MONOMIAL GENERATORS OF COMPLETE PLANAR IDEALS

arXiv:1701.03503v3 [math.AG] 29 Oct 2017

MARIA ALBERICH-CARRAMIÑANA, JOSEP ÀLVAREZ MONTANER, AND GUILLEM BLANCO
Abstract. We provide an algorithm that computes a set of generators for any complete
ideal in a smooth complex surface. More interestingly, these generators admit a presentation as monomials in a set of maximal contact elements associated to the minimal
log-resolution of the ideal. Furthermore, the monomial expression given by our method
is an equisingularity invariant of the ideal. As an outcome, we provide a geometric
method to compute the integral closure of a planar ideal and we apply our algorithm to
some families of complete ideals.

1. Introduction
Let (X, O) be a germ of smooth complex surface and OX,O the ring of germs of holomorphic functions in a neighbourhood of O, which we identify with C{x, y} by taking local
coordinates, and let m be the maximal ideal at O. Let π : X ′ → X be a proper birational
morphism that can be achieved as a sequence of blow-ups along a set of points. Given an
effective Z-divisor D in X ′ we may consider its associated (sheaf) ideal π∗ OX ′ (−D) whose
stalk at O we simply denote as HD . This type of ideals were systematically studied by
Zariski in [32]. They are complete ideals of OX,O and m-primary whenever D has exceptional support. Among the class of divisors defining the same complete ideal we may find
a unique maximal representative which happens to have the property of being antinef.
Actually, Zariski [32] showed that that above correspondence is in fact an isomorphism of
semigroups between the set of complete m-primary ideals and the set of antinef divisors
with exceptional support.
The aim of this work is to make computationally explicit this correspondence. Namely,
given a proper birational morphism π : X ′ → X and an antinef divisor D in X ′ we provide
an algorithm that gives us a system of generators of the ideal HD . Moreover, our method
captures the topological type of D and gives an output that discerns between the equisingularity class of HD and its analytic type: it produces a set of monomial expressions,
which are invariant of the equisingularity class of HD (or equivalently invariant of the
topological type of D); and when the set of variables in these monomial expressions are
substituted by a set of maximal contact elements of the resolution π, they give a system
of generators of HD . Our approach uses two main ingredients: the Zariski decomposition
of complete ideals into simple ones, and subtle properties in the theory of adjacent ideals,
which relay on results obtained in the study of sandwiched singularities and the Nash
conjecture of arcs on these singularities.
All three authors are supported by Spanish Ministerio de Economı́a y Competitividad MTM2015-69135P. MAC and JAM are also supported by Generalitat de Catalunya 2014SGR-634 project and they are
with the Barcelona Graduate School of Mathematics (BGSMath). MAC is also with the Institut de
Robòtica i Informàtica Industrial (CSIC-UPC).
1

