scholarly journals Hirzebruch-Type Inequalities Viewed as Tools in Combinatorics

10.37236/8335 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Piotr Pokora
Keyword(s):  
Projective Plane ◽  
Plane Curve ◽  
Combinatorial Problems ◽  
Complex Projective Plane ◽  
Open Problems ◽  
Line Arrangements ◽  
Dirac Conjecture ◽  
Complex Projective

The main purpose of this survey is to provide an introduction, algebro-topological in nature, to Hirzebuch-type inequalities for plane curve arrangements in the complex projective plane. These inequalities gain more and more interest due to their utility in many combinatorial problems related to point or line arrangements in the plane. We would like to present a summary of the technicalities and also some recent applications, for instance in the context of the Weak Dirac Conjecture. We also advertise some open problems and questions.

ON FINITE BRANCHED UNIFORMIZATIONS OF THE PROJECTIVE PLANE

2013 ◽  
Vol 24 (02) ◽  
pp. 1350017
Author(s):  
A. MUHAMMED ULUDAĞ ◽  
CELAL CEM SARIOĞLU
Keyword(s):  
Projective Plane ◽  
Fundamental Group ◽  
Free Product ◽  
Plane Curve ◽  
Complex Projective Plane ◽  
Galois Covering ◽  
Cyclic Groups ◽  
Galois Coverings ◽  
Complex Projective ◽  
Branching Index

We give a brief survey of the so-called Fenchel's problem for the projective plane, that is the problem of existence of finite Galois coverings of the complex projective plane branched along a given divisor and prove the following result: Let p, q be two integers greater than 1 and C be an irreducible plane curve. If there is a surjection of the fundamental group of the complement of C into a free product of cyclic groups of orders p and q, then there is a finite Galois covering of the projective plane branched along C with any given branching index.

scholarly journals Characteristic Varieties for a Class of Line Arrangements

2011 ◽  
Vol 54 (1) ◽  
pp. 56-67 ◽  
Author(s):  
Thi Anh Thu Dinh
Keyword(s):  
Projective Plane ◽  
Complex Projective Plane ◽  
Line Arrangement ◽  
Line Arrangements ◽  
Irreducible Components ◽  
Characteristic Varieties ◽  
Non Local ◽  
Complex Projective

AbstractLet be a line arrangement in the complex projective plane ℙ2, having the points of multiplicity ≥ 3 situated on two lines in , say H0 and H∞. Then we show that the non-local irreducible components of the first resonance variety are 2-dimensional and correspond to parallelograms ℙ in ℂ2 = ℙ2 \ H∞ whose sides are in and for which H0 is a diagonal.

scholarly journals On complex line arrangements and their boundary manifolds

2015 ◽  
Vol 159 (2) ◽  
pp. 189-205 ◽  
Author(s):  
V. FLORENS ◽  
B. GUERVILLE-BALLÉ ◽  
M.A. MARCO-BUZUNARIZ
Keyword(s):  
Projective Plane ◽  
Fundamental Group ◽  
Fundamental Groups ◽  
Complex Line ◽  
Complex Projective Plane ◽  
Line Arrangement ◽  
Line Arrangements ◽  
Regular Neighbourhood ◽  
Complex Projective

AbstractLet ${\mathcal A}$ be a line arrangement in the complex projective plane $\mathds{C}\mathds{P}^2$. We define and describe the inclusion map of the boundary manifold, the boundary of a closed regular neighbourhood of ${\mathcal A}$, in the exterior of the arrangement. We obtain two explicit descriptions of the map induced on the fundamental groups. These computations provide a new minimal presentation of the fundamental group of the complement.

