zbMATH — the first resource for mathematics

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

MSC:
 51E20 Combinatorial structures in finite projective spaces 14G15 Finite ground fields in algebraic geometry
Full Text:
References:
 [1] Bruen, H. F., (Principles of Geometry, Volume 3, (1923), Cambridge University Press) [2] A. E. Brouwer, letter. [3] Bruen, A. A.; Hirschfeld, J. W.P., Intersections in projective space I: combinatorics, Math. Z., 193, 215-225, (1986) · Zbl 0579.51010 [4] Campbell, A. D., Pencils of conics in the Galois fields of order 2n, Amer. J. Math., 49, 401-406, (1972) · JFM 53.0122.05 [5] Cicchese, M., Sulle cubiche di un piano di Galois, Rend. Mat. e Appl., 24, 291-330, (1965) · Zbl 0139.38302 [6] Cicchese, M., Sulle cubiche di un piano lineare S_{2,q} con q≡ 1 (mod 3), Rend. Mat., 4, 249-283, (1971) · Zbl 0206.50102 [7] Dickson, L. E., On families of quadratic forms in a general field, Q. J. Pure App. Math, 39, 316-333, (1908) · JFM 39.0148.01 [8] Ebert, G. L., Partitioning projective geometries into caps, Canad. J. Math., 37, 1163-1175, (1985) · Zbl 0571.51002 [9] H. van Halteren, Construction of sets of type (m, n)_{2} in PG(3, q) using combinations of disjoint quadrics, M.Sc. thesis, University of Nijmegen. [10] Hirschfeld, J. W.P, Projective Geometries over Finite Fields, (1979), Oxford University Press · Zbl 0418.51002 [11] Hirschfeld, J. W.P, Finite Projective Spaces of Three Dimensions, (1985), Oxford University Press · Zbl 0574.51001 [12] Kestenband, B. C., Projective geometries that are disjoint unions of caps, Canad. J. Math., 32, 1299-1305, (1980) · Zbl 0414.51001 [13] Kestenband, B. C., Hermitian configurations in odd-dimensional projective geometries, Canad. J. Math., 33, 500-512, (1981) · Zbl 0464.51005 [14] Schoof, R., Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A, 46, 183-211, (1987) · Zbl 0632.14021 [15] Semple, J. G.; Kneebone, G. J., Algebraic Projective Geometry, (1952), Oxford University Press · Zbl 0046.38103 [16] Todd, J. A., Projective and Analytical Geometry, (1946), Pitman London · Zbl 0061.30608
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.