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M. ALBERICH-CARRAMIÑANA, J. ÀLVAREZ MONTANER, AND G. BLANCO

Our main result is an algorithm (see Algorithm 3.5) which we briefly describe. We
start by fixing a set {f0 , . . . , fg } of maximal contact elements associated to π. These are
irreducible curves that are geometrically adapted to the proper birational morphism in the
sense that they are parameterized by the dead-ends of the dual graph of the divisor D (see
Subsection 2.4 for details). Once we have this fixed set we consider the decomposition
D = ρ1 D1 + · · · + ρr Dr of the divisor D corresponding to the Zariski decomposition
ρ1
ρr
HD = HD
· · · HD
, where the ideals HDi are simple and complete (see [32], [9] for details).
r
1
For each ideal HDi we compute in a very specific unique way an adjacent ideal below HDi ,
i.e. a codimension one ideal HDb i
HDi , and we prove in Proposition 4.4 that we may
pick a maximal contact element (or a power of it) belonging to HDi but not belonging to
HDb i . These adjacent ideals are no longer simple so we may repeat the same procedure.
This process finishes after finitely many steps since these adjacent ideals have smaller
support (see Lemma 4.1). We visualize the steps of our algorithm as a tree of ideals with
leaves corresponding to the maximal ideal. From the elements of maximal contact that
we pick at each step we construct a system of generators of HD .
As an outcome of our method we obtain a set of generators of a planar complete ideal
given by monomials in this fixed set of maximal contact elements (see Theorem 4.5). That
a
is, we obtain an expression of the form HD = (f α1 , . . . , f αr ), where f α = f0a0 · · · fg g for
α = (a0 , . . . , ag ) ∈ Zg+1
≥0 . More interestingly, if we fix another set of maximal contact
′
′
elements {f0 , . . . , fg }, the algorithm returns the same monomial expression (in this new
set) for the generators of HD since it only uses the information of the equisingularity
types of the maximal contact elements. Actually, the monomial expression that we have
for HD also works for any other ideal HD′ ⊆ OX,O whose associated divisor D ′ has the
same weighted dual graph as D. Hence the monomial expression given by our method is
an equisingularity invariant of the ideal.
As a first application of our algorithm we provide a method to compute the integral
closure of any ideal a ⊆ OX,O . More precisely, given a log-resolution π : X ′ → X of the
ideal a, let D be the effective Cartier divisor such that a · OX ′ = OX ′ (−D). Then, the
integral closure a is the ideal HD . We should point out that, given a set of generators of
the ideal a we may compute the divisor D using the algorithm that we developed in [2].
There are some general algorithms to compute the integral closure as those proposed
by Vasconcelos [30, 31], de Jong [11], Leonard and Pelikaan [21] or Singh and Swanson
[27] which have been implemented in computer algebra systems such as Macaulay2 [17]
or Singular [10]. When comparing these algorithms with our method in the case of
planar ideals we notice that the system of generators that we produce is not minimal
in general. However, our method reflects more closely some geometric properties. For
instance, if two different ideals belong to the same equisingularity class, meaning that
they have the similar weighted cluster of base points [9, 8.3] or equivalently their minimal
log-resolution have the same weighted dual graph, their integral closure have the same
monomial expression in the respective set of maximal contact elements.
The usefulness of our method becomes still more apparent when dealing with families
of complete ideals dominated by the same log-resolution. This is the case of multiplier
ideals which is an invariant of singularities that has received a lot of attention in recent
years. Combining the algorithms given in [2] and [5] with Algorithm 3.5 we develop a
method that, given a set of generators of a planar ideal, returns a set of generators of

MONOMIAL GENERATORS OF COMPLETE PLANAR IDEALS

3

the corresponding multiplier ideals. Since multiplier ideals are invariant up to integral
closure, we obtain a result that resembles a formula given by Howald [18] in the sense
that the multiplier ideals of a monomial ideal (in the set of maximal contact elements) is
monomial as well.
Another interesting family of complete ideals was considered by Teissier in [28]. These
ideals are described by valuative conditions given by the intersection multiplicity of the
elements of OX,O with a fixed germ of plane curve. With the help of Algorithm 3.5 we
may provide an explicit system of generators for these ideals.
Finally, we would like to mention that Casas-Alvero [8] proposes a geometrical procedure to obtain a minimal set of generators of the complete ideal HD based on a construction of a filtration by complete ideals, namely a chain of consecutive adjacent ideals
between HD and mHD . The main drawback in order to computationally implement this
method is that one does not know a priori which log-resolution of HD will dominate all
the log-resolutions of each complete ideal appearing in the filtration, since it depends on
the choices of the adjacent ideal at each step. In contrast, our procedure fixes a proper
birational morphism π from the very beginning and we always keep working with complete ideals whose log-resolution is dominated by π. The further points which must be
blown up after π in the Casas-Alvero’s method can not be chosen following an intrinsic
unique way which works for any ideal whose log-resolution is dominated by π. Moreover
these choices in Casas-Alvero’s method determine the generators of HD : the generators
must go through certain infinitely near points and have to skip some others. Both sets
of points are among those blown up by π and by the further blowing-ups. Therefore an
hypothetical computational implementation of Casas-Alvero’s method would suffer from a
lack of robustness and invariance, which are the strengths of our method. For discounted,
the fact that the monomial expression given by our method is an equisingularity invariant
of the ideal is not achieved by any other previous method.
The algorithms developed in this paper have been implemented in the computer algebra
system Magma [7] and are available at
github.com/gblanco92/IntegralClosureDim2
2. Preliminaries
The aim of this section is to introduce all the background that we will use throughout
this work.
2.1. Proper birational morphisms and infinitely near points. Let X be a smooth
complex surface and OX,O ∼
= C{x, y} the ring of germs of holomorphic functions in a
neighborhood of a smooth point O ∈ X and m = mX,O ⊆ OX,O the maximal ideal at O.
Given a proper ideal a ⊆ OX,O we have a decomposition a = (a) · a′ , where a ∈ OX,O is
the greatest common divisor of the elements of a and a′ is m-primary.
Let π : X ′ → X a proper birational morphism, with X ′ a smooth surface. Any such
proper birational morphism can be achieved as a sequence of blow-ups
π : X ′ = Xr+1 −
→ Xr −
→ ··· −
→ X1 = X
with Xi+1 = Blpi Xi for a point pi ∈ Xi . Denote by E = Exc(π) the exceptional locus. Since we are in a local framework, we shall assume throughout this work that E

