Blowups and cohomology bases for De Concini-Procesi models of subspace arrangements

Gaiffi, Giovanni

doi:10.1007/s000290050013

Cited by 15 publications

(22 citation statements)

References 6 publications

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“…A monomial basis of algebra M T was first constructed in [15] (see [15], Remark 3.11) and then generalized in [8]. Combining the monomial bases of M T for all flags T we obtain the following.…”

Section: Proposition 22 Implies That M (X) Can Be Taken As a Model Omentioning

confidence: 99%

Small rational model of subspace complement

Yuzvinsky

2002

Trans. Amer. Math. Soc.

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Abstract. This paper concerns the rational cohomology ring of the complement M of a complex subspace arrangement. We start with the De ConciniProcesi differential graded algebra that is a rational model for M . Inside it we find a much smaller subalgebra D quasi-isomorphic to the whole algebra. D is described by defining a natural multiplication on a chain complex whose homology is the local homology of the intersection lattice L whence connecting the De Concini-Procesi model with the Goresky-MacPherson formula for the additive structure of H * (M ). The algebra D has a natural integral version that is a good candidate for an integral model of M . If the rational local homology of L can be computed explicitly we obtain an explicit presentation of the ring H * (M, Q). For example, this is done for the cases where L is a geometric lattice and where M is a k-equal manifold.

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Section: Proposition 22 Implies That M (X) Can Be Taken As a Model Omentioning

confidence: 99%

Small rational model of subspace complement

Yuzvinsky

2002

Trans. Amer. Math. Soc.

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“…In [4] a new exponential (non recursive) formula for the Betti numbers of the models Y G(1,1,n) has been found, using the following combinatorial approach. In [39] (see also [24]) a monomial basis of H * (Y G(1,1,n) ) was described; the elements of this basis can be represented by graphs, that are some oriented rooted trees on n leaves, with exponents attached to the internal vertices. Now let us focus on the trees that have k internal vertices; in [26] a bijection between these trees and the partitions of {1, ..., n + k − 1} into k parts of cardinality ≥ 2 has been described (this is in fact a variant of a bijection shown in [17]).…”

Section: A Combinatorial Approachmentioning

confidence: 99%

“…As we mentioned in the Introduction, some recursive formulas for the Poincarè series of these varieties are well known (see for instance [33], [39], [24]).…”

Section: The Braid Casementioning

confidence: 99%

Exponential formulas for models of complex reflection groups

Gaiffi

2016

European Journal of Combinatorics

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In this paper we find exponential formulas for the Betti numbers of the De Concini-Procesi minimal wonderful models YG(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 An-1(=G(1, 1, n)), Bn (=G(2, 1, n)) and Dn(=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 Dn

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“…In [5] they showed, using a description of the cohomology rings of the projective wonderful models to give an explicit presentation of a Morgan algebra, that the mixed Hodge numbers and the rational homotopy type of the complement of a complex subspace arrangement depend only on the intersection Date: June 8, 2018. lattice (viewed as a ranked poset). The cohomology rings of the models of subspace arrangements were then studied in [20], [12], were some integer bases were provided, and also, in the real case, in [7], [19]. Some combinatorial objects (nested sets, building sets) turned out to be relevant in the description of the boundary of the models and of their cohomology rings: their relation with discrete geometry were pointed out in [8], [13]; the case of complex reflection groups was dealt with in [14] from the representation theoretic point of view and in [2] from the homotopical point of view.…”

Section: Introductionmentioning

confidence: 99%