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On the Characterization of certain Sets of Points in Finite Projective Geometry of Dimension Three

On the Characterization of certain Sets of Points in Finite Projective Geometry of Dimension Three ON THE CHARACTERIZATION OF CERTAIN SETS OF POINTS IN FINITE PROJECTIVE GEOMETRY OF DIMENSION THREE DAVID G. GLYNN 1. Introduction Let PG {d, q) be the projective geometry of dimension d over the finite field GF(q) of q elements. A subset of points of PG{d,q) is said to be of type (1, n, q + l) if every line meets it in 1, n or q +1 points. In [6], Tallini Scafati classified all subsets of type (I, n, q+l) in PG(d , q) for d ^ 2, q > 4, except for the case n = \q + l. This case was completed by Hirschfeld and Thas, (see [2] and [3]), except for a certain set K in PG (3, q), q even. The problem for d = 3, q = 4 was solved by Hirschfeld and Hubaut, (see [1]), and in [5], Sherman gave an algebraic solution to the remaining problem of sets of type (1, 3 , 5) in PG(d, 4) for d ^ 4. In this note, the set K of Hirschfeld and Thas is shown to be the projection of a non-singular quadric of PG (4, q). This confirms the conjecture made at the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On the Characterization of certain Sets of Points in Finite Projective Geometry of Dimension Three

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References (9)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/15.1.31
Publisher site
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Abstract

ON THE CHARACTERIZATION OF CERTAIN SETS OF POINTS IN FINITE PROJECTIVE GEOMETRY OF DIMENSION THREE DAVID G. GLYNN 1. Introduction Let PG {d, q) be the projective geometry of dimension d over the finite field GF(q) of q elements. A subset of points of PG{d,q) is said to be of type (1, n, q + l) if every line meets it in 1, n or q +1 points. In [6], Tallini Scafati classified all subsets of type (I, n, q+l) in PG(d , q) for d ^ 2, q > 4, except for the case n = \q + l. This case was completed by Hirschfeld and Thas, (see [2] and [3]), except for a certain set K in PG (3, q), q even. The problem for d = 3, q = 4 was solved by Hirschfeld and Hubaut, (see [1]), and in [5], Sherman gave an algebraic solution to the remaining problem of sets of type (1, 3 , 5) in PG(d, 4) for d ^ 4. In this note, the set K of Hirschfeld and Thas is shown to be the projection of a non-singular quadric of PG (4, q). This confirms the conjecture made at the

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jan 1, 1983

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