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M. ALBERICH-CARRAMIÑANA, J. ÀLVAREZ MONTANER, AND G. BLANCO

is connected. The set K of points which have been blown up gives an index set for the
exceptional components {Ep }p∈K in E. We say that these points are a cluster of infinitely
near to the origin and we may establish a proximity relation between them. Namely, we
say that a point q ∈ K is proximate to p ∈ K if and only if q belongs to the exceptional divisor Ep corresponding to p as proper or infinitely near point. We will denote
this relation as q → p and we collect all these relations by means of the proximity matrix
P = (Pp,q ) defined as:

if p = q,
 1
−1
if q → p,
Pp,q :=
 0
otherwise.
The proximity matrix is related to the negative-definite intersection matrix N = (Ep · Eq )
by the formula N = −P ⊤ P . An infinitely near point q is proximate to just one or two
points. In the former case we say that q is a free point, and in the later it is a satellite
point. Besides, one can establish a partial ordering in K. Namely, q ≤ p if and only if p
is infinitely near to q.
A log-resolution of a proper ideal a is a proper birational morphism π : X ′ → X where
X ′ is smooth such that there is an effective Cartier divisor F satisfying a·OX ′ = OX ′ (−F ),
and the divisor F + Exc(π) has simple
P normal crossings. The weighted dual graph of π is
the dual graph of the divisor F = 1≤i≤s vi (F )Ei weighted by the values {vi (F )}1≤i≤s ,
where Ei = Epi with 1 ≤ i ≤ r are the exceptional components, and Ei with r < i ≤ s
are the non-exceptional irreducible components of F .
Definition 2.1. Two proper ideals a and a′ are equisingular, or belong to the same
equisingularity class if one of the following equivalent conditions hold:
• They have respectively minimal log-resolutions with equal weighted dual graph.
• The divisors F and F ′ associated to their respective minimal log-resolutions are
homeomorphic (topologically equivalent).
• The weighted cluster of base points of a and a′ are similar in the sense of [9, 8.3]
in case a and a′ are m-primary.
Notice that when dealing with principal ideals a = (f ) and a′ = (g) the notion of
equisingularity of the ideals is equivalent to the classical notion of equisingularity (or
topological equivalence) of the germs f = 0 and g = 0. Equisingular ideals have equisingular generic members. Moreover, equisingular m-primary complete ideals have besides
equal codimension.
L
2.2. Divisor basis. Let Λπ := p∈K ZEp be the lattice of integral divisors in X ′ with
exceptional support. We have two different basis of this Z-module given by the strict
transforms and the total transforms of the exceptional components. For simplicity we will
also denote the strict transforms by Ep and the total transforms by Ep . In particular, any
divisor D ∈ Λπ can be presented in two different ways
X
X
D=
vp (D)Ep =
ep (D)E p ,
p∈K

p∈K

where the weights vp (D) (resp. ep (D)) are the values (resp. multiplicities) of D. The
relation between values and multiplicities is given by the proximity relations
X
(2.1)
vq (D) = eq (D) +
vp (D).
q→p

MONOMIAL GENERATORS OF COMPLETE PLANAR IDEALS

5

that provide a base change formula e⊤ = P · v⊤ , where we collect the multiplicities and
values in the vectors e = (ep (D))p∈K and v = (vp (D))p∈K , respectively.
Aside from the total and strict transform basis {Ep }p∈K and {Ep }p∈K of the lattice Λπ
of exceptional divisors, we may also consider the branch basis {Bp }p∈K defined as the dual
of {−Ep }p∈K with respect to the intersection form. Any divisor D ∈ Λπ has a presentation
X
(2.2)
D=
ρp (D)Bp ,
p∈K

