Cohomology bases for the De Concini-Procesi models of hyperplane arrangements and sums over trees. (English) Zbl 0989.14006

Yuzvinsky, Sergey

Invent. Math. 127, No. 2, 319-335 (1997).

Introduction: Let \(M_{0,n+1}\) be the moduli space of \(n+1\)-tuples of distinct points on \(\mathbb{C}\mathbb{P}^1\) modulo projective automorphisms. Since a projective automorphism of \(\mathbb{C}\mathbb{P}^1\) is uniquely defined by the images of three points, \(M_{0,n+1}\) can be regarded as the complement in \(\mathbb{C} \mathbb{P}^{n-2}\) 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 \(M_{0,n+1} \times\mathbb{C}\) as the complement in \(\mathbb{C}\mathbb{P}^{n-1}\) 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 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 \(\overline M\) of the complement \(M\) of any complex projective hyperplane (even subspace) arrangement. They also gave a presentation for the cohomology ring \(H^* (\overline M)\).
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 \(\mathbb{Z}\)-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., \(\overline M=\overline M_{0,n+1}\), and the Hilbert series has been computed by Manin (loc. cit.). We recover this result in our computation.
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:

14F25	Classical real and complex (co)homology in algebraic geometry
52C35	Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
14D20	Algebraic moduli problems, moduli of vector bundles
14N10	Enumerative problems (combinatorial problems) in algebraic geometry

Keywords:

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

Citations:

Zbl 0768.14002; Zbl 0842.14038; Zbl 0848.18004; Zbl 0871.14022