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Variétés de Richardson : multiplicités et désingularisation

Abstract : A Richardson variety is the intersection of a direct Schubert variety with an opposite Schubert variety inside a fiag variety. ln this thesis, we are interested in the singularities of Richardson varieties. A result of Kreiman and Lakshmibai gives the multiplicity at a T-fixed point on a Richardson variety. In chapter 1, we prove that in characteristic zero, their formula is true for an arbitrary point, provided the fiag variety is cominuscule. Next, we consider a desingularization of a Richardson variety in the full fiag variety of type An, obtained as a subvariety of a Bott-Samelson variety. There is a natural family of line bundles on Bott-Samelson varieties, and their spaces of sections have been studied by Lakshmibai and Magyar, who give a basis of these spaces indexed by combinatorial objects called standard tableaux. We prove in chapter II that this basis is compatible with the desingularization of the Richardson variety when the line bundle is very ample. In this way, we obtain a basis indexed by particular tableaux, the so-called w0-standard ones.
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  • HAL Id : tel-03416030, version 1

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Michaël Balan. Variétés de Richardson : multiplicités et désingularisation. Mathématiques [math]. Université de Valenciennes et du Hainaut-Cambrésis, 2011. Français. ⟨NNT : 2011VALE0027⟩. ⟨tel-03416030⟩

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