Sets of class in
Introduction
A set of points of has class if every -space meets in points for some . If each value occurs as an intersection number, then we say has type .
A ruled cubic surface in is a variety 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 , let be a line and let be a non-degenerate conic in a plane with . Let be the non-homogeneous coordinates of , respectively. That is, without loss of generality, we can write and . Let be a projectivity in that maps to . Consider the set of lines of joining a point of to the corresponding point of according to . Then the points on these lines form a variety 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 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 is a set of points satisfying the following intersection properties.
- (I)
has type ,
- (II)
has type ,
- (III)
has type .
In Section 2.7, we exhibit a set of points of type 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 , odd, , a set of points is a ruled cubic surface if and only if has class , and every plane that contains four points of contains at least points of .
To prove this, we first show for all that if is a set of points of class such that every plane that contains four points of contains at least points of , then each 3-space meets in either a line, a -arc, a -arc, 2 intersecting lines, a line meeting a -arc, or three lines. We then restrict to being odd so that we can apply a characterisation from [1]. A complete characterisation when is even remains an open question.
Section snippets
Proof of Theorem 1
Throughout this section, we let be a set of points in of class satisfying:
if a plane contains four points of , then it contains at least points of .
We proceed with a series of lemmas, first looking at how lines meet . In Section 2.1, we show that is a set of class , that is, every line of meets in , , , , or points. In Section 2.3, we show that no line meets in exactly three points, so is a set of class
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.
References (5)
- et al.
A characterisation of Baer subplanes
J. Geom.
(2020) - et al.
Spreads induced by varieties of and Baer subplanes
Boll. Unione Mat. Ital. B
(1981)