117 research outputs found
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 systÚmes 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 modÚles, et d'une ontologie du domaine. Le GO donne accÚs 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
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