Face enumeration for line arrangements in a 2-torus

2014/04/07 by Karthik Chandrashekhar, Chandrashekhar, Karthik, Priyavrat Deshpande +1 · 1 citation
Mathematics · #32S22 #52C35 #53C22 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.AT #math.CO #msc:32S22 #msc:52C35 #msc:53C22

paper · pdf · doi:10.48550/arxiv.1404.1665

12 pages, 6 figures

arxiv created 2014/04/07 · openalex publication_date 2014/04/07 · arxiv updated 2014/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A toric arrangement is a finite collection of codimension-1 subtori in a torus. These subtori stratify the ambient torus into faces of various dimensions. Let f_i denote the number of i-dimensional faces; these so-called face numbers satisfy the Euler relation ∑_i (-1)ⁱ f_i = 0. However not all tuples of natural numbers satisfying this relation arise as face numbers of some toric arrangement. In this paper we focus on toric arrangements in a 2-dimensional torus and obtain a characterization of face numbers. In particular we show that the convex hull of these face numbers is a cone. Finally we extend some of these results to arrangements of geodesics in surfaces of higher genus.

Citations

• On the number of connected components in complements to arrangements of submanifolds

Cited by

• Strata of toric hyperplane arrangements, zonotope lattice points, and the Bondal-Thomsen collection

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