IMJ-PRG CNRS - UPMC - Paris Diderot

Séminaire sur les Singularités

Année 2017- 2018

Archive avant 2015

Hébergé par le projet Géométrie et Dynamique de l’IMJ

Organisateurs : Hussein MOURTADA, Matteo RUGGIERO, Bernard TEISSIER

Mardi 16h-18h
Bâtiment Sophie Germain, salle 2015

Adam Parusinski

Generic Zariski equisingularity of surfaces and Lipschitz stratification

mardi 22 mai 2018 à 16:00

Consider a (generic) Zariski equisingular family of surface singularities in C^3. We show that a natural stratification of such family given by the family of generic polar curves and the singular locus is a Lipschitz stratification in the sense of Mostowski. In particular, such a family is bi-Lipschitz trivial (that for families of isolated singularities has been shown by Neumann and Pichon). Our proof is based on an analysis of the equisingularity type of generic polar curves due to Briançon and Henry.

Version française : Equisingularité générique de Zariski de surfaces et stratification lipschitzienne. Nous étudions une famille de singularités de surfaces dans C^3, que nous supposons équisingulière au sens de Zariski. Nous montrons qu’une stratification naturelle de cette famille donnée par la famille de courbes polaires génériques et le lieu singulier est une stratification lipschitzienne au sens de Mostowski. En particulier, une telle famille est bi-Lipschitz trivial (ce que pour les familles de singularités isolées a été montré par Neumann et Pichon). Notre preuve est basée sur une analyse du type d’équisingularité des courbes polaires génériques due à Briançon et Henry.

Marco Golla - CNRS, Université de Nantes

Planar contact 3-manifolds and normal surface singularities

mardi 13 mars 2018 à 16:00

We give new obstructions to the existence of planar open books for contact structures, in terms of the homology of their fillings, with applications to links of normal surface singularities.
This is based on joint work with Paolo Ghiggini and Olga Plamenevskaya.

Antoine Chambert-Loir

Combinatoire des matroïdes et anneau de Chow de variétés toriques

mardi 27 février 2018 à 16:00

L’exposé sera consacré à une conjecture de log-concavité des coefficients du polynôme caractéristique d’un matroïde. La preuve que viennent d’en proposer
Adiprasito, Huh et Katz considère l’anneau de Chow d’une variété torique associée au matroïde. Bien que cette variété ne soit pas propre, son anneau vérifie dualité de Poincaré, théorème de Lefschetz fort et relations bilinéaires de Hodge-Riemann. La preuve de la log-concavité est alors analogue à celle des inégalités de Khovanskii-Teissier.

Franz-Viktor Kuhlmann - University of Szczecin, Poland

Defects, tame and wild

mardi 20 février 2018 à 16:00

Defects can appear in algebraic extensions of valued fields when the residue characteristic is positive. They constitute a major problem in
the structure theory of valued function fields, which in turn has been shown to be of crucial importance in the open problems of local
uniformization in positive characteristic and the model theory of imperfect valued fields. Therefore it is desirable to gain a deeper
understanding of defects.

The title actually contains a contradiction : extensions with nontrivial defect are wild by definition and not tame. However, the title expresses
the observation that some defects appear to be more harmful than others.
In valued fields of positive characteristic defects have been classified to be dependent when they can be obtained from purely inseparable
extensions by a transformation, and independent otherwise.

There are several reasons to believe that the dependent defects are the more harmful ones. For example, Cutkosky and Piltant gave an example of
an extension of two-dimensional valued function fields consisting of a tower of two Artin-Schreier defect extensions where strong
monomialization fails. Work of Cutkosky, ElHitti and Ghezzi has shown that both of them have dependent defect.

Recently we extended the classification to the case of valued fields of mixed characteristic, overcoming the obstacle that they do not have
inseparable extensions. Building on this, we introducedsemitame fields which generalize the class of tame fields ; the latter
have played an important role in results on local uniformization and the model theory of perfect valued fields. Semitame fields are close to
Gabber’s deeply ramified fields, and thus also to perfectoid fields (but the latter do not fit our purposes well). We hope that important results
on tame fields and the structure theory of their valued function fields can be generalized to the case of semitame fields.

If time permits, I will also mention another result, which relates to a continuation of Cutkosky’s and Piltant’s study of valued function fields
in dimension 2. While such function fields may admit infinite towers of defect extensions, there are only finitely many essentially distinct
defects, in a sense that will be made precise in the talk.

This is joint work with Anna Blaszczok.

Antoni Rangachev - Deformations of isolated singularities and local volumes

Séminaire sur les Singularités

mardi 21 novembre 2017 à 16:00

In this talk I will introduce a class of singularities that generalizes the class of smoothable singularities : these are all singularities that admit deformations to singularities with deficient conormal spaces. I will discuss how this new class arises from problems in differential equisingularity and how it relates to the local volume of a line bundle. If time permits, I will discuss which known classes of singularities have deficient conormal spaces.