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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 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 |
3. 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 |
4. CJM 2000 (vol 52 pp. 1018)
Essential Dimensions of Algebraic Groups and a Resolution Theorem for $G$-Varieties Let $G$ be an algebraic group and let $X$ be a generically free $G$-variety.
We show that $X$ can be transformed, by a sequence of blowups with smooth
$G$-equivariant centers, into a $G$-variety $X'$ with the following
property the stabilizer of every point of $X'$ is isomorphic to a
semidirect product $U \sdp A$ of a unipotent group $U$ and a
diagonalizable group $A$.
As an application of this result, we prove new lower bounds on essential
dimensions of some algebraic groups. We also show that certain
polynomials in one variable cannot be simplified by a Tschirnhaus
transformation.
Categories:14L30, 14E15, 14E05, 12E05, 20G10 |