Luyckx, Deirdre

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Ghent University Academic Bibliography

On regular hyperbolic fibrationsDeirdre LuyckxDepartment of Pure Mathematics and Computer AlgebraGhent UniversityKrijgslaan 281, S259000 GentBelgiumdluyckx@cage.ugent.beJoint work with Matthew Brown, Gary EbertA hyperbolic fibration of PG(3, q) is a collection of q − 1 hyperbolicquadrics Q+i (3, q), i = 1, 2, . . . , q − 1, and two lines L0, L∞ in PG(3, q) thatpartition the points of PG(3, q), see [1]. A hyperbolic fibration is calledregular if the lines L0 and L∞ form a conjugate (skew) pair with respect toeach of the polarities associated with the q−1 hyperbolic quadricsQ+i (3, q) ofthe fibration. A regular hyperbolic fibration agrees on L0, respectively L∞,if the extension to GF(q2) of each quadric Q+i (3, q), i = 1, 2, . . . , q−1, meetsL0, respectively L∞, in the same pair of points {x, x} which are conjugatewith respect to the extension GF(q2) of GF(q).In [2], Baker, Ebert and Penttila show (algebraically) that a regularhyperbolic fibration that agrees on one of its lines corresponds to a flock ofa quadratic cone in PG(3, q) with one conic (plane) specified, and conversely.For this correspondence, a unified geometric construction will be given forall q. Further it will be explained that all hyperbolic fibrations that agreeon one of their lines, are necessarily regular. If q is even, an even strongerresult holds: in this case it will be pointed out that all hyperbolic fibrationsare regular.References[1] R. D. Baker, J. M. Dover, G. L. Ebert, and K. L. Wantz. Hyperbolicfibrations of PG(3, q). European J. Combin., 20(1):1–16, 1999.[2] R. D. Baker, G. L. Ebert, and Tim Penttila. Hyperbolic fibrations andq-clans. Des. Codes Cryptogr., 34(2-3):295–305, 2005.

On regular hyperbolic fibrations

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On regular hyperbolic fibrations

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Ghent University Academic Bibliography