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 $(n_4)$ configurations, still open for few remaining values of $n$. Our approach is based on quasi-configurations: point-line incidence structures where each point is incident to at least $3$ lines and each line is incident to at least $3$ points. We investigate the existence problem for these quasi-configurations, with a particular attention to $3|4$-configurations where each element is $3$- or $4$-valent. We use these quasi-configurations to construct the first $(37_4)$ and $(43_4)$ configurations. The existence problem of finding $(22_4)$, $(23_4)$, and $(26_4)$ 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 $(n_4)$ configurations, still open for few remaining values of $n$. Our approach is based on quasi-configurations: point-line incidence structures where each point is incident to at least $3$ lines and each line is incident to at least $3$ points. We investigate the existence problem for these quasi-configurations, with a particular attention to $3|4$-configurations where each element is $3$-or $4$-valent. We use these quasi-configurations to construct the first $(37_4)$ and $(43_4)$ configurations. The existence problem of finding $(22_4)$, $(23_4)$, and $(26_4)$ configurations remains open

On some points-and-lines problems and configurations

We apply an old method for constructing points-and-lines configurations in the plane to study some recent questions in incidence geometry.Comment: 14 pages, numerous figures of point-and-line configurations; to appear in the Bezdek-50 special issue of Periodica Mathematica Hungaric

Conditional probabilities via line arrangements and point configurations

We study the connection between probability distributions satisfying certain conditional independence (CI) constraints, and point and line arrangements in incidence geometry. To a family of CI statements, we associate a polynomial ideal whose algebraic invariants are encoded in a hypergraph. The primary decompositions of these ideals give a characterisation of the distributions satisfying the original CI statements. Classically, these ideals are generated by 2-minors of a matrix of variables, however, in the presence of hidden variables, they contain higher degree minors. This leads to the study of the structure of determinantal hypergraph ideals whose decompositions can be understood in terms of point and line configurations in the projective space.Comment: 24 pages, 5 figure

Enumerating topological (nk)(n_k)-configurations

An $(n_k)$-configuration is a set of $n$ points and $n$ lines in the projective plane such that their point-line incidence graph is $k$-regular. The configuration is geometric, topological, or combinatorial depending on whether lines are considered to be straight lines, pseudolines, or just combinatorial lines. We provide an algorithm for generating, for given $n$ and $k$, all topological $(n_k)$-configurations up to combinatorial isomorphism, without enumerating first all combinatorial $(n_k)$-configurations. We apply this algorithm to confirm efficiently a former result on topological $(18_4)$-configurations, from which we obtain a new geometric $(18_4)$-configuration. Preliminary results on $(19_4)$-configurations are also briefly reported.Comment: 18 pages, 11 figure

From Pappus Theorem to parameter spaces of some extremal line point configurations and applications

In the present work we study parameter spaces of two line point configurations introduced by B\"or\"oczky. These configurations are extremal from the point of view of Dirac-Motzkin Conjecture settled recently by Green and Tao. They have appeared also recently in commutative algebra in connection with the containment problem for symbolic and ordinary powers of homogeneous ideals and in algebraic geometry in considerations revolving around the Bounded Negativity Conjecture. Our main results are Theorem A and Theorem B. We show that the parameter space of what we call $B12$ configurations is a three dimensional rational variety. As a consequence we derive the existence of a three dimensional family of rational $B12$ configurations. On the other hand the moduli space of $B15$ configurations is shown to be an elliptic curve with only finitely many rational points, all corresponding to degenerate configurations. Thus, somewhat surprisingly, we conclude that there are no rational $B15$ configurations.Comment: 17 pages, v.2. title modified, material reorganized, introduction new rewritten, discussion more streamline