50 research outputs found

### The integer cohomology of toric Weyl arrangements

A referee found an error in the proof of the Theorem 2 that we could not fix.
More precisely, the proof of Lemma 2.1 is incorrect. Hence the fact that
integer cohomology of complement of toric Weyl arrangements is torsion free is
still a conjecture.
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A toric arrangement is a finite set of hypersurfaces in a complex torus,
every hypersurface being the kernel of a character. In the present paper we
prove that if \Cal T_{\wdt W} is the toric arrangement defined by the
\textit{cocharacters} lattice of a Weyl group \wdt W, then the integer
cohomology of its complement is torsion free

### Blocking Sets in the complement of hyperplane arrangements in projective space

It is well know that the theory of minimal blocking sets is studied by
several author. Another theory which is also studied by a large number of
researchers is the theory of hyperplane arrangements. We can remark that the
affine space $AG(n,q)$ is the complement of the line at infinity in $PG(n,q)$.
Then $AG(n,q)$ can be regarded as the complement of an hyperplane arrangement
in $PG(n,q)$! Therefore the study of blocking sets in the affine space
$AG(n,q)$ is simply the study of blocking sets in the complement of a finite
arrangement in $PG(n,q)$. In this paper the author generalizes this remark
starting to study the problem of existence of blocking sets in the complement
of a given hyperplane arrangement in $PG(n,q)$. As an example she solves the
problem for the case of braid arrangement. Moreover she poses significant
questions on this new and interesting problem

### The integer cohomology of toric Weyl arrangements

A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper we prove that if T(W) is the toric arrangement defined by the cocharacters lattice of a Weyl group W, then the integer cohomology of its complement is torsion free.Arrangement of hyperplanes, toric arrangements, CW complexes, Salvetti complex, Weyl groups, integer cohomology

### A stability-like theorem for cohomology of pure braid groups of the series A, B and D

Consider the ring R:=\Q[\tau,\tau^{-1}] of Laurent polynomials in the
variable $\tau$. The Artin's Pure Braid Groups (or Generalized Pure Braid
Groups) act over $R,$ where the action of every standard generator is the
multiplication by $\tau$. In this paper we consider the cohomology of such
groups with coefficients in the module $R$ (it is well known that such
cohomology is strictly related to the untwisted integral cohomology of the
Milnor fibration naturally associated to the reflection arrangement). We give a
sort of \textit{stability} theorem for the cohomologies of the infinite series
$A$, $B$ and $D,$ finding that these cohomologies stabilize, with respect to
the natural inclusion, at some number of copies of the trivial $R$-module \Q.
We also give a formula which compute this number of copies.Comment: 17 pages; added reference for section

### Social choice among complex objects: Mathematical tools

Here the reader can find some basic definitions and notations in order to better understand the model for social choise described by L. Marengo and S. Settepanella in their paper: Social choice among complex objects. The interested reader can refer to [Bou68], [Massey] and [OT92] to go into more depth.Arrangements, simplicial complexes, CW complexes,fundamental group, Salvetti's complex.

### Homology graph of real arrangements and monodromy of Milnor Fiber

We study the first homology group of the Milnor fiber of sharp arrangements
in the real projective plane. Our work relies on the minimal Salvetti complex
of the deconing arrangement and its boundary map. We describe an algorithm
which computes possible eigenvalues of the first monodromy operator. We prove
that, if a condition on some intersection points of lines is satisfied, then
the only possible non trivial eigenvalues are cubic roots of the unity.
Moreover we give sufficient conditions for just eigenvalues of order 3 or 4 to
appear in cases in which this condition is not satisfied.Comment: 35 pages, 15 figure

### The homotopy type of toric arrangements

A toric arrangement is a finite set of hypersurfaces in a complex torus,
every hypersurface being the kernel of a character. In the present paper we
build a CW-complex homotopy equivalent to the arrangement complement, with a
combinatorial description similar to that of the well-known Salvetti complex.
If the toric arrangement is defined by a Weyl group we also provide an
algebraic description, very handy for cohomology computations. In the last part
we give a description in terms of tableaux for a toric arrangement appearing in
robotics.Comment: To appear on J. of Pure and Appl. Algebra. 16 pages, 3 picture

### Braid groups in complex Grassmannians

We describe the fundamental group and second homotopy group of ordered
$k-$point sets in $Gr(k,n)$ generating a subspace of fixed dimension.Comment: 10 page