652 research outputs found
Arrangements and local systems
We use stratified Morse theory to construct a complex to compute the
cohomology of the complement of a hyperplane arrangement with coefficients in a
complex rank one local system. The linearization of this complex is shown to be
the Orlik-Solomon algebra with the connection operator. Using this result, we
establish the relationship between the cohomology support loci of the
complement and the resonance varieties of the Orlik-Solomon algebra for any
arrangement, and show that the latter are unions of subspace arrangements in
general, resolving a conjecture of Falk. We also obtain lower bounds for the
local system Betti numbers in terms of those of the Orlik-Solomon algebra,
recovering a result of Libgober and Yuzvinsky. For certain local systems, our
results provide new combinatorial upper bounds on the local system Betti
numbers. These upper bounds enable us to prove that in non-resonant systems the
cohomology is concentrated in the top dimension, without using resolution of
singularities.Comment: LaTeX, 14 page
Broken circuit complexes and hyperplane arrangements
We study Stanley-Reisner ideals of broken circuits complexes and characterize
those ones admitting a linear resolution or being complete intersections. These
results will then be used to characterize arrangements whose Orlik-Terao ideal
has the same properties. As an application, we improve a result of Wilf on
upper bounds for the coefficients of the chromatic polynomial of a maximal
planar graph. We also show that for an ordered matroid with disjoint minimal
broken circuits, the supersolvability of the matroid is equivalent to the
Koszulness of its Orlik-Solomon algebra.Comment: 21 page
Note on resonance varieties
We study the irreducibility of resonance varieties of graded rings over an
exterior algebra E with particular attention to Orlik-Solomon algebras. We
prove that for a stable monomial ideal in E the first resonance variety is
irreducible. If J is an Orlik- Solomon ideal of an essential central hyperplane
arrangement, then we show that its first resonance variety is irreducible if
and only if the subideal of J generated by all degree 2 elements has a 2-linear
resolution. As an application we characterize those hyperplane arrangements of
rank less than or equal to 3 where J is componentwise linear. Higher resonance
varieties are also considered. We prove results supporting a conjecture of
Schenck-Suciu relating the Betti numbers of the linear strand of J and its
first resonance variety. A counter example is constructed that this conjecture
is not true for arbitrary graded ideals
The Orlik-Solomon model for hypersurface arrangements
We develop a model for the cohomology of the complement of a hypersurface
arrangement inside a smooth projective complex variety. This generalizes the
case of normal crossing divisors, discovered by P. Deligne in the context of
the mixed Hodge theory of smooth complex varieties. Our model is a global
version of the Orlik-Solomon algebra, which computes the cohomology of the
complement of a union of hyperplanes in an affine space. The main tool is the
complex of logarithmic forms along a hypersurface arrangement, and its weight
filtration. Connections with wonderful compactifications and the configuration
spaces of points on curves are also studied.Comment: 23 pages; presentation simplified, results unchange
On the cohomology of discriminantal arrangements and Orlik-Solomon algebras
We relate the cohomology of the Orlik-Solomon algebra of a discriminantal
arrangement to the local system cohomology of the complement. The Orlik-Solomon
algebra of such an arrangement (viewed as a complex) is shown to be a linear
approximation of a complex arising from the fundamental group of the
complement, the cohomology of which is isomorphic to that of the complement
with coefficients in an arbitrary complex rank one local system. We also
establish the relationship between the cohomology support loci of the
complement of a discriminantal arrangement and the resonant varieties of its
Orlik-Solomon algebra.Comment: LaTeX2e, 16 pages, to appear in Singularities and Arrangements,
Sapporo-Tokyo 1998, Advanced Studies in Pure Mathematic
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