169 research outputs found
Toric and tropical compactifications of hyperplane complements
These lecture notes are based on lectures given by the author at the summer
school "Arrangements in Pyr\'en\'ees" in June 2012. We survey and compare
various compactifications of complex hyperplane arrangement complements. In
particular, we review the Gel'fand-MacPherson construction, Kapranov's visible
contours compactification, and De Concini and Procesi's wonderful
compactification. We explain how these constructions are unified by some ideas
from the modern origins of tropical geometry. The paper contains a few new
arguments intended to make the presentation as self-contained as possible.Comment: 26 page
Equivariant Euler characteristics of discriminants of reflection groups
Let G be a finite, complex reflection group and f its discriminant
polynomial. The fibers of f admit commuting actions of G and a cyclic group.
The virtual character given by the Euler characteristic of the
fiber is a refinement of the zeta function of the geometric monodromy,
calculated in a paper of Denef and Loeser. We compute the virtual character
explicitly, in terms of the poset of normalizers of centralizers of regular
elements of G, and of the subspace arrangement given by proper eigenspaces of
elements of G. As a consequence, we compute orbifold Euler characteristics and
find some new "case-free" information about the discriminant.Comment: 18 page
Moment-angle complexes, monomial ideals, and Massey products
Associated to every finite simplicial complex K there is a "moment-angle"
finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth,
compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study
the cohomology ring, the homotopy groups, and the triple Massey products of a
moment-angle complex, relating these topological invariants to the algebraic
combinatorics of the underlying simplicial complex. Applications to the study
of non-formal manifolds and subspace arrangements are given.Comment: 30 pages. Published versio
On the homotopy Lie algebra of an arrangement
Let A be a graded-commutative, connected k-algebra generated in degree 1. The
homotopy Lie algebra g_A is defined to be the Lie algebra of primitives of the
Yoneda algebra, Ext_A(k,k). Under certain homological assumptions on A and its
quadratic closure, we express g_A as a semi-direct product of the
well-understood holonomy Lie algebra h_A with a certain h_A-module. This allows
us to compute the homotopy Lie algebra associated to the cohomology ring of the
complement of a complex hyperplane arrangement, provided some combinatorial
assumptions are satisfied. As an application, we give examples of hyperplane
arrangements whose complements have the same Poincar\'e polynomial, the same
fundamental group, and the same holonomy Lie algebra, yet different homotopy
Lie algebras.Comment: 20 pages; accepted for publication by the Michigan Math. Journa
Matroid connectivity and singularities of configuration hypersurfaces
Consider a linear realization of a matroid over a field. One associates with
it a configuration polynomial and a symmetric bilinear form with linear
homogeneous coefficients. The corresponding configuration hypersurface and its
non-smooth locus support the respective first and second degeneracy scheme of
the bilinear form. We show that these schemes are reduced and describe the
effect of matroid connectivity: for (2-)connected matroids, the configuration
hypersurface is integral, and the second degeneracy scheme is reduced
Cohen-Macaulay of codimension 3. If the matroid is 3-connected, then also the
second degeneracy scheme is integral. In the process, we describe the behavior
of configuration polynomials, forms and schemes with respect to various matroid
constructions.Comment: 64 pages, 4 figure
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