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Étude des fibres singulières des systèmes de Mumford impairs et pairs

Abstract : This thesis is dedicated to the study and to the description of the fibres of the momentum map of the (even or odd) Mumford system of degree g>0. These fibres are parameterized by hyperelliptic curves. Mumford proved that each fiber over a smooth curve is isomorphic to the Jacobian of the curve, minus its theta divisor. We give a geometrical as well as an algebraic description of the fibers over any singular curve. The geometrical description uses in an essential way the g vector field of the Mumford system. They define a stratification of each fiber where each stratum is isomorphic to a particular stratum, called the maximal stratum, of a fiber of a Mumford system of degree at most g. The algebraic description uses the theory of subresultants, which is applied to the polynomials which parametrize the points of phase space. We show that every stratum is isomorphic with an affine part of the generalized Jacobian of a singular hyperelliptic curve. We also prove that the Mumford vector fields are translation invariant on these generalized Jacobians.
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Submitted on : Tuesday, January 15, 2019 - 9:45:11 AM
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Yasmine Fittouhi. Étude des fibres singulières des systèmes de Mumford impairs et pairs. Géométrie algébrique [math.AG]. Université de Poitiers, 2017. Français. ⟨NNT : 2017POIT2252⟩. ⟨tel-01981449⟩



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