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Zbl 0989.14006
Yuzvinsky, Sergey
Cohomology bases for the De Concini-Procesi models of hyperplane arrangements and sums over trees.
(English)
[J] Invent. Math. 127, No.2, 319-335 (1997). ISSN 0020-9910; ISSN 1432-1297/e

Introduction: Let $M_{0,n+1}$ be the moduli space of $n+1$-tuples of distinct points on $\bbfC\bbfP^1$ modulo projective automorphisms. Since a projective automorphism of $\bbfC\bbfP^1$ is uniquely defined by the images of three points, $M_{0,n+1}$ can be regarded as the complement in $\bbfC \bbfP^{n-2}$ to the projective hyperplane arrangement given by the polynomial $z_1,\dots, z_{n-1}\prod_{i<j} (z_i-z_j)$. More symmetrically, one can interpret $M_{0,n+1} \times\bbfC$ as the complement in $\bbfC\bbfP^{n-1}$ to the projectivization of the $n$-braid arrangement $\prod_{i<j}(z_i-z_j)$. The space $M_{0,n+1}$ has a canonical compactification $\overline M_{0,n+1}$ that is a closed $2n-4$-dimensional complex manifold whose cohomology ring $R(n)=H^* (\overline M_{0,n+1})$ plays an important part in algebraic geometry, field theory, and theory of operads. A presentation of $R(n)$ was found by {\it S. Keel} [Trans. Am. Math. Soc. 330, No. 2, 545-574 (1992; Zbl 0768.14002)].\par Recently {\it C. de Concini} and {\it 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 $\overline M$ of the complement $M$ of any complex projective hyperplane (even subspace) arrangement. They also gave a presentation for the cohomology ring $H^* (\overline M)$.\par The goal of this paper is to use this presentation in order to construct a monomial basis of the graded ring $R=H^*(\overline M)$ as a free $\bbfZ$-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 {\it Yu. I. Manin} [Prog. Math. 129, 401-417 (1995; Zbl 0871.14022)]. Arrangement of type $A_{n-1}$ is a braid arrangement, i.e., $\overline M=\overline M_{0,n+1}$, and the Hilbert series has been computed by Manin (loc. cit.). We recover this result in our computation.\par In section 2 we construct a set $\Delta$ 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 $\Delta$ for reflection arrangements of classical types. In section 5 we use this description to compute the Hilbert series of $R$ for these arrangements.
MSC 2000:
*14F25 Classical real and complex cohomology
52C35 Arrangements of points, flats, hyperplanes
14D20 Algebraic moduli problems
14N10 Enumerative problems (classical algebraic geometry)

Keywords: hyperplane arrangements; basis of cohomology ring; Poincaré pairing; Hilbert series; reflection arrangements; summation over trees

Cited in: Zbl 1247.14057 Zbl 1137.20039

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