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Résolutions coniques des variétés discriminants e applications à la géométrie algébrique complexe et réelle

Abstract : There exist many situations when geometric or topological objects of some kind (like configurations of points on a plane or smooth mappings between manifolds or complex projective hypersurfaces etc.) are parametrized by the elements of a certain vector space. A (generalized) discriminant is the subset of such a vector space formed by the elements corresponding to the objects that are singular in some given sense. Via the Alexander duality, the cohomology groups of the discriminant complement are isomorphic to the Borel-Moore homology groups of the discriminant itself. The latter groups can often be computed using a certain natural resolution of the constant sheaf on the discriminant; due to their construction, such resolutions are sometimes called conical.

In this thesis we generalise the method of conical resolutions that was proposed by V. A. Vassiliev in order to study the cohomology of spaces of smooth complex projective hypersurfaces. Our construction
is based on inclusion relations between singular loci, rather than between the corresponding linear systems. This enables one to perform some computations that seem to be out of reach of the original approach. To illustrate our method, we compute the rational cohomology of the space of plane smooth complex quintic curves, of the space of smooth bielliptic curves on a nondegenerate quadric in the complex projective 3-space and of the space of smooth cubics in the real projective plane.

The thesis contains an appendix, where the following result is proven. Suppose the circle is equipped with an atlas, where all transition maps are fractional linear; then this circle bounds an orientable surface with an atlas where all transition maps are also fractional linear (only this time with complex coefficients) and are compatible in the obviuous way with the transition maps on the boundary. It is also shown that the classification of projective structures on the circle given long ago by N. Kuiper is not quite correct and a correct one is given.
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Contributor : Alexey Gorinov <>
Submitted on : Sunday, April 9, 2006 - 8:48:31 PM
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  • HAL Id : tel-00012101, version 1



Alexey Gorinov. Résolutions coniques des variétés discriminants e applications à la géométrie algébrique complexe et réelle. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2004. Français. ⟨tel-00012101⟩



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