Hébergé par le projet Géométrie et Dynamique de l’IMJ

Organisateurs : Hussein MOURTADA, Matteo RUGGIERO, Bernard TEISSIER

Mardi 16h-18h

Bâtiment Sophie Germain, salle 1016

Hébergé par le projet Géométrie et Dynamique de l’IMJ

Organisateurs : Hussein MOURTADA, Matteo RUGGIERO, Bernard TEISSIER

Mardi 16h-18h

Bâtiment Sophie Germain, salle 1016

In his famous proof of the existence of a resolution of singularities in characteristic zero (1964) Hironaka also introduced several constructive/algorithmic ideas, most prominently the notion of standard bases which have become important tools in computational algebraic geometry nowadays. Other constructive aspects, however, seemed to be without further practical applications.

In this talk I shall explain solutions to two tasks of very different flavour in computational algebraic geometry, which profit each in their own way from Hironaka’s work : A massively parallel algorithm to decide non-singularity of a variety arises from the termination criterion of desingularization and makes use of descent in ambient dimension by means of hypersurfaces of maximal contact. On the other hand, when counting subrings (in the sense of order zeta-functions) p-adic integrals arise for which the domain of integration seems rather inaccessible at first glance. However, a suitable desingularization transforms the task into a number of easier problems each of which can be tackled by standard methods.

Suppose that $(K,\nu)$ is a valued field and $(L,\omega)$ is a finite extension, where $L=K[z]/(f)$. Further suppose that $A$ is a local domain which is dominated by $\nu$ and the unitary polynomial $f(z)$ is in $A[z]$. We consider the problem of computing a generating sequence for $\omega$ in $A[z]/(f)$ and computing the structure of the associated graded ring of $A[z]/(f)$ along $\omega$ as an extension of the associated graded ring of $A$ along $\nu$.

The problem of constructing generating sequences in a Noetherian local domain $A$ which is dominated by a valuation is extremely difficult, and little is known about this problem in general. It is well understood in the case that $A$ has dimension one, and for regular local rings of dimension two. It is known for certain valuations dominating two dimensional quotient singularities and for certain valuations dominating three dimensional regular local rings.

The problem of computing generating sequences for an extension $\omega$ of $\nu$ to R_*\nu*[z]/(f) where R_*\nu* is the valuation ring of $\nu$ and computing the structure of the associated graded ring of R_*\nu*[z]/(f) along $\omega$ as an extension of the associated graded ring of R_*\nu* along $\nu$ has been solved, in papers of MacLane for discrete valuations, and for general valuations by Vaquié.

In joint work with Hussein Mourtada and Bernard Teissier, we establish several theorems showing that MacLane’s algorithm can often be used to compute the generators and relations of extensions of associated graded rings along a valuation.

We assume that $A$ contains an algebraically closed field $k$ such that $A/m_A\cong k$ and that the residue field of $\nu$ is isomorphic to $k$.

If the characteristic $p$ of $k$ does not divide the degree of $f$, then we give a very simple algorithm in $A[z]/(f)$ if $\omega$ is the unique extension of $\nu$.

The associated graded ring of $A[z]/(f)$ along $\omega$ is then a finitely generated and presented module over the associated graded ring of $A$ along $\nu$.

If any of the above assumptions are removed, then we give examples showing that the conclusions of the theorem do not hold

We obtain generalizations of this theorem to the case when the extension of the valuation is not unique and when we have not restriction on the degree of the field extension, which we will discuss.

First we will discuss the numerical semigroup rings, which are subsemigroups of the natural numbers. In this case, these rings are also the homogeneous coordinate rings of the monomial curves. Given two semigroups A and B, we will construct the minimal homogenous resolution of the semigroup C obtained by gluing A and B. From this construction, we can read off the formulae for the invariants of C in terms of A and B. There is not yet a complete generalization of gluing in higher dimensions. We will give some partial results in higher dimensions as well as some instances where a gluing is impossible.

Milnor fibers and Milnor monodromies are the most basic invariants of the hypersurface singularities.

I will explain the way to study them, with mixed Hodge modules.

The cohomologies of Milnor fibers have canonical mixed Hodge structures.

If a singular point is an isolated singular point, their weight filtrations are the monodromy weight filtrations, which have the data of the Jordan normal forms of the Milnor monodromies.

To compute them, we express the mixed Hodge structures of the cohomologies of the Milnor fibers in terms of nearby cycle sheaves as mixed Hodge modules.

Then, thanks to the power of the functorial properties of the mixed Hodge modules, we can compute the mixed Hodge structures and the Milnor monodromies of the cohomologies of the Milnor fibers.

In this talk, first, I will introduce the definition of the Milnor fibrations, Milnor fibers and Milnor monodromies of hypersurface singular points.

Second, I will briefly explain the basic notions of mixed Hodge structures and mixed Hodge modules.

Finally, I will demonstrate how to compute the mixed Hodge structures and the Milnor monodromies of the cohomologies of Milnor fibers, with nearby cycle sheaves as mixed Hodge modules.