Displaying similar documents to “Elementary introduction to representable functors and Hilbert schemes”

The fundamental groupoid scheme and applications

Hélène Esnault, Phùng Hô Hai (2008)

Annales de l’institut Fourier

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We define a linear structure on Grothendieck’s arithmetic fundamental group π 1 ( X , x ) of a scheme X defined over a field k of characteristic 0. It allows us to link the existence of sections of the Galois group Gal ( k ¯ / k ) to π 1 ( X , x ) with the existence of a neutral fiber functor on the category which linearizes it. We apply the construction to affine curves and neutral fiber functors coming from a tangent vector at a rational point at infinity, in order to follow this rational point in the universal covering...

Integral models for moduli spaces of G -torsors

Martin Olsson (2012)

Annales de l’institut Fourier

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Given a finite tame group scheme G , we construct compactifications of moduli spaces of G -torsors on algebraic varieties, based on a higher-dimensional version of the theory of twisted stable maps to classifying stacks.

Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber

Marco Antei (2010)

Journal de Théorie des Nombres de Bordeaux

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We show that the natural morphism ϕ : π 1 ( X η , x η ) → π 1 ( X , x ) η between the fundamental group scheme of the generic fiber X η of a scheme X over a connected Dedekind scheme and the generic fiber of the fundamental group scheme of X is always faithfully flat. As an application we give a necessary and sufficient condition for a finite, dominated pointed G -torsor over X η to be extended over X . We finally provide examples where ϕ : π 1 ( X η , x η ) → π 1 ( X , x ) η is an isomorphism.

On the S-fundamental group scheme

Adrian Langer (2011)

Annales de l’institut Fourier

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We introduce a new fundamental group scheme for varieties defined over an algebraically closed (or just perfect) field of positive characteristic and we use it to study generalization of C. Simpson’s results to positive characteristic. We also study the properties of this group and we prove Lefschetz type theorems.

Vertex algebras and the formal loop space

Mikhail Kapranov, Eric Vasserot (2004)

Publications Mathématiques de l'IHÉS

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We construct a certain algebro-geometric version ℒ ( X ) of the free loop space for a complex algebraic variety X. This is an ind-scheme containing the scheme ℒ 0 ( X ) of formal arcs in X as studied by Kontsevich and Denef-Loeser. We describe the chiral de Rham complex of Malikov, Schechtman and Vaintrob in terms of the space of formal distributions on ℒ ( X ) supported in ℒ 0 ( X ) . We also show that ℒ ( X ) possesses a factorization structure: a certain non-linear version of a vertex algebra structure. This explains...

Tame stacks in positive characteristic

Dan Abramovich, Martin Olsson, Angelo Vistoli (2008)

Annales de l’institut Fourier

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We introduce and study a class of algebraic stacks with finite inertia in positive and mixed characteristic, which we call tame algebraic stacks. They include tame Deligne-Mumford stacks, and are arguably better behaved than general Deligne-Mumford stacks. We also give a complete characterization of finite flat linearly reductive schemes over an arbitrary base. Our main result is that tame algebraic stacks are étale locally quotient by actions of linearly reductive finite group schemes. ...