where ρp (D) = −D · Ep is the excess at p and the relation between excesses and multiplicities is given by ρ⊤ = P ⊤ e⊤ , where ρ = (ρp (D))p∈K denote the vectors of excesses.
The support
Supp(D) of a divisor D ∈ Λπ is the union of irreducible divisors of D, i.e.
P
if D = α vα (D)Eα is the expression of D in the strict transform basis then Supp(D) =
{Eα | vα (D) 6= 0}. WeP
will also consider the same construction in the total transform
basis. Namely, let D = α eα (D)E α be the expression of D in the total transform basis,
then we define its support in the total transform basis as SuppE (D) = {E α | eα (D) 6= 0}.
To avoid confusion with the usual notion of support, we will always be explicit when
referring to the support in the total transform basis.
Remark 2.2. Let ξ : f = 0 be the germ of curve defined by an element f ∈ OX,O satisfying
that π dominates a resolution of the principal ideal (f ). The total transform of ξ is the
pull-back ξ := π ∗ f and its strict transform ξ ′ is the closure of π −1 (ξ − {O}). Then we
have a presentation
X
X
X
ξ = ξ′ +
vp (f )Ep = ξ ′ +
ep (f )Ep = ξ ′ +
ρp (f )Bp ,
p∈K

p∈K

p∈K

where vp (f ) := vp (D), ep (f ) := ep (D) and ρp (f ) := ρp (D) for D = Div(π ∗ f )exc .
Throughout this work we will be interested in germs of curves associated to branch
basis divisors.
Construction 2.3. Given a point p ∈ K, let fp ∈ OX,O be an irreducible element such
that its strict transform by the resolution π intersects transversely Ep at a smooth point
of E. We have that Bp = Div(π ∗ fp )exc . Conversely, any g ∈ OX,O with Bp = Div(π ∗ g)exc
is irreducible and its strict transform by the resolution π intersects transversely Ep at a
smooth point of E.
Since the elements fp ∈ OX,O are irreducible, the points q ∈ K such that eq (Bp ) 6= 0
are totally ordered. Furthermore, the resolution of any fp ∈ OX,O , p ∈ K is dominated
by π.QMoreover, any f whose resolution is dominated by π can be written as a product
ρ
f = p fp p of suitable elements fp obtained as in Construction 2.3.
Remark 2.4. Let ξ : f = 0, η : g = 0 be germs of curves such that π dominates their
resolutions and set D = Div(π ∗ f )exc and C = Div(π ∗ g)exc . Noether’s intersection formula
[9,
P for their intersection multiplicity at O as [ξ.η]O =
P Theorem 3.3.1] gives
P an expression
)(
e
(D)
·
e
(C)
=
(
e
(D)E
p
p
p ep (C)E p ) = D · C. Hence, vp (f ) = [ξ.ηp ]O = D · Bp ,
p p
p p
where ηp : fp = 0, with fp ∈ OX,O as in Construction 2.3.

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M. ALBERICH-CARRAMIÑANA, J. ÀLVAREZ MONTANER, AND G. BLANCO

P
2.3. Complete ideals and antinef divisors. Given an effective divisor D = p∈K vp Ep
∈ Λπ , we may consider its associated ideal sheaf π∗ OX ′ (−D). Its stalk at O is
(2.3)

HD = {f ∈ OX,O | vp (f ) ≥ vp for all Ep ≤ D},

This ideal HD is complete, see [32], and m-primary since D has only exceptional support.
Complete ideals are closed under all standard operations on ideals, except addition: the
intersection, product and quotient of complete ideals is complete.
Recall that an effective divisor D ∈ Div(X ′ ) is called antinef if ρp = −D · Ep ≥ 0, for
every exceptional component Ep , p ∈ K. This notion is equivalent, in the total transform
basis, to
X
(2.4)
ep (D) ≥
eq (D), for all p ∈ K.
q→p

These are usually called proximity inequalities, see [9, §4.2]. By means of the relation
given in Equation (2.3), Zariski [32] establishes an isomorphism of semigroups between
the set of ideals HD and the set of antinef divisors in Λπ .
e called
Given a non-antinef divisor D, one can compute an equivalent antinef divisor D,
the antinef closure, under the equivalence relation that both divisors define the same
e and via the so called unloading procedure. This is
ideal, i.e. π∗ OX ′ (−D) = π∗ OX ′ (−D),
an inductive procedure which was already described in the work of Enriques [13, IV.II.17]
(see [9, §4.6] for more details). The version that we present here is the one considered in
[5].
Algorithm 2.5. (Unloading procedure [5])
P
Input: A divisor D = dp Ep ∈ Λπ .
e of D.
Output: The antinef closure D

