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In this thesis work we study the fundamental group-scheme of curve-connected varieties or associated to them Curve-connected varieties are the generalization of rationally connected varieties whose definition was conceived by J Koll r These notions are the closest ones in algebraic geometry to the notion of arc connectedness in topology because over an algebraically closed field uncountable over any pair of two very general points in a curve-connected variety resp chain-connected there exists a curve resp chain of curves with a morphism to the variety whose image contains the two points mentioned before Depending on the type of curves we consider we have the notions of g-connectedness resp chain g-connectedness where we consider exclusively curves resp chains of curves with irreducible components are smooth and projective curves of genus g and the notion of C-connectedness for a fixed curve C where over any two very general points we can contain them in the image of a morphism from C to the variety Using classical and recent results from the theory of fundamental group-schemes which classifies torsors under the action of an affine group-scheme notably Nori fundamental group-scheme and the S-fundamental group-scheme we try to describe the Nori fundamental group-scheme of certain types of curve-connected varieties for which the rationally connected case is known and some associated varieties To obtain these results we use all the aspects that play a role in the theory of the fundamental group-scheme affine group-schemes tannakian categories of vector bundles over proper varieties and the theory of affine torsors Moreover we build new fundamental group-schemes associated to tannakian categories of vector bundles over varieties where we can join any pair of points by a chain of curves belonging to arbitrary families of curves generalizing a recent construction of I Biswas P H Hai and J P Dos Santos which could provide a new framework for the study of fundamental group-schemes of curve-connected varieties More specifically we propose two different approaches to understand these fundamental group-schemes apply the new framework for fundamental group-schemes described in the paragraph above for g-connected varieties and to utilize the maximal rationally connected fibration and describe the behaviour of the fundamental group over it Inspired by the second approach we describe the fundamental group-scheme of fibrations over elliptic curves with rationally connected fibers inspired by the description of elliptically connected varieties in characteristic zero made by F Gounelas These varieties are not necessarily elliptically connected in positive characteristic but the description of their fundamental group-schemes is possible with the homotopy exact sequence

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Fundamental group-scheme of some curve-connected varieties and associated ones

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