Greenberg approximation and the geometry of arc spaces

We study the differential properties of generalized arc schemes, and geometric versions of Kolchin's Irreducibility Theorem over arbitrary base fields. As an intermediate step, we prove an approximation result for arcs by algebraic curves

Birational self-maps and piecewise algebraic geometry

Let X be a smooth projective complex variety, of dimension 3, whose Hodge numbers h^{3,0}(X), h^{1,0}(X) both vanish. Let f: X--> X be a birational map that induces an isomorphism on (dense) open subvarieties U,V of X. Then we show that the complex reduced varieties (X \ U), (X \ V) are piecewise isomorphic

The Grid Observatory

International audienceThe goal of the Grid Observatory project (GO) is to contribute to an experimental theory of large grid systems by integrating the collection of data on the behaviour of the flagship European Grid Infrastructure (EGI) and its users, the development of models, and an ontology for the domain knowledge. The GO gives access to a database of grid usage traces available to the wider computer science community without the need of grid credentials. The paper presents the architecture of the digital curation process enacted by the GO and examples of their exploitation.L'objectif du projet Grid Observatoiry (GO) est de contribuer à une théorie expérimentale de systèmes globalisés à grande échelle en intégrant l'acquisition de données sur le comportement de l'infrastructure de la grille européenne phare (EGI) et de ses utilisateurs, avec le développement de modèles, et d'une ontologie du domaine. Le GO donne accès à une base de données des traces d'utilisation de la grille, mise à la disposition de la communauté scientifique. L'article présente l'architecture du processus de conservation numérique adoptée par le GO et des exemples de l'exploitation des traces collectées

Motivic Serre invariants, ramification, and the analytic Milnor fiber

We show how formal and rigid geometry can be used in the theory of complex singularities, and in particular in the study of the Milnor fibration and the motivic zeta function. We introduce the so-called analytic Milnor fiber associated to the germ of a morphism f from a smooth complex algebraic variety X to the affine line. This analytic Milnor fiber is a smooth rigid variety over the field of Laurent series C((t)). Its etale cohomology coincides with the singular cohomology of the classical topological Milnor fiber of f; the monodromy transformation is given by the Galois action. Moreover, the points on the analytic Milnor fiber are closely related to the motivic zeta function of f, and the arc space of X. We show how the motivic zeta function can be recovered as some kind of Weil zeta function of the formal completion of X along the special fiber of f, and we establish a corresponding Grothendieck trace formula, which relates, in particular, the rational points on the analytic Milnor fiber over finite extensions of C((t)), to the Galois action on its etale cohomology. The general observation is that the arithmetic properties of the analytic Milnor fiber reflect the structure of the singularity of the germ f.Comment: Some minor errors corrected. The original publication is available at http://www.springerlink.co

Singular Support of a Vertex Algebra and the Arc Space of Its Associated Scheme

Book Subtitle: In Honour of the 75th Birthday of Tony JosephSeries Title: Progress in Mathematics (vol. 330)Attached to a vertex algebra V are two geometric objects. The associated scheme of V isthespectrum of Zhu's Poisson algebra Rv.Thesingular support of V is the spectrum of the associated graded algebra gr(V) with respect to Li's canonical decreasing filtration. There is a closed embedding from the singular support to the arc space of the associated scheme, which is an isomorphism in many interesting cases. In this note we give an example of a non-quasi-lisse vertex algebra whose associated scheme is reduced, for which the isomorphism is not true as schemes but true as varieties