On line and pseudoline configurations and ball-quotients

In this note we show that there are no real configurations of $d\geq 4$ lines in the projective plane such that the associated Kummer covers of order $3^{d-1}$ are ball-quotients and there are no configurations of $d\geq 4$ lines such that the Kummer covers of order $4^{d-1}$ are ball-quotients. Moreover, we show that there exists only one configuration of real lines such that the associated Kummer cover of order $5^{d-1}$ is a ball-quotient. In the second part we consider the so-called topological $(n_{k})$-configurations and we show, using Shnurnikov's inequality, that for $n < 27$ there do not exist $(n_{5})$-configurations and and for $n < 41$ there do not exist $(n_{6})$-configurations.Comment: 7 pages, one figure. This is the final version, incorporating the suggestions of the referee, to appear in ARS Mathematica Contemporane

On the Sylvester-Gallai and the orchard problem for pseudoline arrangements

We study a non-trivial extreme case of the orchard problem for $12$ pseudolines and we provide a complete classification of pseudoline arrangements having $19$ triple points and $9$ 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

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

On topological and geometric (194)(19_4) configurations

An $(n_k)$ configuration is a set of $n$ points and $n$ lines such that each point lies on $k$ lines while each line contains $k$ points. The configuration is geometric, topological, or combinatorial depending on whether lines are considered to be straight lines, pseudolines, or just combinatorial lines. The existence and enumeration of $(n_k)$ configurations for a given $k$ has been subject to active research. A current front of research concerns geometric $(n_4)$ configurations: it is now known that geometric $(n_4)$ configurations exist for all $n \ge 18$, apart from sporadic exceptional cases. In this paper, we settle by computational techniques the first open case of $(19_4)$ configurations: we obtain all topological $(19_4)$ configurations among which none are geometrically realizable.Comment: 13 pages, 7 figure

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

On the Finding of Final Polynomials

Final polynomials have been used to prove non-representability for oriented matroids, i.e. to decide whether geometric embeddings of combinatorial structures exist. They received more attention when Dress and Sturmfels, independently, pointed out that non-representable oriented matroids always possess a final polynomial as a consequence of an appropriate real version of Hilbert's Nullstellensatz. We discuss the more difficult problem of determining such final polynomials algorithmically. We introduce the notion of bi-quadratic final polynomials, and we show that finding them is equivalent to solving an LP-Problem. We apply a new theorem about symmetric oriented matroids to a series of cases of geometrical interest

On topological and geometric (194)(19_4) configurations

International audienceAn $(n_k)$ configuration is a set of $n$ points and $n$ lines such that each point lies on $k$ lines while each line contains $k$ points. The configuration is geometric, topological, or combinatorial depending on whether lines are considered to be straight lines, pseudolines, or just combinatorial lines. The existence and enumeration of $(n_k)$ configurations for a given $k$ has been subject to active research. A current front of research concerns geometric $(n_4)$ configurations: it is now known that geometric $(n_4)$ configurations exist for all $n \ge 18$, apart from sporadic exceptional cases. In this paper, we settle by computational techniques the first open case of $(19_4)$ configurations: we obtain all topological $(19_4)$ configurations among which none are geometrically realizable

Enumerating topological (nk)(n_k)-configurations

International audienceAn $(n_k)$-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

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