Contemporary Mathematics
On the Log-Canonical Threshold for Germs of Plane Curves
E. Artal Bartolo, Pi. Cassou-Nogu`es, I. Luengo, and A. Melle-Hern´andez
Dedicated to e ung Tr´ ang on the Occasion of His Sixteenth Birthday
Abstract. In this article we show that for a given, reduced or non reduced,
germ of a complex plane curve, there exists a local system of coordinates such
that its log-canonical threshold at the singularity can be explicitly computed
from the intersection of the boundary of its Newton polygon in such coordi-
nates (degenerated or not) with the diagonal line.
1. Introduction
Let f be the germ of an analytic function at a point p on a complex d-
dimensional manifold X such that f(p) = 0. Let π : Y X be an embedded resolu-
tion of the hypersurface f−1{0} defined by the zero locus of f. Let Ei, i I, be the
irreducible components of the divisor π−1(f−1{0}) and let Ip := {i I | p π(Ei)}.
For each j I, we denote by Nj the multiplicity of Ej in the divisor of the function
f π and we denote by νj 1 the multiplicity of Ej in the divisor of π∗(ω) where
ω is a non-vanishing holomorphic d-form in a neighbourhood of p X. The pair
(νi, Ni) is called the numerical data of the irreducible component Ei.
The log-canonical threshold of f at p is defined by
cp(f) := min
i∈Ip
νi
Ni
,
see [8, Proposition 8.5]. It does not depend on the resolution π since −cp(f) is the
closest root to the origin of the Bernstein-Sato polynomial bf,p(s) of f at p, see [8,
Theorem 10.6] or [9, 17]. Since f(p) = 0 then bf,p(s) = (s +
1)˜f,p(s)
b where
˜
b f,p(s)
is the reduced Bernstein-Sato polynomial of f at p introduced by M. Saito in [13].
Let Rf,p be the set of roots of
˜
b
f,p
(−s) and αf,p := min Rf,p.
The following result by M. Saito, Corollary 3.3 in [13] gives a bound for cp(f)
in the non-degenerate case. We introduce a preliminary notation.
2000 Mathematics Subject Classification. 14B05,32S05,32S10.
Key words and phrases. Log-canonical threshold, Eisenbud-Neumman diagrams, topological
zeta function.
First and second author are partially supported by MTM2007-67908-C02-01; the last three
authors are partially supported by the grant MTM2007-67908-C02-02.
1
Contemporary Mathematics
Volume 474, 2008
1
http://dx.doi.org/10.1090/conm/474/09250
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