20 research outputs found
Classification of Moduli Spaces of Arrangements of 9 Projective Lines
In this paper, we present a proof that the list of the classification of
arrangements of 9 lines by Nazir and Yoshinaga is complete.Comment: Changed notations in the definition of moduli space to improve
clarity. Results unchange
Improvement of inequalities for the -structures and some geometrical connections
summary:The main results are the inequalities (1) and (6) for the minimal number of -structure classes,which improve the ones from [3], and also some geometrical connections, especially the inequality (13)
Lines, Circles, Planes and Spheres
Let be a set of points in , no three collinear and not
all coplanar. If at most are coplanar and is sufficiently large, the
total number of planes determined is at least . For similar conditions and
sufficiently large , (inspired by the work of P. D. T. A. Elliott in
\cite{Ell67}) we also show that the number of spheres determined by points
is at least , and this bound is best
possible under its hypothesis. (By , we are denoting the
maximum number of three-point lines attainable by a configuration of
points, no four collinear, in the plane, i.e., the classic Orchard Problem.)
New lower bounds are also given for both lines and circles.Comment: 37 page
A generalisation of Sylvester's problem to higher dimensions
In this article we consider to be a set of points in -space with the property that any points of span a hyperplane and not all the points of are contained in a hyperplane. The aim of this article is to introduce the function , which denotes the minimal number of hyperplanes meeting in precisely points, minimising over all such sets of points with .Postprint (published version