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On (q + t)-arcs of type (0, 2, t) in a desarguesian plane of order q

Published online by Cambridge University Press:  24 October 2008

Gábor Korchmáros  and
Francesco Mazzocca
Gábor Korchmáros
Affiliation:
Department of Mathematics, University of Basilicata, 85100 Potenza, Italy
Francesco Mazzocca
Affiliation:
Department of Mathematics and its Applications, University of Napoli, via Mezzocannone 8, 80134 Napoli, Italy
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Extract

This paper is concerned with certain point-sets T in a projective plane PG (2, q) over GF (q) which have only three characters with respect to the lines. We assume throughout this paper that for any line l of π

where

It is easily seen that if t = 1 then T is a (q + 1)-arc, i.e. an oval; otherwise T is a (q+t, t)-arc of type (0, 2, t). Therefore (q+t, t)-arcs of type (0, 2, t) appear to be a generalization of ovals and there are interesting connections between ovals and (q + t, t)-arcs of type (0, 2, t) from various points of view. Our purpose is to investigate such particular (k, t)-arcs using some ideas of B. Segre developed for ovals in three fundamental papers [16, 17, 18]. For these papers and more recent results in this direction the reader is referred to [6], chapter 10 and [9]. General results concerning (k, n)-arcs may be found in [6], chapter 12; see also [4, 7, 20, 23].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 3 , November 1990 , pp. 445 - 459
Copyright
Copyright © Cambridge Philosophical Society 1990

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