Geometry and combinatorics of toric arrangements

Abstract

A toric arrangement is a finite set of hypersurfaces in a complex torus, each hypersurface being the kernel of a character. In the first chapter we focus on the case of toric arrangements defined by root systems: by describing the action of the Weyl group, we get precise counting formulae for the layers (connected components of intersections) of the arrangement, and then we compute the Euler characteristic of its complement. In the second chapter we introduce a multiplicity Tutte polynomial M(x,y), which generalizes the ordinary one and has applications to zonotopes, multigraphs and toric arragements. We prove that M(x,y) satisfies a deletion-restriction formula and has positive coefficients. The characteristic polynomial and the Poincaré polynomial of a toric arrangement are shown to be specializations of the associated polynomial M(x,y). Furthermore, M(x,1) counts integral points in the faces of a zonotope, while M(1,y) is the graded dimension of the related discrete Dahmen-Micchelli space. In the third chapter we build wonderful models for toric arrangements. We develop the "toric analogue" of the combinatorics of nested sets, which allows to prove that the model is smooth, and to give a precise description of the normal crossing divisor.