On the homotopy type of the complement of an arrangement that is a 2-generic section of the parallel connection of an arrangement and a pencil of lines

2015/07/16 by Kristopher Williams, Williams, Kristopher
Mathematics · #14N20 (Secondary) #32S22 #52C35 (Primary) #Advanced Combinatorial Mathematics #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Mathematics and Applications #math.AG #math.AT #msc:14N20 #msc:32S22 #msc:52C35

paper · pdf · doi:10.48550/arxiv.1507.04706

13 pages, 5 figures

arxiv created 2015/07/16 · openalex publication_date 2015/07/16 · arxiv updated 2015/07/17 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

Let A be a complexified-real arrangement of lines in ℂ². Let H be any line in A . Then, form a new complexified-real arrangement B_H = A ∪ C where C ∪ \H\ is a pencil of lines with multiplicity m≥ 3 , the intersection point in the pencil is not a multiple point in A, and every line in C intersects every line in A∖ \H\ in points of multiplicity two. In this article, we show that for H₁, H₂ ∈ A we may have that B_H1 and B_H2 do not have diffeomorphic complements, but the complements of the arrangements will always be homotopy equivalent.

Related

• On the difference between real and complex arrangements
• Topological line arrangements and their topological invariants
• Connectedness and combinatorial interplay in the moduli space of line arrangements
• Configurations of points and topology of real line arrangements
• On the connectivity of the realization spaces of line arrangements