Hirzebruch-type inequalities and plane curve configurations

2017 ◽  
Vol 28 (02) ◽  
pp. 1750013 ◽  
Author(s):  
Piotr Pokora
Keyword(s):  
Projective Plane ◽  
Type Inequality ◽  
Plane Curve ◽  
Complex Projective Plane ◽  
Characteristic Numbers ◽  
Complex Projective

In this paper, we come back to a problem proposed by F. Hirzebruch in the 1980s, namely whether there exists a configuration of smooth conics in the complex projective plane such that the associated desingularization of the Kummer extension is a ball quotient. We extend our considerations to the so-called [Formula: see text]-configurations of curves in the projective plane and we show that in most cases for a given configuration the associated desingularization of the Kummer extension is not a ball quotient. Moreover, we provide improved versions of Hirzebruch-type inequality for [Formula: see text]-configurations. Finally, we show that the so-called characteristic numbers (or [Formula: see text] numbers) for [Formula: see text]-configurations are bounded from above by [Formula: see text]. At the end of the paper we give some examples of surfaces constructed via Kummer extensions branched along conic configurations.

Introduction

2017 ◽  
Author(s):  
Paula Tretkoff
Keyword(s):  
Projective Plane ◽  
General Type ◽  
Complex Surface ◽  
Explicit Computation ◽  
Complex Projective Plane ◽  
Surfaces Of General Type ◽  
Line Arrangements ◽  
Finite Covers ◽  
Ball Quotients ◽  
Complex Projective

This chapter explains that the book deals with quotients of the complex 2-ball yielding finite coverings of the projective plane branched along certain line arrangements. It gives a complete list of the known weighted line arrangements that can produce such ball quotients, and then provides a justification for the existence of the quotients. The Miyaoka-Yau inequality for surfaces of general type, and its analogue for surfaces with an orbifold structure, plays a central role. The book also examines the explicit computation of the proportionality deviation of a complex surface for finite covers of the complex projective plane ramified along certain line arrangements. Candidates for ball quotients among these finite covers arise by choosing weights on the line arrangements such that the proportionality deviation vanishes.

scholarly journals ALEXANDER POLYNOMIALS OF COMPLEX PROJECTIVE PLANE CURVES

2018 ◽  
Vol 97 (3) ◽  
pp. 386-395 ◽  
Author(s):  
QUY THUONG LÊ
Keyword(s):  
Projective Plane ◽  
Plane Curve ◽  
Alexander Polynomial ◽  
Plane Curves ◽  
Complex Projective Plane ◽  
Multiplier Ideals ◽  
Alexander Polynomials ◽  
Irreducible Components ◽  
Complex Projective

We compute the Alexander polynomial of a nonreduced nonirreducible complex projective plane curve with mutually coprime orders of vanishing along its irreducible components in terms of certain multiplier ideals.

scholarly journals On 3-syzygy and unexpected plane curves

2021 ◽  
Author(s):  
Grzegorz Malara ◽  
Piotr Pokora ◽  
Halszka Tutaj-Gasińska
Keyword(s):  
Projective Plane ◽  
Plane Curves ◽  
Complex Projective Plane ◽  
Geometric Properties ◽  
Complex Projective

AbstractIn this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study 3-syzygy curve arrangements and we present examples that admit unexpected curves.

Spectral Sequences

2020 ◽  
pp. 45-56
Author(s):  
Loring W. Tu
Keyword(s):  
Projective Plane ◽  
Spectral Sequence ◽  
Fiber Bundles ◽  
Spectral Sequences ◽  
Complex Projective Plane ◽  
Computational Tool ◽  
Short Introduction ◽  
Complex Projective

This chapter focuses on spectral sequences. The spectral sequence is a powerful computational tool in the theory of fiber bundles. First introduced by Jean Leray in the 1940s, it was further refined by Jean-Louis Koszul, Henri Cartan, Jean-Pierre Serre, and many others. The chapter provides a short introduction, without proofs, to spectral sequences. As an example, it computes the cohomology of the complex projective plane. The chapter then details Leray's theorem. A spectral sequence is like a book with many pages. Each time one turns a page, one obtains a new page that is the cohomology of the previous page.

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