Sets of class [q+1,2q+1,3q+1]3 in PG(4,q)

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Abstract

The article gives a combinatorial characterisation of a ruled cubic surface in PG(4,q). In PG(4,q), q odd, q9, a set K of q2+2q+1 points is a ruled cubic surface if and only if K has class [q+1,2q+1,3q+1]3 such that every plane that contains four points of K contains at least q+1 points of K.

Introduction

A set K of points of PG(n,q) has class [x1,,xr]s if every s-space meets K in xi points for some i=1,,r. If each value xi occurs as an intersection number, then we say K has type (x1,,xr)s.

A ruled cubic surface in PG(4,q) is a variety V23 of dimension 2 and order 3. The points of a ruled cubic surface form a scroll that rules a line and a non-degenerate conic according to a projectivity. That is, in PG(4,q), let be a line and let C be a non-degenerate conic in a plane π with π=. Let θ,ϕ be the non-homogeneous coordinates of , C respectively. That is, without loss of generality, we can write ={(1,θ,0,0,0)θFq{}} and C={(0,0,1,ϕ,ϕ2)ϕFq{}}. Let σ be a projectivity in PGL(2,q) that maps (1,θ) to (1,ϕ). Consider the set of q+1 lines of PG(4,q) joining a point of to the corresponding point of C according to σ. Then the points on these lines form a variety V23 of dimension 2 and order 3, that is, a ruled cubic surface. The ruling lines (called generators) are pairwise disjoint, and together with are the only lines on the surface. The surface contains q2 conics, called conic directrices, which pairwise meet in a point. Each conic directrix contains one point of each generator and is disjoint from . For more details, see [2], in particular, all ruled cubic surfaces are projectively equivalent.

In this article, we give a characterisation of ruled cubic surfaces by looking at the intersection numbers with 2- and 3-spaces. It follows from [2], [3] that a ruled cubic surface K is a set of q2+2q+1 points satisfying the following intersection properties.

  • (I)

    K has type (q+1,2q+1,3q+1)3,

  • (II)

    K has type (0,1,2,3,q+1,2q+1)2,

  • (III)

    K has type (0,1,2,q+1)1.

In Section 2.7, we exhibit a set of q2+2q+1 points of type (q+1,2q+1,3q+1)3 which is not a ruled cubic surface. That is, intersection sizes with 3-spaces is not enough to characterise a ruled cubic surface. This article uses geometric techniques to prove the following combinatorial characterisation of ruled cubic surfaces.

Theorem 1

In PG(4,q), q odd, q9, a set K of q2+2q+1 points is a ruled cubic surface if and only if K has class [q+1,2q+1,3q+1]3, and every plane that contains four points of K contains at least q+1 points of K.

To prove this, we first show for all q that if K is a set of points of class [q+1,2q+1,3q+1]3 such that every plane that contains four points of K contains at least q+1 points of K, then each 3-space meets K in either a line, a (q+1)-arc, a (q+1)3-arc, 2 intersecting lines, a line meeting a (q+1)-arc, or three lines. We then restrict q to being odd so that we can apply a characterisation from [1]. A complete characterisation when q is even remains an open question.

Section snippets

Proof of Theorem 1

Throughout this section, we let K be a set of q2+2q+1 points in PG(4,q) of class [q+1,2q+1,3q+1]3 satisfying:

  • []

    if a plane contains four points of K, then it contains at least q+1 points of K.

We proceed with a series of lemmas, first looking at how lines meet K. In Section 2.1, we show that K is a set of class [0,1,2,3,q,q+1]1, that is, every line of PG(4,q) meets K in 0, 1, 2, 3, q or q+1 points. In Section 2.3, we show that no line meets K in exactly three points, so K is a set of class [0,1,2,q,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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