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On the Cremona group and it's subgroups

Abstract : The thesis consists of three parts: 1. Group theory. Here we emphasize the role of the mutation, which is some element of the Thompson group T. In particular using mutations we get a new presentation of this group in terms of generators and relations. 2. Birational geometry. We study the action of the Cremona group and some of it's subgroups on the projective plane. In particular we are interested in the subgroup Symp of the Cremona group, that preserve the logarithmic Poisson bracket, and in it's subgroup H generated by cluster mutations and by SL(2;Z). We construct a projective system of surfaces, on which this groups act by regular automorphisms, and then we deduce a linear presentation of H in the inductive limit of Picard groups of rational surfaces. 3. Homological algebra. For an algebraic variety we construct a triangulated category which depends only on it's birational class. Using the techniques of the quotients of dg-categories we compute such a triangulated category for a rational surface. As a consequence we obtain an action of the Cremona group on the non-commutative ring by outer automorphisms. We give applications of this results to the formulas of non-commutative cluster mutations.
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Alexandr Usnich. On the Cremona group and it's subgroups. Algebraic Geometry [math.AG]. Université Pierre et Marie Curie - Paris VI, 2008. English. ⟨NNT : 2008PA066376⟩. ⟨tel-00812808⟩

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