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Variations on Van Kampen's method

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Abstract

We give a detailed account of the classical Van Kampen method for computing presentations of fundamental groups of complements of complex algebraic curves, and of a variant of this method, working with arbitrary projections (even with vertical asymptotes).

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... The Zariski-Van Kampen method allow one to give a finite presentation for the fundamental group of the complement to a projective plane curve. It is hence a constructive method and in some cases it is even effective, i.e. it has been implemented in the case of line arrangements, curves with easy singularities and equations on the Gaussian integers Z[ √ −1] (see [14,11]). A very nice approach to this method can be found in the unpublished notes written by I.Shimada in [71]. ...
... EXAMPLE 3.5. The braid monodromy of the wiring diagram of Figure 3 is given as follows: (11) β 1 = σ 2 2 β 2 = (σ 2 ) * σ 2 1 β 3 = ((σ 2 )(σ 1 )) * (σ 2 σ 3 ) 3 β 4 = ((σ 2 )(σ 1 )(σ 3 σ 2 σ 3 )) * σ 2 1 One can prove the following result. THEOREM 3.6. ...
... H := { 1 2 3 · · · 12 = 0}, where 1 = {y = 0}, 2 = {(x + ω 2 y + ω 2 z) = 0}, 3 = {(x + ωy + ωz) = 0}, 4 = {(x + ω 2 y + ωz) = 0}, 5 = {(x + ωy + ω 2 z) = 0}, 6 = {(x + ω 2 y + z) = 0}, 7 = {(x + ωy + z) = 0}, 8 = {x = 0}, 9 = {(x + y + z) = 0}, 10 = {z = 0}, 11 = {(x + y + ωz) = 0}, 12 = {(x + y + ω 2 z) = 0}, where ω is a root of z 2 + z + 1 = 0. Assume 10 is the line at infinity and project from the quadruple point P := [1 : −1 : 0]. The lines 9 , 11 , and 12 become vertical. ...
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These are the notes from a one-week course on Braid Monodromy of Algebraic Curves given at the Universite de Pau et des Pays de l'Adour during the Premiere Ecole Franco-Espagnole: Groupes de tresses et topologie en petite dimension in October 2009. This is intended to be an introductory survey through which we hope we can briefly outline the power of the concept monodromy as a common area for group theory, algebraic geometry, and topology of projective curves. The main classical results are stated in §2, where the Zariski-van Kampen method to compute a presentation for the fundamental group of the complement to projective plane curves is presented. In §1 these results are prefaced with a review of basic concepts like fundamental groups, locally trivial fibrations, branched and unbranched coverings and a first peek at monodromy. Descriptions of the main motivations that have lead mathematicians to study these objects are included throughout this first chapter. Finally, additional tools and further results that are direct applications of braid monodromy will be considered in §3. While not all proofs are included, we do provide either originals or simplified versions of those that are relevant in the sense that they exhibit the techniques that are most used in this context and lead to a better understanding of the main concepts discussed in this survey. Nothing here is hence original, other than an attempt to bring together different results and points of view. It goes without saying that this is not the first, and hopefully not the last, survey on the topic. For other approaches to braid monodromy we refer to the following beautifully-written papers [73, 20, 6]. We finally wish to thank the organizers and the referee for their patience and understanding in the process of writing and correcting these notes.
... By standard arguments (see e.g. [13] proposition 2.2, or [5]) we know that the induced map P = π 1 (X, x 0 ) → π 1 (X 0 , x 0 ) is surjective, and that its kernel K is normally generated by the meridians around the hyperplanes in A c L . Since the following diagram is commutative 1 ...
Preprint
We investigate the extensions of the Hecke algebras of finite (complex) reflection groups by lattices of reflection subgroups that we introduced, for some of them, in our previous work on the Yokonuma-Hecke algebras and their connections with Artin groups. When the Hecke algebra is attached to the symmetric group, and the lattice contains all reflection subgroups, then these algebras are the diagram algebras of braids and ties of Aicardi and Juyumaya. We prove a stucture theorem for these algebras, generalizing a result of Espinoza and Ryom-Hansen from the case of the symmetric group to the general case. We prove that these algebras are symmetric algebras at least when W is a Coxeter group, and in general under the trace conjecture of Brou\'e, Malle and Michel.
... By standard arguments (see e.g. [13] proposition 2.2, or [5]) we know that the induced map P = π 1 (X, x 0 ) → π 1 (X 0 , x 0 ) is surjective, and that its kernel K is normally generated by the meridians around the hyperplanes in A c L . Since the following diagram is commutative 1 ...
Article
We investigate the extensions of the Hecke algebras of finite (complex) reflection groups by lattices of reflection subgroups that we introduced, for some of them, in our previous work on the Yokonuma-Hecke algebras and their connections with Artin groups. When the Hecke algebra is attached to the symmetric group, and the lattice contains all reflection subgroups, then these algebras are the diagram algebras of braids and ties of Aicardi and Juyumaya. We prove a stucture theorem for these algebras, generalizing a result of Espinoza and Ryom-Hansen from the case of the symmetric group to the general case. We prove that these algebras are symmetric algebras at least when W is a Coxeter group, and in general under the trace conjecture of Brou\'e, Malle and Michel.
... It was recently shown that alternative Garside structures exist: see [10] for the braid groups, and [7, 6] for some other Artin–Tits groups associated with finite Coxeter groups. Very recently, quasi-Garside structures (a variant in which one does not require that the divisors of the Garside element be finite in number) have been found on some Artin–Tits groups associated with infinite Coxeter groups, firstly the free groups [5] and, conjecturally, all Artin–Tits groups (N. Brady, J. Crisp, A. Kaul, J. McCammond).) ...
... Note that an explicit description of the maps π k → π can be obtained via the braid monodromy of a generic projection of the curve (accurate packages have been developed to compute braid monodromies of curves with equations over the rationals by Bessis [9] and Carmona [13]). For notation the reader is referred to the discussion previous to Theorem 5.2(1). ...
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In the present paper, Alexander polynomials of plane algebraic curves twisted by linear representations are considered. They are shown to divide the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is given by the determinant of the Blanchfield intersection form. Specializing to the classical case, this gives a divisibility formula in the sense of Libgober's divisibility theorem. Examples of twisted polynomials for some algebraic curves are explicitly calculated showing that they can detect Zariski pairs of equivalent Alexander polynomials and that they are sensitive to nodal degenerations.
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Empirical properties of generating systems for complex reflection groups and their braid groups have been observed by Orlik-Solomon and Brou\'e-Malle-Rouquier, using Shephard-Todd classification. We give a general existence result for presentations of braid groups, which partially explains and generalizes the known empirical properties. Our approach is invariant-theoretic and does not use the classification. The two ingredients are Springer theory of regular elements and a Zariski-like theorem. Comment: 21 pages
VKCURVE, software package for GAP3, source and documentation available at
  • D Vk
  • J Bessis
  • Michel
[VK] D. Bessis, J. Michel, VKCURVE, software package for GAP3, source and documentation available at: http://www.math.jussieu.fr/ ∼ jmichel/vkcurve.html