ISSN:
1572-9168
Keywords:
Primary: 51E20
;
Secondary 51E24, 05B25
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Atube of even orderq=2 d is a setT={L, $$\mathcal{L}$$ } ofq+3 pairwise skew lines in PG(3,q) such that every plane onL meets the lines of $$\mathcal{L}$$ in a hyperoval. Thequadric tube is obtained as follows. Take a hyperbolic quadricQ=Q 3 + (q) in PG(3,q); letL be an exterior line, and let $$\mathcal{L}$$ consist of the polar line ofL together with a regulus onQ. In this paper we show the existence of tubes of even order other than the quadric one, and we prove that the subgroup of PГL(4,q) fixing a tube {L, $$\mathcal{L}$$ } cannot act transitively on $$\mathcal{L}$$ . As pointed out by a construction due to Pasini, this implies new results for the existence of flat π.C 2 geometries whoseC 2-residues are nonclassical generalized quadrangles different from nets. We also give the results of some computations on the existence and uniqueness of tubes in PG(3,q) for smallq. Further, we define tubes for oddq (replacing ‘hyperoval’ by ‘conic’ in the definition), and consider briefly a related extremal problem.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01266318
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