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Intersections in projective space. II: Pencils of quadrics. (English) Zbl 0644.51006
[For part I see the authors, Math. Z. 193, 215-225 (1986; Zbl 0579.51010)].
The authors give a complete classification of those pencils \({\mathcal B}\) of quadrics in the projective space PG(3,q) over a finite field GF(q), where \({\mathcal B}\) contains at least one non-singular quadric and where the base curve of \({\mathcal B}\) is not absolutely irreducible. Up to projective equivalence, there are 27 pencils of this type if q is odd, and 23 pencils if q is even. As consequences of this classification, one obtains partitions of PG(3,q) by lines (i.e. spreads of lines) and by quadrics [cp. G. L. Ebert, Can. J. Math. 37, 1163-1175 (1985; Zbl 0571.51002)]. Furthermore, the authors give the dual pencil for each pencil \({\mathcal B}\) as above, and they comment briefly on the case where the base curve is irreducible.
Reviewer: Th.Grundhöfer

51E20 Combinatorial structures in finite projective spaces
14G15 Finite ground fields in algebraic geometry
Full Text: DOI
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