ovoids ofQ(4,q) are known. Next we prove that each generalized quadrangle of order (q 2,q) arising from a flock of a quadratic cone has an ovoid. Finally, we give the following characterization of the classical generalized quadranglesQ(5,q): IfS is a generalized quadrangle of order (q,q 2),q even, having a subquadrangleS′ isomorphic toQ(4,q) and if inS′ each ovoid consisting of all points collinear with a given pointx ofS\S′ is an elliptic quadric, thenS is isomorphic toQ(5,q).
Materialart:
Digitale Medien
URL:
http://dx.doi.org/10.1007/BF01278475
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