Wonderful Models for Generalized Dowling Arrangements
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Wonderful Models for Generalized Dowling Arrangements Giovanni Gaiffi1 · Viola Siconolfi2 Received: 7 December 2018 / Accepted: 4 December 2019 / © Springer Nature B.V. 2020
Abstract 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 (Trans. Amer. Math. Soc. 325(1), 1–37, 1991). 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. Keywords Wonderful models · Dowling lattice · Subspace arrangement
1 Introduction The De Concini-Procesi wonderful models of subspace arrangements were introduced in [5] and [4] and play since then a crucial role in the study of configuration spaces and in various other fields of mathematics. The relevance of their combinatorial properties and their relation with discrete geometry were pointed out for instance in [8, 9, 13, 15, 18]; their relations with Bergman fans, toric and tropical geometry were enlightened in [10] and [6]; the connections between the geometry of these models and the Chow rings of matroids were pointed out first in [11] and then in [1], where they also played a crucial role in the study of some log-concavity problems.
Viola Siconolfi
[email protected] Giovanni Gaiffi [email protected] 1
Dipartimento di Matematica, Universit`a di Pisa, Pisa, Italy
2
Dipartimento di Matematica, Universit`a di Roma Tor Vergata, Rome, Italy
Order
Let us recall their definition: given a subspace arrangement G in (Cn )∗ , we consider for each D ∈ G its annihilator D ⊥ in Cn and the projective space P(Cn /D ⊥ ). Let A(G ) be the ⊥ complement of D∈G D in Cn ; we can then define the following embedding: i : A(G ) → Cn ×
P(Cn /D ⊥ ).
D∈G
The wonderful model YG is defined as the closure of the image of i. If G is a building set (this is a combinatorial property that will be discussed in Section 2), YG turns out to be a smooth variety such that YG − i(A(G )) is a divisor with normal crossings. The geometric and topological properties of a wonderful model are deeply connected with its initial combinatorial data. For instance, the poset of the intersections of the irreducible components in the boundary YG − i(A(G )) is the nested set poset associated to G (see Sections 2 and 3). Moreover, the integer cohomology ring of YG can be described using some functions with integer values called ’admissible functions’ defined on G (see [5, 20]). The hyperplane arrangements associated to real and complex refl
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