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Courbes rationnelles et hypersurfaces de l'espace projectif

Abstract : An algebraic variety is unirational if it is dominated by a projective space; it is separably unirational if one can take separable the previous morphism. This last property has interest only in positive characteristic. By writing again the demonstration of Paranjape and Srinivas of the unirationality of the hypersurfaces of very small degree in front of dimension, we notice that it shows in fact the separable unirationality. We are interested also in the separability of the morphisms provided by various traditional constructions of unirationality of cubic hypersurfaces.

In the third part, we study the separable rational connectedness: a smooth projective variety X on a algebraically closed field is separably rationally connected if there is a very free rational curve (i.e. with an ample tangent bundle) on X. We test on the Fermat's hypersurfaces of dimension N-1 and degree q+1, where q is a power of the characteristic of the ground field, the conjecture that all the smooth hypersurfaces of dimension N-1 and degree smaller than N are separably rationally connected. We show that for N larger than 2q-1, the Fermat's hypersurface of degree q+1 contains a very free rational curve definite on the prime subfield; it is thus separably rationally connected.
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Contributor : Denis Conduché <>
Submitted on : Thursday, November 23, 2006 - 11:52:00 PM
Last modification on : Friday, June 19, 2020 - 9:10:04 AM
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  • HAL Id : tel-00115879, version 1



Denis Conduché. Courbes rationnelles et hypersurfaces de l'espace projectif. Mathématiques [math]. Université Louis Pasteur - Strasbourg I, 2006. Français. ⟨NNT : 2006STR13243⟩. ⟨tel-00115879⟩



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