Waldhausen, Gruppen mit Zentrum and 3-dimensionale Mannigfaltien, Topology 6 (1967), 505-517. Waldhausen, On irreducible 3-manifolds that are sufficiently large, Ann. Waldhausen, Ein Klasse von 3-dimensionalen Mannigfaltigkeiten, Invent. computable from the intersection lattice), and freeness of the module behaves well under certain operations on hyperplane arrangements which are used to. Rybnikov, On the fundamantal group of the complement of a complex hyperplane arrangement, DIMACS Ser. Based on this decomposition we compute the. We show a fundamental structure of the intersection lattices by decomposing the poset ideals as direct products of smaller lattices corresponding to smaller dimensions. These arrangements are known to be equivalent to discriminantal arrangements. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin, 1992. We consider hyperplane arrangements generated by generic points and study their intersection lattices. Solomon, Unitary reflection groups and cohomology, Invent. More precisely, let K be a field, U a finite-dimensional vector space over K and H a finite collection of. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. intersection lattice of a hyperplane arrangement. Neumann, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. natorial structure of a hyperplane arrangement is encoded in its characteristic polynomial, which is defined recursively through the intersection lattice of. The intersection lattice of the type B Coxeter arrangement is iso- morphic to the signed partition lattice. Mumford, The topology of normal singularities of an algebraic surfaces and a criterion for simplicity, Inst. Moishezon, Simply connected algebraic surfaces of general type, Invent. Yau, Topological invariance of intersection lattices of arrangements in CP2, Bull. Yau, Diffeomorphic type of the complement of arrangement of hyperplanes, Compositio Math. Yau, Topological and differential structures of the complement of an arrangement of hyperplanes, Proc. The second lattice is the more complicated. Thus the minimal element is the empty intersection Rn and the maximal element of L is the intersection of all the hyperplanes, that is, the zero vector. These arrangements are known to be equivalent to discriminantal arrangements. The intersection lattice L consists of all the intersections of the hyperplanes in H ordered with respect to reverse inclusion. Hirzebruch, Arrangements of lines and algebraic surfaces, In: " Arithmetic and Geometry," Vol. We consider hyperplane arrangements generated by generic points and study their intersection lattices. Gel'Fand, General theory of hypergeometric functions, Soviet. Falk, Homotopy types of line arrangements, Invent. Falk, On the algebra associated with a geometric lattice, Adv. Falk, The cohomology andfundamental group of a hyperplane complement, Contemp. Falk, Arrangement and cohomology, preprint, 1996. Mostow, Monodromy of hypergeometric functions and non-lattice integral monodromy, Inst. A hyperplane arrangementA is a finite collection of codimension1 linear subspaces in a complex affine space Cn. Brieskorn, Sur les groups de tresses, In: "Séminaire Bourbaki 1971/73", Lecture Notes in Math. Arnol'D, The cohomology ring of the colored braid group, Mat. ![]() However, that seems neither necessary nor sufficient for affine hyperplanes. (A hyperplane is the intersection of the two half-spaces it bounds.) Let. In the central case, the intersection is empty if and only if the normal vectors positively span the origin (if I remember correctly). It is possible for central arrangements (all hyperplanes contain 0) since the normal vectors determine everything, but I think for affine arrangements there is not enough information. of the intersection lattice has motivated much work in the field. its intersection lattice has a one dimensional moduli space, and it is free but not recursively free. I am not sure, but I think that from the normal vectors and the intersection semilattice it is not possible to say which half-space intersections are empty. A hyperplane arrangement in mathbb Pn is free if R/J is CohenMacaulay (CM). (I assume $R_r$ means $B_r$.) This is separate from the intersection semilattice. The example suggests that the question is about the intersections of half-spaces. It is defined as the set of non-empty intersections of subsets of A, so nothing in the semilattice is empty. D.15.1.38 arrLattice, computes the intersection lattice / poset. The intersection poset of an affine arrangement is a semilattice, not necessarily a lattice. arrGet access to a single/multiple hyperplane(s) - arrMinus deletes given hyperplanes.
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