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 Lˆ e D˜ 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

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 Lˆ e D˜ 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