T-Hypersurfaces with a Maximal Number of Connected Components (Jules Chenal, Université de Lille)

08.04.2024 14:30 – 16:00

Abstract:
A T-hypersurface is a combinatorial hypersurface of the real locus of a projective toric variety Y . It is constructed from a primitive triangulation K of a moment polytope P of Y and a 0-cochain ε on K with coefficients in the field with two elements F2, called a sign distribution. O. Viro showed that when K is convex the T-hypersurface is ambiantly isotopic to a real algebraic hypersurface of Y . A. Renaudineau and K. Shaw gave upper bounds on the Betti numbers of T-hypersurfaces in terms of the Hodge numbers of a generic section of the ample line bundle L associated with the moment polytope. In particular, the number of connected components of a T-hypersurface cannot exceed the geometric genus of a generic section of L plus one. We want to present a generalisation of B. Haas' theorem that characterises the couples (K;ε) leading to T- hypersurfaces realising the Renaudineau-Shaw upper bound on the number of connecte components. In contrast with the dimension 2 we find that the upper bound is not always attainable on every primitive triangulations. For some of those on which it is not attainable we provide a sharper upper bound. Finally we use our characterisation to show that there always exist a triangulation and a sign distribution on the standard simplex that reach the Renaudineau-Shaw upper bound.

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Bâtiment: Conseil Général 7-9

Salle 1-15, Séminaire "Fables géométriques"

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