Title :

On the First Page of the Renaudineau-Shaw Spectral Sequence

Author(s) :

Chenal, Jules [Auteur]
Université de Lille
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Publication date :

2024-03

English keyword(s) :

Topology of real varieties
Viro's Patchwork

HAL domain(s) :

Mathématiques [math]/Géométrie algébrique [math.AG]
Mathématiques [math]/Topologie algébrique [math.AT]

English abstract : [en]

A T-hypersurface is a combinatorial hypersurface of the real locus of a projective toricvariety Y . It is constructed from a primitive triangulation K of a moment polytope Pof Y and a 0-cochain ε on K with coefficients in ...
Show more >A T-hypersurface is a combinatorial hypersurface of the real locus of a projective toricvariety Y . It is constructed from a primitive triangulation K of a moment polytope Pof Y and a 0-cochain ε on K with coefficients in the field with two elements F2, called asign distribution. O. Viro showed that when K is convex the T-hypersurface is ambiantlyisotopic to a real algebraic hypersurface of Y . A. Renaudineau and K. Shaw gave upperbounds on the Betti numbers of T-hypersurfaces in terms of the Hodge numbers of ageneric section of the ample line bundle L associated with the moment polytope. Inparticular, the number of connected components of a T-hypersurface cannot exceed thegeometric genus of a generic section of L plus one. In this article we investigate wether thisupper bound is attainable. We are able to characterise the couples (K; ε) leading to Thypersurfacesrealising the Renaudineau-Shaw upper bound on the number of connectedomponents. This theorem generalises B. Haas’ theorem for T-curves. In contrast withthis results we find that the upper bound is not always attainable on every primitivetriangulations. For some of those on which it is not attainable we provide a sharper upperbound. Finally we use our characterisation to show that there always exist a triangulationand a sign distribution on the standard simplex that reach the Renaudineau-Shaw upperbound. We also study the growth of the expected number of connected components of aT-hypersurface as we dilate the moment polytope by d (i.e. we tensorise the line bundled-times with itself) and show that it is always of the order of dn where n is the dimensionof P.Show less >

Language :

Anglais

ANR Project :

Aspects tropicaux des singularités

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL