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Regular hyperbolic fibrations. (English) Zbl 0991.51006

A hyperbolic fibration is a partition of the point set of the 3-dimensional projective space PG\((3,q)\) over the field of \(q\) elements into two lines and \(q-1\) hyperbolie quadrics. If the two lines are conjugate with respect to every quadric in the set, then the fibration is called regular. There seem to be no examples (yet) of non-regular fibrations. As is easily seen, fibrations give rise to \(2^{q-1}\) spreads, which produce translation planes. The paper under review uses the fibration context to study these spreads. In particular, two (interrelated) questions seem to be prominent here: (1) how many isomorphism classes of spreads does a fibration define? (2) are there extra reguli in the associated spread that are not a regulus of a hyperbolic quadric of the partition? Both questions, applied to the examples, get a lot of attention (and solutions!) in the paper. In particular, it seems that the framework of hyperbolic fibrations is very suitable to study the spreads related to so-called \(j\)-planes [introduced by N. L. Johnson, R. Pomareda, and F. W. Wilke in J. Comb. Theory, Ser. A 56, No. 2, 271-284 (1991; Zbl 0724.51008)].
One of the most interesting features appears at the end, where it is pointed out that there is a strong relation with \(q\)-clans and flocks of quadratic cones, as remarked by several participants of the Fourth Isle of Thorns conference in Sussex. This relationship is not worked out in the paper under review, but the authors promise to develop this connection in a subsequent paper. I am looking forward to it.

MSC:

51E23 Spreads and packing problems in finite geometry
51A40 Translation planes and spreads in linear incidence geometry

Citations:

Zbl 0724.51008

Software:

Magma
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References:

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