Braid Monodromy of Algebraic Curves

Lattice extensions of Hecke algebras

“…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…”

Kummer covers and braid monodromy

“…This problem was considered in [18] using a computer-based approach. It can be seen as the union of the four completely reducible fibers in a pencil of smooth cubics whose base points are the nine inflection points.…”

“…Our purpose in this section is to obtain the generic braid monodromy of the Hesse arrangement. This problem was considered in [18] using a computer-based approach. For our approach, note that the Hesse arrangement can be obtained from the nodal cubic in ğ 7.5 using a Kummer cover π 3 : it is the preimage of the cubic and the three ramification lines.…”

“…On the other hand, the generic braid monodromy of a plane projective curve is a powerful invariant that provides a way to compute the fundamental group of its complement, and it was originally described as a formalization of the Zariski-van Kampen method [42,26] (see [35,23]; also [18] and references therein for a detailed exposition on the subject). However, generic braid monodromies are much more powerful invariants, since in fact they encode the topology of the embedding of the curve, as well as the isomorphism problem for surfaces whose branching locus over P 2 is a given curve (see [13,29,37,15,14,28,9,21,1], to name only a few).…”

Un exemple de feuilletage modulaire déduit d’une solution algébrique de l’équation de Painlevé VI

“…Pour une introduction aux connexions logarithmiques plates, voir [27]. Enfin, pour les calculs de groupes fondamentaux, on propose [30] ou [11]. Par commodité, on a employé fréquem-ment un logiciel de calcul formel pour nos discussions.…”