Displaying similar documents to “The fundamental groupoid scheme and applications”

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.

Whittaker and Bessel functors for G 𝕊 p 4

Sergey Lysenko (2006)

Annales de l’institut Fourier

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The theory of Whittaker functors for G L n is an essential technical tools in Gaitsgory’s proof of the Vanishing Conjecture appearing in the geometric Langlands correspondence. We define Whittaker functors for G 𝕊 p 4 and study their properties. These functors correspond to the maximal parabolic subgroup of G 𝕊 p 4 , whose unipotent radical is not commutative. We also study similar functors corresponding to the Siegel parabolic subgroup of G 𝕊 p 4 , they are related with Bessel models for G 𝕊 p 4 and...

Elementary introduction to representable functors and Hilbert schemes

Stein Strømme (1996)

Banach Center Publications

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The purpose of this paper is to define and prove the existence of the Hilbert scheme. This was originally done by Grothendieck in [4]. A simplified proof was given by Mumford [11], and we will basically follow that proof, with small modifications.

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.

Universal covering spaces and fundamental groups in algebraic geometry as schemes

Ravi Vakil, Kirsten Wickelgren (2011)

Journal de Théorie des Nombres de Bordeaux

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In topology, the notions of the fundamental group and the universal cover are closely intertwined. By importing usual notions from topology into the algebraic and arithmetic setting, we construct a fundamental group family from a universal cover, both of which are schemes. A geometric fiber of the fundamental group family (as a topological group) is canonically the étale fundamental group. The constructions apply to all connected quasicompact quasiseparated schemes. With different methods...