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On regular {v, n}‐arcs in finite projective spaces
Author(s) -
Ueberberg Johannes
Publication year - 1993
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.3180010603
Subject(s) - mathematics , combinatorics , projective plane , blocking set , projective space , projective line , order (exchange) , collineation , subspace topology , affine plane (incidence geometry) , line (geometry) , linear subspace , plane (geometry) , dimension (graph theory) , affine space , space (punctuation) , arc (geometry) , finite geometry , affine transformation , projective test , pure mathematics , mathematical analysis , geometry , linguistics , philosophy , finance , economics , correlation
Abstract A regular {v, n}‐arc of a projective space P of order q is a set S of v points such that each line of P has exactly 0,1 or n points in common with S and such that there exists a line of P intersecting S in exactly n points. Our main results are as follows: (1) If P is a projective plane of order q and if S is a regular {v, n} ‐arc with n ≥ √q + 1, then S is a set of n collinear points, a Baer subplane, a unital, or a maximal arc. (2) If P is a projective space of order q and if S is a regular {v, n} ‐arc with n ≥ √q + 1 spanning a subspace U of dimension at least 3, then S is a Baer subspace of U , an affine space of order q in U , or S equals the point set Of U . © 1993 John Wiley & Sons, Inc.