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This paper presents a bridge between the theories of wonderful models associated with toric arrangements and wonderful models associated with hyperplane arrangements. In a previous work, the same authors noticed that the model of the toric arrangement of type $A_{n-1}$ is isomorphic to the one of the hyperplane arrangement of type $A_{n}$; it is natural to ask if there exist similar isomorphisms between other families of arrangements. The aim of this paper is to study one such family, namely the family of arrangements defined by graphs. The main result states that there is indeed an isomorphism between the model of a toric arrangement defined by a graph $\Gamma$ and the model of a hyperplane arrangement defined by the cone of $\Gamma$, provided that a suitable building set is chosen., Comment: 19 pages, 2 figures

In this paper we find monomial bases for the integer cohomology rings of compact wonderful models of toric arrangements. In the description of the monomials various combinatorial objects come into play: building sets, nested sets, and the fan of a suitable toric variety. We provide some examples computed via a SageMath program and then we focus on the case of the toric arrangements associated with root systems of type A. Here the combinatorial description of our basis offers a geometrical point of view on the relation between some Eulerian statistics on the symmetric group., Comment: 39 pages, 7 figures

We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of combinatorial data. This is obtained by introducing a differential graded algebra over Q whose minimal model is equivalent to the Sullivan minimal model of the arrangement.

For any triple given by a positive integer n, a finite group G, and a faithful representation V of G, one can describe a subspace arrangement whose intersection lattice is a generalized Dowling lattice in the sense of Hanlon. In this paper we construct the minimal De Concini-Procesi wonderful model associated to this subspace arrangement and give a description of its boundary. Our aim is to point out the nice poset provided by the intersections of the irreducible components in the boundary, which provides a geometric realization of the nested set poset of this generalized Dowling lattice. It can be represented by a family of forests with leaves and labelings that depend on the triple (n,G,V). We will study it from the enumerative point of view in the case when G is abelian., Comment: In this revised version the bijection in Section 5 has been improved and holds also in the case when G is not abelian

Some projective wonderful models for the complement of a toric arrangement in a n-dimensional algebraic torus T were constructed in [3]. In this paper we describe their integer cohomology rings by generators and relations., Comment: A new section (Section 9) has been added, to include the presentation of the cohomology rings of all the strata in the boundary

In this paper we illustrate an algorithmic procedure which allows to build projective wonderful models for the complement of a toric arrangement in a n-dimensional algebraic torus T. The main step of the construction is a combinatorial algorithm that produces a toric variety by subdividing in a suitable way a given smooth fan.

An important piece of information in the theory of the arithmetic Galois action on the geometric fundamental groups of schemes is that divisorial inertia is acted on cyclotomically. We detail in this note the content of this fact in the case of the profinite braid groups arising from complex reflection groups, naturally viewing them as the geometric fundamental groups of the attending classifying spaces. We also include the case of the full (non colored) braid groups, whose completed classifying spaces are Deligne-Mumford stacks rather than schemes., Comment: 26 pages, added references

In this paper we find some exponential formulas for the Betti numbers of the De Concini-Procesi minimal wonderful models Y_{G(r,p,n)} associated to the complex reflection groups G(r,p,n). Our formulas are different from the ones already known in the literature: they are obtained by a new combinatorial encoding of the elements of a basis of the cohomology by means of set partitions with weights and exponents. We also point out that a similar combinatorial encoding can be used to describe the faces of the real spherical wonderful models of type A_{n-1}=G(1,1,n), B_n=G(2,1,n) and D_n=G(2,2,n). This provides exponential formulas for the f-vectors of the associated nestohedra: the Stasheff's associahedra (in this case closed formulas are well known) and the graph associahedra of type D_n., Comment: with respect to v.1: misprint corrected in Example 3.1

The De Concini-Procesi wonderful models of the braid arrangement of type $A_{n-1}$ are equipped with a natural $S_n$ action, but only the minimal model admits an `hidden' symmetry, i.e. an action of $S_{n+1}$ that comes from its moduli space interpretation. In this paper we explain why the non minimal models don't admit this extended action: they are `too small'. In particular we construct a {\em supermaximal} model which is the smallest model that can be projected onto the maximal model and again admits an extended $S_{n+1}$ action. We give an explicit description of a basis for the integer cohomology of this supermaximal model. Furthermore, we deal with another hidden extended action of the symmetric group: we observe that the symmetric group $S_{n+k}$ acts by permutation on the set of $k$-codimensionl strata of the minimal model. Even if this happens at a purely combinatorial level, it gives rise to an interesting permutation action on the elements of a basis of the integer cohomology.

