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Hypersurfaces cubiques spéciales

Abstract : The rationality problem of smooth cubic hypersurfaces of dimension four (cubic fourfolds for short) is one of the most challenging open problem in algebraic geometry. In this thesis, we were interested in special cubic fourfolds, that is cubic fourfolds containing a surface which is not homologous to a complete intersection. These cubic fourfolds are parametrized by a countable union of divisors C_d in the moduli space of cubic fourfolds C.First, we work on the theory of intersection of these divisors C_d in C. More precisely, we consider some classes of cubic fourfolds that are birational to fibrations over P^2, where the fibers are rational surfaces.The rationality of these fibered cubic fourfolds is strongly related to the rationality of these surfaces over the function field of P^2 and to the existence of rational sections of the associated fibration. By intersecting the divisors parametrizing these fibered cubic fourfolds with other ones whose elements are known to be rational, via lattice theory, we provide explicit description of these intersections in terms of irreducible components and we exhibit new examples of rational fibered cubic fourfolds such that the associated fibration doesn't have sections. Furthermore, we extend this intersection by giving necessary condition for up to 20 divisors to intersect. We apply this construction to build a countable infinity of one dimensional families of cubic fourfolds with finite dimensional Chow motives of abelian type inside every divisor C_d. This also implies abelianity and finite dimensionality of the motive of certain related Hyperkähler varieties.Then, we consider universal cubic fourfolds over certain divisors C_d in C. We propose two methods to prove their unirationality, for the divisors C_d, in the range 8<=d<=42.Finally, we study another family of Fano fourfolds, Gushel-Mukai fourfolds, and develop a general method to show the unirationality of the moduli spaces of m pointed (cubic or Gushel-Mukai) fourfolds.
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Submitted on : Monday, November 29, 2021 - 4:10:07 PM
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  • HAL Id : tel-03455416, version 1

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Hanine Awada. Hypersurfaces cubiques spéciales. Topologie géométrique [math.GT]. Université Montpellier, 2021. Français. ⟨NNT : 2021MONTS047⟩. ⟨tel-03455416⟩

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