Repeat:
· Define Θ := {Ep ∈ Λπ | ρp = −D · Ep < 0}.
l m
· Let np = Eρp2 for each Ep ∈ Θ. Notice that (D + np Ep ) · Ep ≤ 0.
p
P
e =D+
· Define a new divisor as D
np Ep .
Ep ∈Θ

e is antinef.
Until the resulting divisor D

A fundamental result of Zariski [32] establishes the unique factorization of complete
ideals into simple complete ideals, an ideal being simple if it is not the product of ideals
different from the unit. A reinterpretation of this result in a more geometrical context is
given by Casas-Alvero in [9, §8.4].
P
Theorem 2.6. [32], [9, §8.4] Let D ∈ Λπ an antinef divisor expressed as D = p∈K ρp Bp
in the branch basis. Then,
Y ρ
(2.5)
HD =
HBpp ,
p∈K

with HBp being simple complete ideals for any p ∈ K.
In the sequel, we will call simple divisor the unique antinef divisor defining a simple
complete ideal. As a corollary of Theorem 2.6, simple divisors in Λπ will always be equal
to Bp for some p ∈ K.

MONOMIAL GENERATORS OF COMPLETE PLANAR IDEALS

7

2.4. Maximal contact elements. Let ξ : f = 0 be a germ of an irreducible element
such that π dominates the resolution of (f ). Its equisingularity class, or topological
equivalence class, is determined by the set of characteristic exponents {m1 /n, . . . , mg /n},
where n = eO (f ) introduced in [33]. Then, f has a Puiseux series of the form
X
X
X
X
aj xj/n ,
aj xj/n +
aj xj/n + · · · +
aj xj/n +
(2.6) s(x) =
j∈(ng−1 )
mg−1 ≤j<mg

j∈(n1 )
m1 ≤j<m2

j∈(n0 )
m0 ≤j<m1

j∈(ng )
j≥mg

where m0 = 0, and ni = gcd(n, m1 , . . . , mi ) so that, in particular, n0 = n and ng = 1.
Another way of describing the equisingularity class is using its semigroup (see [34]).
Consider vξ : OX,O → Z the valuation induced by the intersection multiplicity of ξ
with another element ζ : h = 0, h ∈ OX,O , h 6∈ (f ). It is defined by vξ (ζ) = [ξ.ζ]O =
ordt h(tn , s(tn )), and does not depend on the equation defining the germ ζ or the parameterization of ξ. Then, the semigroup of ξ is
Σ(ξ) = {vξ (ζ) ∈ Z | ζ : h = 0, h ∈ OX,O , h 6∈ (f )},
and it is finitely generated. Namely, Σ(ξ) = hn, m̌1 , . . . , m̌g i where
(2.7)

m̌i =

i−1
X
(nj−1 − nj )mj
j=1

ni−1

+ mi ,

for i = 1, . . . , g.

Throughout this work we will be interested in those elements fi ∈ OX,O such that
vξ (γi ) = [ξ.γi ]O = m̌i , with γi : fi = 0 for i = 0, . . . , g. They will be called maximal contact
elements of ξ. Several choices for each fi giving raise to different germs can be made; for
instance, if ξ is not tangent to the y-axis, then f0 = x and f1 = y + a1 x + a2 x2 + · · · .
In general, these elements can be explicitly computed. Namely, if the equation defining
ξ is in Weierstrass form, the maximal contact elements correspond to its approximate
roots (see [22], [23]). Alternatively, if one has a Puiseux series of ξ as in (2.6), then the
equations of the maximal contact elements fi have Puiseux series:
X
X
aj xj/n + · · · +
aj xj/n + · · · for i = 1, . . . g,
(2.8)
si (x) =
j∈(n0 )
m0 ≤j<m1

j∈(ni−1 )
mi−1 ≤j<mi

where the non-explicit terms are assumed not to increase the polydromy order, [9, §1.2],
n/ni−1 of si , and either f0 = x or f0 = y depending on whether ξ is tangent to the x-axis
or the y-axis respectively, see [9, §5.8]. Notice that the multiplicity at the origin of these
maximal contact elements is eO (fi ) = n/ni−1 for i = 1, . . . , g.
We introduce here the notion of maximal contact elements for a proper birational
morphism π : X ′ −→ X. The maximal contact elements of π with exceptional divisor E
are those fp ∈ OX,O with p ∈ K considered in Construction 2.3, such that the vertex p
is a dead-end of the dual graph of Eexc , i.e. the dual graph remains connected when the
vertex is removed. In particular, a dead-end vertex will always be a free point or the origin
O, since satellite points are always proximate to two points. A set of maximal contact
elements {fi }i∈I contains a unique fpi = fi for each dead-end vertex pi ∈ K. Since these
elements are determined by a finite number of valuative conditions, the elements fi can
always be chosen to be polynomials instead of power series. Moreover, these polynomials
can be chosen in an intrinsic unique way which is valid for any π. This will be a key
feature for computational issues.

