68 research outputs found
Quasi-configurations: building blocks for point-line configurations
We study point-line incidence structures and their properties in the
projective plane. Our motivation is the problem of the existence of
configurations, still open for few remaining values of . Our approach is
based on quasi-configurations: point-line incidence structures where each point
is incident to at least lines and each line is incident to at least
points. We investigate the existence problem for these quasi-configurations,
with a particular attention to -configurations where each element is -
or -valent. We use these quasi-configurations to construct the first
and configurations. The existence problem of finding
, , and configurations remains open.Comment: 12 pages, 9 figure
Quasi-configurations: building blocks for point-line configurations
International audienceWe study point-line incidence structures and their properties in the projective plane. Our motivation is the problem of the existence of configurations, still open for few remaining values of . Our approach is based on quasi-configurations: point-line incidence structures where each point is incident to at least lines and each line is incident to at least points. We investigate the existence problem for these quasi-configurations, with a particular attention to -configurations where each element is -or -valent. We use these quasi-configurations to construct the first and configurations. The existence problem of finding , , and configurations remains open
On the Sylvester-Gallai and the orchard problem for pseudoline arrangements
We study a non-trivial extreme case of the orchard problem for
pseudolines and we provide a complete classification of pseudoline arrangements
having triple points and double points. We have also classified those
that can be realized with straight lines. They include new examples different
from the known example of B\"or\"oczky. Since Melchior's inequality also holds
for arrangements of pseudolines, we are able to deduce that some combinatorial
point-line configurations cannot be realized using pseudolines. In particular,
this gives a negative answer to one of Gr\"unbaum's problems. We formulate some
open problems which involve our new examples of line arrangements.Comment: 5 figures, 11 pages, to appear in Periodica Mathematica Hungaric
Enumerative and Algebraic Aspects of Slope Varieties
The slope variety of a graph G is an algebraic variety whose points correspond to the slopes arising from point-line configurations of G. We start by reviewing the background material necessary to understand the theory of slope varieties. We then move on to slope varieties over finite fields and determine the size of this set. We show that points in this variety correspond to graphs without an induced path on four vertices. We then establish a bijection between graphs without an induced path on four vertices and series-parallel networks. Next, we study the defining polynomials of the slope variety in more detail. The polynomials defining the slope variety are understood but we show that those of minimal degree suffice to define the slope variety set theoretically. We conclude with some remarks on how we would define the slope variety for point-line configurations in higher dimensions
Zoology of Atlas-groups: dessins d'enfants, finite geometries and quantum commutation
Every finite simple group P can be generated by two of its elements. Pairs of
generators for P are available in the Atlas of finite group representations as
(not neccessarily minimal) permutation representations P. It is unusual but
significant to recognize that a P is a Grothendieck's dessin d'enfant D and
that most standard graphs and finite geometries G-such as near polygons and
their generalizations-are stabilized by a D. In our paper, tripods P -- D -- G
of rank larger than two, corresponding to simple groups, are organized into
classes, e.g. symplectic, unitary, sporadic, etc (as in the Atlas). An
exhaustive search and characterization of non-trivial point-line configurations
defined from small index representations of simple groups is performed, with
the goal to recognize their quantum physical significance. All the defined
geometries G' s have a contextuality parameter close to its maximal value 1.Comment: 19 page
Quantum contextual finite geometries from dessins d'enfants
We point out an explicit connection between graphs drawn on compact Riemann
surfaces defined over the field of algebraic numbers ---
so-called Grothendieck's {\it dessins d'enfants} --- and a wealth of
distinguished point-line configurations. These include simplices,
cross-polytopes, several notable projective configurations, a number of
multipartite graphs and some 'exotic' geometries. Among them, remarkably, we
find not only those underlying Mermin's magic square and magic pentagram, but
also those related to the geometry of two- and three-qubit Pauli groups. Of
particular interest is the occurrence of all the three types of slim
generalized quadrangles, namely GQ(2,1), GQ(2,2) and GQ(2,4), and a couple of
closely related graphs, namely the Schl\"{a}fli and Clebsch ones. These
findings seem to indicate that {\it dessins d'enfants} may provide us with a
new powerful tool for gaining deeper insight into the nature of
finite-dimensional Hilbert spaces and their associated groups, with a special
emphasis on contextuality.Comment: 18 page
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