184 research outputs found
The freeness of Ish arrangements
International audienceThe Ish arrangement was introduced by Armstrong to give a new interpretation of the -Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be freeL’arrangement Ish a été introduit par Armstrong pour donner une nouvelle interprétation des nombres -Catalan de Garsia et Haiman. Armstrong et Rhoades ont montré qu’il y avait des ressemblances frappantes entre l’arrangement Shi et l’arrangement Ish et ont posé des conjectures. L’une d’elles est de savoir si l’arrangement Ish est un arrangement libre ou pas. Dans cet article, nous vérifions que l’arrangement Ish est supersoluble et donc libre. De plus, on donne une condition nécessaire et suffisante pour que l’arrangement Ish réduit soit libre
Ad-nilpotent ideals and The Shi arrangement
We extend the Shi bijection from the Borel subalgebra case to parabolic
subalgebras. In the process, the -deleted Shi arrangement
naturally emerges. This arrangement interpolates between the Coxeter
arrangement and the Shi arrangement , and breaks
the symmetry of in a certain symmetrical way. Among other
things, we determine the characteristic polynomial
of explicitly for and . More generally, let
be an arbitrary arrangement between and
. Armstrong and Rhoades recently gave a formula for
for . Inspired by their result, we obtain
formulae for for , and .Comment: The third version, quasi-antichains are shown to be in bijection with
elements of L(Cox). arXiv admin note: text overlap with arXiv:1009.1655 by
other author
Characteristic Polynomial of Arrangements and Multiarrangements
This thesis is on algebraic and algebraic geometry aspects of complex hyperplane arrangements and multiarrangements. We start by examining the basic properties of the logarithmic modules of all orders such as their freeness, the cdga structure, the local properties and close the first chapter with a multiarrangement version of a theorem due to M. Mustata and H. Schenck.
In the next chapter, we obtain long exact sequences of the logarithmic modules of an arrangement and its deletion-restriction under the tame conditions. We observe how the tame conditions transfer between an arrangement and its deletion-restriction.
In chapter 3, we use some tools from the intersection theory and show that the intersection cycle of a certain projective variety has a closed answer in terms of the characteristic polynomial. This result is used to compute the leading parts of the Hilbert polynomial and Hilbert series of the logarithmic ideal. As a consequence, we recover some of the classical results of the theory such as the Solomon-Terao formula for tame arrangements. This is done by computing the Hilbert series in two different ways. We also introduce the notion of logarithmic Orlik-Terao ideal and show that the intersection lattice parametrizes a primary decomposition. The chapter is closed by a generalization of logarithmic ideals to higher orders. It is shown that these ideals detect the freeness of the corresponding logarithmic modules.
The last chapter is a generalization of the notion of logarithmic ideal to multiarrangements. Some of the basic properties of these ideals are investigated. It is shown that one obtains a natural resolution of this ideal by logarithmic modules under the tame condition. In the final section it is shown that the intersection cycle of the logarithmic ideal of a free multiarrangement is obtained from its characteristic polynomial, similar to simple arrangements
Flow-driven compaction of a fibrous porous medium
A combined theoretical and experimental study is presented for the flow-induced compaction of a one-dimensional fibrous porous medium near its gel point for deformation at low and high rates. The theory is based on a two-phase model in which the permeability is a function of local solid fraction, and the deformation of the solid is resisted by both a compressive yield stress and a rate-dependent bulk viscosity. All three material properties are parameterized and calibrated for cellulose fibers using sedimentation, permeation, and filtration experiments. It is shown that the incorporation of rate-dependence in the solid stress significantly improves the agreement between theory and experiment when the drainage flow is relatively rapid. The model is extended to rates outside the range where it was calibrated to understand the dynamics of a standard test for pulp suspensions: the Canadian Standard Freeness test. The model adequately captures all of the experimental findings, including the score of the freeness test, which is found to be sensitively controlled by the bulk solid viscosity and to a lesser degree by the permeability law, but depends only weakly on the compressive yield stress
On some operators acting on line arrangements and their dynamics
We study some natural operators acting on configurations of points and lines
in the plane and remark that many interesting configurations are fixed points
for these operators. We review ancient and recent results on line or point
arrangements though the realm of these operators. We study the first dynamical
properties of the iteration of these operators on some line arrangements.Comment: 33 page
Construction of free curves by adding lines to a given curve
In the present note we construct new families of free plane curves starting
from a curve and adding high order inflectional tangent lines of , lines
joining the singularities of the curve , or lines in the tangent cone of
some singularities of . These lines have in common that the intersection
consists of a small number of points. We introduce the notion of a
supersolvable plane curve and conjecture that such curves are always free, as
in the known case of line arrangements. Some evidence for this conjecture is
given as well, both in terms of a general result in the case of quasi
homogeneous singularities and in terms of specific examples. We construct a new
example of maximizing curve in degree 8 and the first and unique known example
of maximizing curve in degree 9.Comment: v.2 Example 6.4 is new: here we construct a new example of maximizing
curve in degree 8 and the first and unique known example of maximizing curve
in degree
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