Combinatorics of toric arrangements

Abstract

In this paper we build an Orlik-Solomon model for the canonical gradation of the cohomology algebra with integer coefficients of the complement of a toric arrangement. We give some results on the uniqueness of the representation of arithmetic matroids, in order to discuss how the Orlik-Solomon model depends on the poset of layers. The analysis of discriminantal toric arrangements permits us to isolate certain conditions under which two toric arrangements have diffeomorphic complements. We also give combinatorial conditions determining whether the cohomology algebra is generated in degree one.

Orlik-Solomon type presentations for the cohomology algebra of toric arrangements

“…Unlike the case of hyperplanes and matroids, we do not know whether such posets are fully determined by the arithmetic matroid. For an in-depth discussion of this question see [Pag17]. The paper [DR18] introduces group actions on semimatroids as an attempt for a unified axiomatization of posets of layers and multiplicity functions.…”

“…A crucial aspect that emerged in [DP05] and was confirmed by subsequent research in the topology of toric arrangements is that the matroid data naturally associated with every toric arrangement is not fine enough to encode meaningful geometric and topological invariants of the arrangement's complement. The quest for a suitable enrichment of matroid theory has been pursued from different points of view, i.e., by modeling the algebraic-arithmetic structure of the set of characters defining the arrangement [DM13, BM14,FM16] or by studying the properties of the pattern of intersections [DR18,Pag17] (see §1.2 and Section 2.4).…”

On projective wonderful models for toric arrangements and their cohomology

“…Toric arrangements have been studied since the early 1990s, and over the last two decades several aspects have been investigated: in particular, as far as the topology of the complement is concerned, De Concini and Procesi [12] determined the generators of the cohomology modules over C in the divisorial case, as well as the ring structure in the case of totally unimodular arrangements; d'Antonio and Delucchi, generalizing an algebraic complex first introduced by Moci and Settepanella [25], provided a presentation of the fundamental group for the complement of a divisorial complexified arrangement [5,6]; Callegaro, Delucchi and Pagaria computed the graded cohomology ring with integer coefficients (see [3,4,26]); the cohomology ring itself was computed by Callegaro et al [2].…”

Two examples of toric arrangements

“…The graded algebra with rational coefficients Gr H(M (A), Q) can be obtained from the Leray spectral sequence as shown in [Bib16] and [Dup15]. The graded algebra with integer coefficients Gr H(M (A), Z) was studied in [CD17] and from the combinatorial point of view, in [Pag17]. Recently, presentations of the cohomology algebras H(M (A), Q) and H(M (A), Z) in the spirit of [OS80] was obtained in [CDDMP18], generalizing [DCP05].…”