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Lieu singulier des variétés duales : approche géométrique et applications aux variétés homogènes.

Abstract : We are indebted to Friedrich Knop an amazing theorem that establishes a link between simple Lie algebras of type ADE, and simple singularities of the same type. The result is the following: we consider the projectivization the orbit of highest weight vector for the adjoint action of a simple Lie group on its Lie algebra (this variety is called the adjoint variety). Then there exists a hyperplane section tangent to the orbit with a unique singular point of the same type as the type of the Lie algebra. This theorem is the starting point of this thesis. To get a better understand of this relationship, we study the geometry of duals of adjoint varieties. In the first chapter we prove a dual version of Knop's theorem. Our theorem provides the discriminant of a simple singularity from the dual of the adjoint variety. The hyperplane considered by Knop can be interpreted as a very singular point of the dual. In the second chapter we consider the singular locus of the dual of a smooth projective variety. We show that the existence of certain stratum of maximal dimension is equivalent to the existence of hyperplane section of the original variety admitting singular points of a given type. Then we insist on the importance of two stratas which have a geometrical meaning: the dual variety of the tangential variety and the dual of the secant variety. Finally, in the last chapter we apply those results to study the normality of the duals of homogeneous varieties.
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https://tel.archives-ouvertes.fr/tel-00737441
Contributor : Frédéric Holweck <>
Submitted on : Monday, October 1, 2012 - 5:29:18 PM
Last modification on : Monday, October 19, 2020 - 11:06:18 AM
Long-term archiving on: : Wednesday, January 2, 2013 - 8:50:08 AM

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  • HAL Id : tel-00737441, version 1

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Holweck Frédéric. Lieu singulier des variétés duales : approche géométrique et applications aux variétés homogènes.. Géométrie algébrique [math.AG]. Université Paul Sabatier - Toulouse III, 2004. Français. ⟨tel-00737441⟩

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