Factorisations explicites de g(y) - h(z)
Families of elliptic curves with genus 2 covers of degree 2.
We study genus 2 covers of relative elliptic curves over an arbitrary base in which 2 is invertible. Particular emphasis lies on the case that the covering degree is 2. We show that the data in the "basic construction" of genus 2 covers of relative elliptic curves determine the cover in a unique way (up to isomorphism).A classical theorem says that a genus 2 cover of an elliptic curve of degree 2 over a field of characteristic ≠ 2 is birational to a product of two elliptic curves over the projective...
Familles de Hurwitz et cohomologie non abélienne
Nous nous intéressons à la question de l’existence de familles de Hurwitz au-dessus d’un espace de modules de revêtements de la droite. On sait que de telles familles existent dans le cas où les revêtements n’ont pas d’automorphismes. Dans le cas général, il y a une obstruction cohomologique, de nature non-abélienne. Nous donnons une double description de cette obstruction : la première en termes de gerbe, l’outil le mieux adapté à des situations cohomologiques non-abéliennes et la deuxièmes en...
Familles de revêtements de la droite projective
Field of moduli versus field of definition for cyclic covers of the projective line
We give a criterion, based on the automorphism group, for certain cyclic covers of the projective line to be defined over their field of moduli. An example of a cyclic cover of the complex projective line with field of moduli that can not be defined over is also given.
Fields of moduli of three-point -covers with cyclic -Sylow, II
We continue the examination of the stable reduction and fields of moduli of -Galois covers of the projective line over a complete discrete valuation field of mixed characteristic , where has a cyclic-Sylow subgroup of order . Suppose further that the normalizer of acts on via an involution. Under mild assumptions, if is a three-point -Galois cover defined over , then the th higher ramification groups above for the upper numbering of the (Galois closure of the) extension vanish,...
Formal deformation of curves with group scheme action
We study equivariant deformations of singular curves with an action of a finite flat group scheme, using a simplified version of Illusie's equivariant cotangent complex. We apply these methods in a special case which is relevant for the study of the stable reduction of three point covers.
Fundamental group of a class of rational cuspidal curves.
Fundamental groupoids of -graphs.
Fundamental groups of some special quadric arrangements.
Continuing our work on the fundamental groups of conic-line arrangements (Amram et al., 2003), we obtain presentations of fundamental groups of the complements of three families of quadric arrangements in P2. The first arrangement is a union of n conics, which are tangent to each other at two common points. The second arrangement is composed of n quadrics which are tangent to each other at one common point. The third arrangement is composed of n quadrics, n-1 of them are tangent to the n-th one...