zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Cohomology bases for the De Concini-Procesi models of hyperplane arrangements and sums over trees. (English) Zbl 0989.14006

Introduction: Let M 0,n+1 be the moduli space of n+1-tuples of distinct points on 1 modulo projective automorphisms. Since a projective automorphism of 1 is uniquely defined by the images of three points, M 0,n+1 can be regarded as the complement in n-2 to the projective hyperplane arrangement given by the polynomial z 1 ,,z n-1 i<j (z i -z j ). More symmetrically, one can interpret M 0,n+1 × as the complement in n-1 to the projectivization of the n-braid arrangement i<j (z i -z j ). The space M 0,n+1 has a canonical compactification M ¯ 0,n+1 that is a closed 2n-4-dimensional complex manifold whose cohomology ring R(n)=H * (M ¯ 0,n+1 ) plays an important part in algebraic geometry, field theory, and theory of operads. A presentation of R(n) was found by S. Keel [Trans. Am. Math. Soc. 330, No. 2, 545-574 (1992; Zbl 0768.14002)].

Recently C. de Concini and C. Procesi [Sel. Math., New Ser. 1, No. 3, 459-494 (1995; Zbl 0842.14038) and 495-535 (1995; Zbl 0848.18004)] generalized the construction. They defined a compactification M ¯ of the complement M of any complex projective hyperplane (even subspace) arrangement. They also gave a presentation for the cohomology ring H * (M ¯).

The goal of this paper is to use this presentation in order to construct a monomial basis of the graded ring R=H * (M ¯) as a free -module. We also study and use the Poincaré pairing on the basis. Then we use this basis to compute the Hilbert series of R for the reflection arrangements of classical types B n (=C n ) and D n . We reduce the computation to summation over trees and apply the method of Yu. I. Manin [Prog. Math. 129, 401-417 (1995; Zbl 0871.14022)]. Arrangement of type A n-1 is a braid arrangement, i.e., M ¯=M ¯ 0,n+1 , and the Hilbert series has been computed by Manin (loc. cit.). We recover this result in our computation.

In section 2 we construct a set Δ of monomials in R and prove that it generates R. In section 3 we use the Poincaré duality to prove that it is linearly independent. In section 4 we give a combinatorial description of Δ for reflection arrangements of classical types. In section 5 we use this description to compute the Hilbert series of R for these arrangements.


MSC:
14F25Classical real and complex cohomology
52C35Arrangements of points, flats, hyperplanes
14D20Algebraic moduli problems, moduli of vector bundles
14N10Enumerative problems (algebraic geometry)