On a Special Configuration of Lines and Points in P^N

Journal of Geometry, 1980

Journal of Geometry, 2009

Let S = (P, L, H) be the finite planar space obtained from the 3-dimensional projective space P G(3, n) of order n by deleting a set of n-collinear points. Then, for every point p ∈ S, the quotient geometry S/p is either a projective plane or a punctured projective plane, and every line of S has size n or n + 1. In this paper, we prove that a finite planar space with lines of size n+1−s and n+1, (s ≥ 1), and such that for every point p ∈ S, the quotient geometry S/p is either a projective plane of order n or a punctured projective plane of order n, is obtained from P G(3, n) by deleting either a point, or a line or a set of n-collinear points. . 51E26.

Georgian Mathematical Journal, 2019

The equidistant set of a collection F of lines in 3-space is the set of those points whose distances to the lines in F are all equal. We present many examples and results related to the lines possibly contained in the equidistant set of F. In particular, we determine the possible numbers of lines in the equidistant set of a collection of n lines for every n > 0 {n>0} . For example, if n = 3 {n=3} , then the possible number of such lines is either 4 or 2 or 1 or 0. In a natural way, our results are connected with properties of special types of (ruled) surfaces. For example, we obtain also results on the number of lines in the intersection of quadratic surfaces.

2000

Fix integers d n 3. Here we show the existence of a nodal and connected tree-like (i.e. with lines as irreducible components) curve Y Pn such that pa(Y ) = 0, deg(Y ) = d and Y with maximal rank. If either n 6= 3 or d is not " exceptional " we prove that we may take Y of

Cornell University - arXiv, 2022

Introduction ix 0.1. General Context ix 0.2. History x 0.3. Questions and Results xi 0.4. Summary xii Chapter 1. Preliminaries 1 1.1. Harmonic points 1 1.2. Segre Embeddings and grids 5 1.3. Intersection of quadric cones in P 3 1.4. Arrangements of lines 1.5. Basic algebraic and geometric definitions used in this book Chapter 2. Weddle and Weddle-like varieties 2.1. The d-Weddle locus for a finite set of points in projective space 2.2. The d-Weddle scheme and two approaches to finding it 2.3. Different sets of six points and their Weddle schemes and loci 2.4. d-Weddle loci for some general sets of points in P 3 2.5. Connections to Lefschetz Properties Chapter 3. Geometry of the D 4 configuration 3.1. The rise of D 4 3.2. The geprociness of D 4 Chapter 4. The geography of geproci sets: a complete numerical classification 4.1. A standard construction 4.2. Classification of nondegenerate (a, b)-geproci sets for a ≤ 3 Chapter 5. Nonstandard geproci sets of points in projective space 5.1. Beyond the standard construction 5.2. The Klein configuration 5.3. The Penrose configuration 5.4. The H 4 configuration Chapter 6. Equivalences of geproci sets 6.1. Realizability over the real numbers 6.2. Projective and combinatorial equivalence of geproci sets Chapter 7. Unexpected cones 7.1. Geproci sets and unexpected cones 7.2. Unexpected cubic cones for B n+1 configurations of points 7.3. Unexpected cones for simplicial skeleta Chapter

arXiv: Algebraic Geometry, 2020

The main goal of the paper is to begin a systematic study on conic-line arrangements in the complex projective plane. We show a de Bruijn-Erdos-type inequality and Hirzebruch-type inequality for a certain class of conic-line arrangements having ordinary singularities. We will also study, in detail, certain conic-line arrangements in the context of the geography of log-surfaces and free divisors in the sense of Saito.

Discrete Mathematics, 2003

A famous result of de Bruijn and Erdős (Indag. Math. 10 (1948) 421-423) states that a ÿnite linear space has at least as many lines as points, with equality only if it is a projective plane or a near-pencil. This result led to the problem of characterizing ÿnite linear spaces for which the di erence between the number b of lines and the number v of points is assigned. In this paper ÿnite linear spaces with b − v 6 m, m being the minimum number of lines on a point, are characterized.

Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers; see for example [BH1, Cu, ELS, HaHu, HoHu, Hu1, Hu2] to cite just a few. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence [BH1, GHvT].

International Journal of Mathematics and Mathematical Sciences, 2001

Let Π = (P ,L,I) be a finite projective plane of order n, and let Π = (P ,L ,I ) be a subplane of Π with order m which is not a Baer subplane (i.e., n ≥ m 2 +m). Consider the substructure Π 0 = (P 0 ,L 0 ,I 0 ) with P 0 = P \{X ∈ P | XIl, l ∈ L }, L 0 = L\L , where I 0 stands for the restriction of I to P 0 ×L 0 . It is shown that every Π 0 is a hyperbolic plane, in the sense of Graves, if n ≥ m 2 +m+1+ m 2 + m + 2. Also we give some combinatorial properties of the line classes in Π 0 hyperbolic planes, and some relations between its points and lines.

The purpose of this note is to study configurations of lines in projective planes over arbitrary fields having the maximal number of intersection points where three lines meet. We give precise conditions on ground fields F over which such extremal configurations exist. We show that there does not exist a field admitting a configuration of 11 lines with 17 triple points, even though such a configuration is allowed combinatorially. Finally, we present an infinite series of configurations which have a high number of triple intersection points.