Related papers: Line arrangements with many triple points

We prove the joints conjecture, showing that for any $N$ lines in ${\Bbb R}^3$, there are at most $O(N^{{3 \over 2}})$ points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given $N^2$ lines…

An arrangement of circles in which circles intersect only in angles of $\pi/2$ is called an \emph{arrangement of orthogonal circles}. We show that in the case that no two circles are nested, the intersection graph of such an arrangement is…

We classify all finite linear spaces on at most 15 points admitting a blocking set. There are no such spaces on 11 or fewer points, one on 12 points, one on 13 points, two on 14 points, and five on 15 points. The proof makes extensive use…

This paper introduces a general modeling framework for a multi-type maximal covering location problem in which the position of facilities in different metric spaces are simultaneously decided to maximize the demand generated by a set of…

There are many examples for point sets in finite geometry, which behave "almost regularly" in some (well-defined) sense, for instance they have "almost regular" line-intersection numbers. In this paper we investigate point sets of a…

In this paper we describe the complement of real line arrangements in the complex plane in terms of the boundary three-manifold of the line arrangement. We show that the boundary manifold of any line arrangement is a graph manifold with…

This paper proves the Hasse principle and weak approximation for varieties defined by the smooth intersection of three quadratics in at least 19 variables, over arbitrary number fields.

We show that the monodromy operator action on the first cohomology group of the Milnor fiber is combinatorially determined for line arrangements with at most triple points and containing at most 18 lines, with one possible exception.

The Sylvester-Gallai theorem says that for any finite set of non-collinear points in $\R^2$, there is some line passing through exactly two points of the set. Over the complex numbers, this theorem fails: there are finite configurations…

We find monodromy formulas for line arrangements which are fibered with respect to the projection from one point. We use them to find $0$-dimensional translated components in the first characteristic variety of the arrangement $\mathcal…

The dual codes of the ternary linear codes of the residual designs of biplanes on 56 points are used to prove the nonexistence of quasi-symmetric 2-$(56,12,9)$ and 2-$(57,12,11)$ designs with intersection numbers 0 and 3, and the…

We study the combinatorics of pseudoline arrangements in the real projective plane. Our focus lies on two classes of arrangements: simplicial arrangements and arrangements whose characteristic polynomials have only real roots. We derive…

In this paper we prove a conjecture about the dimension of linear systems of surfaces of degree d in P^3 through at most eight multiple points in general position.

We prove the sharp upper bound of at most $52$ lines on a complex K3-surface of degree four with a non-empty singular locus. We also classify the configurations of more than $48$ lines on smooth complex quartics.

We investigate finite 3-nets embedded in a projective plane over a (finite or infinite) field of any characteristic p. Such an embedding is regular when each of the three classes of the 3-net comprises concurrent lines, and irregular…

Planes are familiar mathematical objects which lie at the subtle boundary between continuous geometry and discrete combinatorics. A plane is geometrical, certainly, but the ways that two planes can interact break cleanly into discrete sets:…

Let $L$ be a set of $n$ lines in $\reals^d$, for $d\ge 3$. A {\em joint} of $L$ is a point incident to at least $d$ lines of $L$, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum…

We examine the metrics that arise when a finite set of points is embedded in the real line, in such a way that the distance between each pair of points is at least 1. These metrics are closely related to some other known metrics in the…

We discuss several extension properties of matroids and polymatroids and their application as necessary conditions for the existence of different matroid representations, namely linear, folded linear, algebraic, and entropic…

We find a new structural feature of equilibrium complex random networks without multiple and self-connections. We show that if the number of connections is sufficiently high, these networks contain a core of highly interconnected vertices.…