Vol. 14, No. 1, 2015

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Semiarcs with a long secant in PG(2,q)

Bence Csajbók, Tamás Héger and György Kiss

Vol. 14 (2015), No. 1, 1–26
Abstract

A t-semiarc is a point set St with the property that the number of tangent lines to St at each of its points is t. We show that if a small t-semiarc St in PG(2,q) has a large collinear subset K, then the tangents to St at the points of K can be blocked by t points not in K. In fact, we give a more general result for small point sets with less uniform tangent distribution. We show that in PG(2,q) small t-semiarcs are related to certain small blocking sets and give some characterization theorems for small semiarcs with large collinear subsets.

Keywords
finite plane, semiarc, semioval, blocking set, Sz”onyi–Weiner Lemma
Mathematical Subject Classification 2010
Primary: 51E20, 51E21
Milestones
Received: 19 July 2013
Accepted: 6 October 2014
Authors
Bence Csajbók
Tamás Héger
György Kiss