Counting Facets and Incidences.
Coxeter group actions on the complement of hyperplanes and special involutions
We consider both standard and twisted actions of a (real) Coxeter group on the complement to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in and give explicit formulae which describe both actions on the total cohomology in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group , the Weyl groups...
Cutting Hyperplane Arrangements.
Cutting Hyperplanes for Divide-and-Conquer.
Diffeomorphic types of the complements of arrangements of hyperplanes
Distributing points on the sphere. I.
Even astral configurations.
Formality of the complements of subspace arrangements with geometric lattices.
Free Arrangements and Relation Spaces.
Free hyperplane arrangements between A n -1 and Bn.
Gale duality for complete intersections
We show that every complete intersection defined by Laurent polynomials in an algebraic torus is isomorphic to a complete intersection defined by master functions in the complement of a hyperplane arrangement, and vice versa. We call systems defining such isomorphic schemes Gale dual systems because the exponents of the monomials in the polynomials annihilate the weights of the master functions. We use Gale duality to give a Kouchnirenko theorem for the number of solutions to a system of master...
Generalisations of the Tits representation.
Geometrically constructed bases for homology of partition lattices of types , and .
Homotopy types of line arrangements.
Homotopy types of subspace arrangements via diagrams of spaces.
Hyperplane arrangements and Milnor fibrations
There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a set of interlocking fibrations. We use cohomology with coefficients in rank local systems on the complement of the arrangement to gain information on the homology of the other three spaces, and on the monodromy operators of the various fibrations.
Improved inclusion-exclusion inequalities for simplex and orthant arrangements.
Isoperimetric Inequalities for Infinite Hyperplane Systems.
Local cohomology of logarithmic forms
Let be a divisor on a smooth algebraic variety . We investigate the geometry of the Jacobian scheme of , homological invariants derived from logarithmic differential forms along , and their relationship with the property that be a free divisor. We consider arrangements of hyperplanes as a source of examples and counterexamples. In particular, we make a complete calculation of the local cohomology of logarithmic forms of generic hyperplane arrangements.
Local Lipschitz property for the Chebyshev center mapping over N-nets