# Intersections Of Hyperconics And Configurations In Classical Planes

1994

Dissertation

## Degree Name

Doctor of Philosophy

## Abstract

Let {dollar}\pi{dollar} = PG(2,F), where F is a field of characteristic 2 and of order greater than 2. Given a conic, its tangents all pass through a common point, the nucleus. A conic, together with its nucleus, is called a hyperconic. All conics considered are non-degenerate.;First, a relationship is established between hyperconics and certain symmetric unipotent Latin squares for all finite projective planes.;Intersection properties of hyperconics in PG(2,F), Fano configurations containing points of a hyperconic, as well as certain subplanes of PG(2,F) are studied. An open question in {dollar}\pi{dollar} = PG(2,q), q even, is: what is the size and structure of a set or maximum size of hyperovals (or hyperconics) pairwise intersecting in exactly 2 points? In PG(2,4), such a set is shown to have size 16 and to have one of 2 'dual' structures: 16 hyperconics missing a fixed line, or 16 hyperconics through a fixed point.;The former is a 2 {dollar}-{dollar} (16,6,2)-design of grid type which can be obtained from the 5 {dollar}-{dollar}(24,8,1) Mathieu design, and which can be related to singular points of a Kummer surface in PG(2,q) for q odd (see (Bruen 2)).;The latter is shown to be an affine plane in 2 ways: (i) taking the hyperconics which all contain the fixed point, as well as the lines through that fixed point (in the original plane) to be the lines of an AG(2,4); and (ii) taking the hyperconics in the original plane to be the points, and the points (except the fixed point in all 16 hyperconics) in the original plane to be the lines of an AG(2,4).;In PG(2,F) let the field F contain a subfield of order 4. Then, in PG(2,F) we describe certain sets of 6 points no 3 collinear called hexagons. It is then shown how the much studied even intersection property in PG(2,4) can be lifted (extended) to certain sets of hyperconics in PG(2,F).

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