5 research outputs found
On the structure of the fibers of truncation morphisms
Let k be an algebraically closed field and let X be a separated scheme of
finite type over k of pure dimension d. We study the structure of the fibres of
the truncation morphisms from the arc space of X to jet spaces of X and also
between jet spaces. Our results are generalizations of results of Denef,
Loeser, Ein and Mustata. We will use them to find the optimal lower bound for
the poles of the motivic zeta function associated to an arbitrary ideal.Comment: 18 pages, to appear in the Bulletin of the London Mathematical
Societ
Arcs and jets on toric singularities and quasi-ordinary singularities
Abstracts from the workshop held January 29--February 4, 2006Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu
Geometric motivic Poincar\'e series of quasi-ordinary singularities
The geometric motivic Poincar\'e series of a germ of complex
algebraic variety takes into account the classes in the Grothendieck ring of
the jets of arcs through . Denef and Loeser proved that this series has
a rational form. We give an explicit description of this invariant when
is an irreducible germ of quasi-ordinary hypersurface singularity in terms of
the Newton polyhedra of the logarithmic jacobian ideals. These ideals are
determined by the characteristic monomials of a quasi-ordinary branch
parametrizing
Arithmetic motivic Poincaré series of Toric varieties
The arithmetic motivic Poincaré series of a variety V defined over a field of characteristic zero, is an invariant of singularities which was introduced by Denef and Loeser by analogy with the Serre-Oesterlé series in arithmetic geometry. They proved that this motivic series has a rational form which specializes to the Serre-Oesterlé series when V is defined over the integers. This invariant, which is known explicitly for a few classes of singularities, remains quite mysterious. In this paper we study this motivic series when V is an affine toric variety. We obtain a formula for the rational form of this series in terms of the Newton polyhedra of the ideals of sums of combinations associated to the minimal system of generators of the semigroup of the toric variety. In particular, we deduce explicitly a finite set of candidate poles for this invariant