Related papers: k-nets embedded in a projective plane over a field
Let X be a smooth projective variety and let K be the canonical divisor of X. In this paper, we study embeddings of X given by adjoint line bundles of the form K+L, where L is an ample line bundle. When X is a regular surface (i.e. H^1(X,…
This paper completes the classification of nets of conics containing at least one double line in $\mathrm{PG}(2,q)$ for $q$ even. This classification contributes to the classification of partially symmetric tensors in $\mathbb{F}_q^3…
We study extensions of the classic \emph{Line Cover} problem, which asks whether a set of $n$ points in the plane can be covered using $k$ lines. Line Cover is known to be NP-hard, and we focus on two natural generalizations. The first is…
Given a set $S$ of $n$ points in $\mathbb{R}^d$, a $k$-set is a subset of $k$ points of $S$ that can be strictly separated by a hyperplane from the remaining $n-k$ points. Similarly, one may consider $k$-facets, which are hyperplanes that…
In the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially…
The aim of this paper is twofold: First we classify all abstract light dual multinets of order $6$ which have a unique line of length at least two. Then we classify the weak projective embeddings of these objects in projective planes over…
A characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of subspaces of a vector space over a finite field of determined characteristic, and does not in general hold over other characteristics. In this…
In this note we consider the flat bundle U and the kernel K of the Higgs field naturally associated to any (polarized) variation of Hodge structures of weight 1. We study how strict the inclusion of U in K can be in the geometric case. More…
In this paper we investigate light dual multinets labeled by a finite group in the projective plane $PG(2,\mathbb{K})$ defined over a field $\mathbb{K}$. We present two classes of new examples. Moreover, under some conditions on the…
We consider problems where the input is a set of points in the plane and an integer $k$, and the task is to find a subset $S$ of the input points of size $k$ such that $S$ satisfies some property. We focus on properties that depend only on…
We show that a line arrangement in the complex projective plane supports a nontrivial resonance variety if and only if it is the underlying arrangement of a "multinet," a multi-arrangement with a partition into three or more equinumerous…
Given a planar digraph $G$ and a positive even integer $k$, an embedding of $G$ in the plane is k-modal, if every vertex of $G$ is incident to at most $k$ pairs of consecutive edges with opposite orientations, i.e., the incoming and the…
In this paper, we discuss matching extendability of optimal $1$-projective plane graphs (abbreviated as O1PPG), which are drawn on the projective plane $P^2$ so that every edge crosses another edge at most once, and has $n$ vertices and…
$\newcommand{\eps}{\varepsilon}\newcommand{\tldO}{\widetilde{O}}$Consider the problem of constructing weak $\eps$-nets where the stabbing elements are lines or $k$-flats instead of points. We study this problem in the simplest setting where…
We use the theory of oriented matroids to show that any linear embedding of $K_9$, the complete graph on nine vertices, contains a non-split link with three components.
The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics over $\mathbb{C}$ and $\mathbb{R}$ in 1906--1907. The analogous problem for…
A Kakeya set $\mathcal{K}$ in an affine plane of order $q$ is the point set covered by a set $\mathcal{L}$ of $q+1$ pairwise non-parallel lines. Large Kakeya sets were studied by Dover and Mellinger; in [6] they showed that Kakeya sets with…
We show that finding orthogonal grid-embeddings of plane graphs (planar with fixed combinatorial embedding) with the minimum number of bends in the so-called Kandinsky model (which allows vertices of degree $> 4$) is NP-complete, thus…
A popular model to measure the stability of a network is k-core - the maximal induced subgraph in which every vertex has at least k neighbors. Many studies maximize the number of vertices in k-core to improve the stability of a network. In…
The main goal of this paper is to give a general method to compute (via computer algebra systems) an explicit set of generators of the ideals of the projective embeddings of some ruled surfaces, namely projective line bundles over curves…