Innovations in Incidence Geometry

Semiovals from unions of conics

Abstract

A semioval in a projective plane $π$ is a collection of points $O$ with the property that for every point $P$ of $O$, there exists exactly one line of $π$ meeting $O$ precisely in the point $P$. There are many known constructions of and theoretical results about semiovals, especially those that contain large collinear subsets.

In a Desarguesian plane $π$ a conic, the set of zeroes of some nondegenerate quadratic form, is an example of a semioval of size $q + 1$ that also forms an arc (i.e., no three points are collinear). As conics are minimal semiovals, it is natural to use them as building blocks for larger semiovals. Our goal in this work is to classify completely the sets of conics whose union forms a semioval.

Article information

Source
Innov. Incidence Geom., Volume 12, Number 1 (2011), 61-83.

Dates
First available in Project Euclid: 28 February 2019

https://projecteuclid.org/euclid.iig/1551323069

Digital Object Identifier
doi:10.2140/iig.2011.12.61

Mathematical Reviews number (MathSciNet)
MR2942718

Zentralblatt MATH identifier
1292.51006

Keywords
semioval conic

Citation

Dover, Jeremy M.; Mellinger, Keith E. Semiovals from unions of conics. Innov. Incidence Geom. 12 (2011), no. 1, 61--83. doi:10.2140/iig.2011.12.61. https://projecteuclid.org/euclid.iig/1551323069

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