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M. ALBERICH-CARRAMIÑANA, J. ÀLVAREZ MONTANER, AND G. BLANCO

This definition coincides with the one given for an irreducible element f ∈ OX,O . The
elements fp ∈ OX,O such that p is a dead-end of the dual graph of the resolution of (f ) are
exactly those with Puiseux series as in (2.8). Conversely, we can construct equations for
the maximal contact elements fp with p ∈ K of any divisor F parameterized by K: the
coordinates, as points in the projective line, q ∈ Eq′ ∼
= P1C , q → q ′ , for all q ≤ p, eq (fp ) 6= 0
determine, and are determined by, the coefficients of a Puiseux series of any fp ∈ OX,O ,
see [9, §5.7], which will be as in (2.8).
Remark 2.7. From any set of maximal contact elements {fi } one can always recover the
morphism π : X ′ → X as the minimal resolution dominating the resolutions of {fi }.
Example 2.8. Consider the proper birational morphism π : X ′ → X consisting in
blowing-up four points with the configuration given by the following dual graph:

4

13

p3

26

p4

12

6

O

p2

p1

Take, for instance, D = 4EO + 6Ep1 + 12Ep2 + 13Ep3 + 26Ep4 . The dead-end points are
precisely O, p1, p3 . Therefore, a set of maximal contact elements for π is {f0 , f1 , f2 } with
f0 = x + a2,0 x2 + a0,2 y 2 + a1,1 xy + · · · ,
f1 = y + b2 x2 + b3 x3 + · · · ,
X
f2 = y 2 − x3 +
ci,j xi y j ,
3i+2j>6

and different choices of ai,j , bi , ci,j ∈ C will give different sets of maximal contact elements.
For instance, if all the coefficients are chosen to be zero, f0 = x, f1 = y, f2 = y 2 − x3 ,
which are polynomials.
3. An algorithm to compute HD
Let (X, O) be a germ of smooth complex surface and let π : X ′ −→ X be a proper
birational transform. In this section we present an algorithm which computes a set of
generators for the m-primary ideal HD for any divisor D with exceptional support in X ′ ,
i.e. D ∈ Λπ .
We briefly describe the main ideas behind Algorithm 3.5. We start with divisor D ∈ Λπ
which can be assumed to be antinef. It decomposes into simple divisors D = ρq1 Bq1 +
· · · + ρqr Bqr with all ρqi > 0. For each simple divisor Bqi , qi 6= O, we compute the
b i . This new divisor describes a particular
antinef closure of Bqi + EO which we denote D
HBqi such that dim HBqi /HDb i = 1
adjacent ideal HDb i below HBqi , i.e. an ideal HDb i
as C-vector space. Next, we find, among the set of maximal contact elements of π, an
b i is no longer simple but
element f ∈ OX,O belonging to HBqi but not to HDb i . Now, D
has smaller support than Bqi in the total transform basis. Therefore we may repeat the
same procedure with D := Bqj , j < i until D = BO := Div(π ∗ m).
The first part of the algorithm can be represented by a tree where each vertex is an
antinef divisor. The leaves of the tree are all BO = Div(π ∗ m) and the root is the initial

