Face enumeration for line arrangements in a $2$-torus

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 ${\sum }_i(-1{)}^i{f}_i=0$. 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.

12 pages, 6 figures

References in corpus (1)

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