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La tour de Teichmüller--Grothendieck

Abstract : We develop the notion of fundamental groupoid of an algebraic (Deligne--Mumford) stack, from its category of etale coverings: this definition is compatible with T. Oda's. This groupoid computes the profinite fundamental group of the associated analytic orbifold and fits in an exact sequence relating geometric and algebaic fundamental groups. In a second chapter, after defining tangent spaces and divisors with normal crossings in the realm of algebraic stacks, we generalize the notion of a tangencial base-point, well-known for schemes in characteristic zero, to stacks in arbitrary characteristic. In a third chapter, we show that the open stata of the moduli space of stable curves with marked points may be described as quotients of products of moduli spaces of smooth curves of lower dimension. We also explain how tangentials base-points on these stacks stem from ribbon graphs. In a fourth chapter, we stress some links between the tower of fundamental groupoids of the moduli spaces of smooth marked curves with respect to the above-mentionned tangencial base points and Lyubashenko's groupoid, by describing paths (twist, braiding) and proving some relations. Two appendices explain the notions of algebraic stacks and2-categories.
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Contributor : Vincent Zoonekynd <>
Submitted on : Tuesday, February 26, 2002 - 12:30:46 PM
Last modification on : Tuesday, February 26, 2002 - 12:30:46 PM
Long-term archiving on: : Tuesday, September 11, 2012 - 2:50:10 PM


  • HAL Id : tel-00001140, version 1



Vincent Zoonekynd. La tour de Teichmüller--Grothendieck. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2001. Français. ⟨tel-00001140⟩



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