In this paper we show a a proof by explicit bijections of the famous Kirkman-Cayley formula for the number of dissections of a convex polygon. Our starting point is the bijective correspondence between the set of nested sets made by \(k\) subsets of \(\{1,2,...,n\}\) with cardinality \(\geq 2\) and the set of partitions of \(\{1,2,...,n+k-1\}\) into \(k\) parts with cardinality \(\geq 2\)., Comment: With respect to v1: minor changes in the notation and correction of some misprints With respect to v2: added references

There are several real spherical models associated with a root arrangement, depending on the choice of a building set. The connected components of these models are manifolds with corners which can be glued together to obtain the corresponding real De Concini-Procesi models. In this paper, starting from any root system Phi with finite Coxeter group W and any W-invariant building set, we describe an explicit realization of the real spherical model as a union of polytopes (nestohedra) which lie inside the chambers of the arrangement. The main point of this realization is that the convex hull of these nestohedra is a larger polytope, a permutonestohedron, equipped with an action of W or also, depending on the building set, of Aut(Phi). The permutonestohedra are natural generalizations of Kapranov's permutoassociahedra.

Given a subspace arrangement, there are several De Concini-Procesi models associated to it, depending on distinct sets of initial combinatorial data (building sets). The first goal of this paper is to describe, for the root arrangements of types A_n, B_n (=C_n), D_n, the poset of all the building sets which are invariant with respect to the Weyl group action, and therefore to classify all the wonderful models which are obtained by adding to the complement of the arrangement an equivariant divisor. Then we point out, for every fixed n, a family of models which includes the minimal model and the maximal model; we call these models `regular models' and we compute, in the complex case, their Poincar\'e polynomials., Comment: 27 pages, 6 figures

In this paper we focus on maximal complex De Concini-Procesi models associated to root arrangements of types A, B, C, D and we compute inductive formulas for their Poincare' series.

In this paper we deal with the action of the symmetric group on the cohomology of the configuration space $C_n(d)$ of $n$ points in $\mathbb{R}^d$. This topic has been studied by several authors (see the introduction). On the cohomology algebra $H^*(C_n(d); \mathbb{C})$ there is, in addition to the natural $S_n$-action, an extended action of $S_{n+1}$; this was first shown for the case when $d$ is even by Mathieu, Robinson and Whitehouse and the second author. For the case when $d$ is odd it was shown by Mathieu (anyway we will give an elementary algebraic construction of the extended action for this case). The purpose of this article is to present some results that can be obtained, in an elementary way, exploiting the interplay between the extended action and the standard action.

We prove that one can obtain natural bundles of Lie algebras on rank two s-K\"ahler manifolds, whose fibres are isomorphic to so(s+1,s+1), su(s+1,s+1) and sl(2s + 2,\R). In the most rigid case (which includes complex tori and abelian varieties) these bundles have natural flat connections, whose flat global sections act naturally on cohomology. We also present several natural examples of manifolds which can be equipped with an s-K\"ahler structure with various levels of rigidity: complex tori and abelian varieties, cotangent bundles of smooth manifolds and moduli of pointed elliptic curves., Comment: in v2 a new theorem in Section 8 has been added

We determine the structure of the $*$-Lie superalgebra generated by a set of carefully chosen natural operators of an orientable WSD manifold of rank three. This Lie superalgebra is formed by global sections of a natural Lie superalgebra bundle, and turns out to be a product of $\mathbf{sl}(4,\C)$ with the full special linear superalgebras of some graded vector spaces isotypical with respect to a natural action of $\mathbf{so}(3,\R)$. We provide an explicit description of one of the real forms of this superalgebra, which is geometrically natural being made of $\mathbf{so}(3,\R)$-invariant operators which preserve the Poincar\'e (odd Hermitean) inner product on the bundle of forms., Comment: 20 pages, 4 tables. Minor changes in presentation from v1

Given an orientable weakly self-dual manifold X of rank two, we build a geometric realization of the Lie algebra sl(6,C) as a naturally defined algebra L of endomorphisms of the space of differential forms of X. We provide an explicit description of Serre generators in terms of natural generators of L. This construction gives a bundle on X which is related to the search for a natural Gauge theory on X. We consider this paper as a first step in the study of a rich and interesting algebraic structure., Comment: 16 pages, no figures