Families of building sets and regular wonderful models

TL;DR

This paper classifies Weyl group-invariant building sets for root arrangements of types A, B, C, D, describing all associated De Concini-Procesi models and computing their Poincaré polynomials for regular models.

Contribution

It characterizes all Weyl group-invariant building sets for classical root arrangements and introduces a family of regular models including minimal and maximal models, with explicit Poincaré polynomial calculations.

Findings

Classified all Weyl group-invariant building sets for types A, B, C, D.

Identified a family of regular models including minimal and maximal models.

Computed Poincaré polynomials for these regular models.

Abstract

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.