Exponential formulas for models of complex reflection groups

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.

On models of the braid arrangement and their hidden symmetries

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

Enumeration of points, lines, planes, etc

One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erd\H{o}s: Every set of points $E$ in a projective plane determines at least $|E|$ lines, unless all the points are contained in a line. Motzkin and others extended the result to higher dimensions, who showed that every set of points $E$ in a projective space determines at least $|E|$ hyperplanes, unless all the points are contained in a hyperplane. Let $E$ be a spanning subset of a $d$-dimensional vector space. We show that, in the partially ordered set of subspaces spanned by subsets of $E$, there are at least as many $(d-k)$-dimensional subspaces as there are $k$-dimensional subspaces, for every $k$ at most $d/2$. This confirms the "top-heavy" conjecture of Dowling and Wilson for all matroids realizable over some field. The proof relies on the decomposition theorem package for $\ell$-adic intersection complexes.Comment: 18 pages, major revisio

Affine and toric hyperplane arrangements

We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure

DE CONCINI AND PROCESI MODELS OF REFLECTION GROUPS AND COXETER GROUPS

Study of De Concini and Procesi Wonderful models for subspace arrangement related to subspace arrangement generated by reflection groups and Coxeter groups