Expand all Collapse all | Results 1 - 8 of 8 |
1. CJM 2007 (vol 59 pp. 1098)
Ruled Exceptional Surfaces and the Poles of Motivic Zeta Functions In this paper we study ruled surfaces which appear as an exceptional
surface in a succession of blowing-ups. In particular we prove
that the $e$-invariant of such a ruled exceptional surface $E$ is
strictly positive whenever its intersection with the other
exceptional surfaces does not contain a fiber (of $E$). This fact
immediately enables us to resolve an open problem concerning an
intersection configuration on such a ruled exceptional surface
consisting of three nonintersecting sections. In the second part
of the paper we apply the non-vanishing of $e$ to the study of the
poles of the well-known topological, Hodge and motivic zeta
functions.
Categories:14E15, 14J26, 14B05, 14J17, 32S45 |
2. CJM 2007 (vol 59 pp. 1069)
Quotients jacobiens : une approche algÃ©brique Le diagramme d'Eisenbud et Neumann d'un germe est un arbre qui
repr\'esente ce germe et permet d'en calculer les invariants. On donne
une d\'emonstration alg\'ebrique d'un r\'esultat caract\'erisant
l'ensemble des quotients jacobiens d'un germe d'application $(f,g)$
\`a partir du diagramme d'Eisenbud et Neumann de $fg$.
Keywords:SingularitÃ©, jacobien, quotient jacobien, polygone de Newton Categories:14B05, 32S05, 32S50 |
3. CJM 2005 (vol 57 pp. 1314)
Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra In 1999 V. Arnol'd introduced the local contact algebra: studying the
problem of classification of singular curves in a contact space, he
showed the existence of the ghost of the contact structure (invariants
which are not related to the induced structure on the curve). Our
main result implies that the only reason for existence of the local
contact algebra and the ghost is the difference between the geometric
and (defined in this paper) algebraic restriction of a $1$-form to a
singular submanifold. We prove that a germ of any subset $N$ of a
contact manifold is well defined, up to contactomorphisms, by the
algebraic restriction to $N$ of the contact structure. This is a
generalization of the Darboux-Givental' theorem for smooth
submanifolds of a contact manifold. Studying the difference between
the geometric and the algebraic restrictions gives a powerful tool for
classification of stratified submanifolds of a contact manifold. This
is illustrated by complete solution of three classification problems,
including a simple explanation of V.~Arnold's results and further
classification results for singular curves in a contact space. We
also prove several results on the external geometry of a singular
submanifold $N$ in terms of the algebraic restriction of the contact
structure to $N$. In particular, the algebraic restriction is zero if
and only if $N$ is contained in a smooth Legendrian submanifold of
$M$.
Keywords:contact manifold, local contact algebra,, relative Darboux theorem, integral curves Categories:53D10, 14B05, 58K50 |
4. CJM 2002 (vol 54 pp. 55)
On the Milnor Fiber of a Quasi-ordinary Surface Singularity We verify a generalization of (3.3) from \cite{Le} proving
that the homotopy type of the Milnor fiber of a reduced
hypersurface singularity depends only on the embedded
topological type of the singularity. In particular, using
\cite{Za,Li1,Oh1,Gau} for irreducible quasi-ordinary germs,
it depends only on the normalized distinguished pairs of the
singularity. The main result of the paper provides an explicit
formula for the Euler-characteristic of the Milnor fiber in the
surface case.
Categories:14B05, 14E15, 32S55 |
5. CJM 2001 (vol 53 pp. 834)
Zeta Functions and `Kontsevich Invariants' on Singular Varieties Let $X$ be a nonsingular algebraic variety in characteristic zero. To
an effective divisor on $X$ Kontsevich has associated a certain
motivic integral, living in a completion of the Grothendieck ring of
algebraic varieties. He used this invariant to show that birational
(smooth, projective) Calabi-Yau varieties have the same Hodge
numbers. Then Denef and Loeser introduced the invariant {\it motivic
(Igusa) zeta function}, associated to a regular function on $X$, which
specializes to both the classical $p$-adic Igusa zeta function and the
topological zeta function, and also to Kontsevich's invariant.
This paper treats a generalization to singular varieties. Batyrev
already considered such a `Kontsevich invariant' for log terminal
varieties (on the level of Hodge polynomials of varieties instead of
in the Grothendieck ring), and previously we introduced a motivic zeta
function on normal surface germs. Here on any $\bbQ$-Gorenstein
variety $X$ we associate a motivic zeta function and a `Kontsevich
invariant' to effective $\bbQ$-Cartier divisors on $X$ whose support
contains the singular locus of~$X$.
Keywords:singularity invariant, topological zeta function, motivic zeta function Categories:14B05, 14E15, 32S50, 32S45 |
6. CJM 2000 (vol 52 pp. 1149)
Canonical Resolution of a Quasi-ordinary Surface Singularity We describe the embedded resolution of an irreducible quasi-ordinary
surface singularity $(V,p)$ which results from applying the canonical
resolution of Bierstone-Milman to $(V,p)$. We show that this process
depends solely on the characteristic pairs of $(V,p)$, as predicted
by Lipman. We describe the process explicitly enough that a resolution
graph for $f$ could in principle be obtained by computer using only
the characteristic pairs.
Keywords:canonical resolution, quasi-ordinary singularity Categories:14B05, 14J17, 32S05, 32S25 |
7. CJM 1999 (vol 51 pp. 1123)
First Steps of Local Contact Algebra We consider germs of mappings of a line to contact space and
classify the first simple singularities up to the action of
contactomorphisms in the target space and diffeomorphisms of the
line. Even in these first cases there arises a new interesting
interaction of local commutative algebra with contact structure.
Keywords:contact manifolds, local contact algebra, Diracian, contactian Categories:53D10, 14B05 |
8. CJM 1999 (vol 51 pp. 1226)
Semi-Affine Coxeter-Dynkin Graphs and $G \subseteq \SU_2(C)$ The semi-affine Coxeter-Dynkin graph is introduced, generalizing
both the affine and the finite types.
Categories:20C99, 05C25, 14B05 |