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Geometry of singular Fano varieties and projective vector bundles over curves

Abstract : This thesis is devoted to the geometry of Fano varieties and projective vector bundles over a smooth projective curve.In the first part we study the geometry of mildly singular Fano varieties on which there is a prime divisor of Picard number 1. By studying the contractions associated to extremal rays in the Mori cone of these varieties, we provide a structure theorem in dimension 3 for varieties with maximal Picard number. Afterwards, we address the case of toric varieties and we extend the structure theorem to toric varieties of dimension greater than 3 and with maximal Picard number. Finally, we treat the lifting of extremal contractions to universal covering spaces in codimension 1.In the second part we study Newton-Okounkov bodies on projective vector bundles over a smooth projective curve. Inspired by Wolfe's estimates used to compute the volume function on these varieties, we compute all Newton-Okounkov bodies with respect to linear flags and we study how these bodies depend on the Schubert cell decomposition with respect to linear flags which are compatible with the Harder-Narasimhan filtration of the bundle. Moreover, we characterize semi-stable vector bundles over smooth projective curves via Newton-Okounkov bodies.
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Pedro Pablo Montero Silva. Geometry of singular Fano varieties and projective vector bundles over curves. Algebraic Geometry [math.AG]. Université Grenoble Alpes, 2017. English. ⟨NNT : 2017GREAM050⟩. ⟨tel-01753190⟩

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