Poincare' series for maximal De Concini-Procesi models of root arrangements

The homology of real subspace arrangements

Associated to any subspace arrangement is a "De Concini-Procesi model", a certain smooth compactification of its complement, which in the case of the braid arrangement produces the Deligne-Mumford compactification of the moduli space of genus 0 curves with marked points. In the present work, we calculate the integral homology of real De Concini-Procesi models, extending earlier work of Etingof, Henriques, Kamnitzer and the author on the (2-adic) integral cohomology of the real locus of the moduli space. To be precise, we show that the integral homology of a real De Concini-Procesi model is iso…

Geometric realizations and duality for Dahmen-Micchelli modules and De Concini-Procesi-Vergne modules

We give an algebraic description of several modules and algebras related to the vector partition function, and we prove that they can be realized as the equivariant K-theory of some manifolds that have a nice combinatorial description. We also propose a more natural and general notion of duality between these modules, which corresponds to a Poincaré duality-type correspondence for equivariant K-theory.

Families of building sets and regular wonderful models

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…

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). Th…

子空间排列的奇妙模型的胞腔$\boldsymbol{\funnyA}^1$-同调

Cellular $\mathbb{A}^1$-homology of wonderful models of subspace arrangements

该研究计算子空间排列的De Concini–Procesi奇妙模型的胞腔$\funnyA^1$-同调，通过动机爆破计算得到关键附着类，将相关同调用区间上同调表达，并给出辫排列的显式分解与示例。