## Innovations in Incidence Geometry

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

#### 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.

#### Article information

Source
Innov. Incidence Geom., Volume 14, Number 1 (2015), 1-26.

Dates
Received: 19 July 2013
Accepted: 6 October 2014
First available in Project Euclid: 28 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.iig/1551323025

Digital Object Identifier
doi:10.2140/iig.2015.14.1

Mathematical Reviews number (MathSciNet)
MR3450949

Zentralblatt MATH identifier
1351.51007

#### Citation

Csajbók, Bence; Héger, Tamás; Kiss, György. Semiarcs with a long secant in PG(2,q). Innov. Incidence Geom. 14 (2015), no. 1, 1--26. doi:10.2140/iig.2015.14.1. https://projecteuclid.org/euclid.iig/1551323025

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