MONOMIAL GENERATORS OF COMPLETE PLANAR IDEALS

9

divisor D. The second part of the algorithm traverses the tree bottom-up computing, in
each node, the ideal associated to the divisor of that node. Using the notations from the
above paragraph, given any node in the tree with divisor D, the ideal HD is computed
multiplying the ideals in child nodes and adding the element f to the resulting generators.
Before giving a more explicit description of the algorithm, let us first state two technical
results. The first one presents some properties of adjacent ideals based on results obtained
by Fernández-Sánchez (see [14], [15], [16]) in the study of sandwiched singularities and
the Nash conjecture of arcs on these singularities.
Proposition 3.1. Let HD be the complete ideal defined by an antinef divisor D ∈ Λπ .
b the antinef closure of D + EO , obtained from D + EO by unloading on a given1
Consider D
subset of points T ⊆ K. Then, HDb HD are adjacent ideals if and only if ρO (D) = 0.
Furthermore, if HDb HD are adjacent then,
P
P
i) p∈T Ep is the connected component of p∈K, ρp (D)=0 Ep containing EO ;
b = eO (D) + 1, and ep (D) − 1 ≤ ep (D)
b ≤ ep (D) for any p ∈ K, p 6= O;
ii) eO (D)
b > 0;
moreover ρO (D)
b = ep (D) − 1.
iii) if p ∈ K \ T and p is proximate to some point in T then, ep (D)

Proof. Consider the cluster K ′ obtained from K by adding r := ρO (D) + 1 free points
p1 , . . . , pr lying on EO . Let π ′ : Y ′ → X be the composition of π with the sequence of blowups of the points p1 , . . . , pr . Denote by G ∈ Div(Y ′ ) the pullback of any G ∈ Div(X ′ ).
For simplicity, denote the strict and the total transform basis by {Ep }p∈K ′ and {Ep }p∈K ′
respectively in the lattice Λπ′ .

Clearly, both D + EO and D + Ep1 + · · · + Epr are not consistent, whereas D + Ep1 +
· · · + Epi are consistent for all 1 ≤ i < r. Moreover, when applying the unloading
procedure described in Algorithm 2.5, we find that the antinef closures of D + EO and
c′ , and ep (D
c′ ) = 0 for all 1 ≤ i ≤ r. Indeed,
D + Ep1 + · · · + Epr are the same, say it D
i
the first step of the unloading procedure applied to D + EO or D + Ep1 + · · · + Epr gives
c′ is the pullback of the antinef
the same divisor D + E0 + Ep1 + · · · + Epr . Furthermore, D
b of D +EO in Div(X ′ ) and hence they define the same complete ideal H c′ = H b .
closure D
D
D
Now, from [9, §4.7], the codimension of a complete ideal HG defined by a divisor G ∈ Λπ
satisfies
X ep (G)(e
e p (G)
e + 1) X ep (G)(ep (G) + 1)
dim OX,O /HG =
≤
,
2
2
p∈K
p∈K
e is the antinef closure of G. Hence,
where G
HDb = HD+Ep1 +···+Epr

HD+Ep1 +···+Epr−1

...

HD+Ep1

HD

is a chain of adjacent complete ideals, giving dim HBqi /HDb i = r = ρO (D) + 1. Therefore,
HDb HD are adjacent, if and only if ρO (D) = 0.
Finally, from [3, 2.1] and [14, 4.6] claim i) follows. Claim ii) and iii) are consequences
of [15, 4.2] and [16, 2.2] (see also [4, 3.1]).

b − (D + EO ). Notice that T may be
is the set of points p ∈ K that parameterize the support of D
empty.
1T

10

M. ALBERICH-CARRAMIÑANA, J. ÀLVAREZ MONTANER, AND G. BLANCO

Remark 3.2. Although there may be multiple adjacent ideals HDb to a fixed ideal HD , the
b =
adjacent ideal considered in Proposition 3.1 is unique with the property that eO (D)
eO (D) + 1, so we will refer to it as the adjacent ideal to HD . This property turns out to
be crucial for the finiteness of the algorithm.
Remark 3.3. Notice that if D is a simple divisor such that HD 6= m, then ρO (D) = 0
and HDb is always adjacent. Furthermore, the unloading step is always required, i.e. T is
always non empty in this case.
Lemma 3.4. Let HD be the complete ideal defined by an antinef divisor D ∈ Λπ . The
divisor D + E O is antinef and HD+E O = mHD .
Proof. Clearly m = {f ∈ OX,O | eO (f ) ≥ 1}. Thus, m = HE O , and since E O = BO it is
antinef and the result follows by the correspondence between antinef divisor and complete
ideals in section 2.3.

Algorithm 3.5. (Generators for HD )
Input: A proper birational morphism π : X ′ → X and an antinef divisor D ∈ Λπ .
Output: Generators for the ideal HD .
1. Compute and fix a set of maximal contact elements {fi } with i ∈ I of π.
2. Set D (0) := D and proceed from step (0.1).
Step (i):
i.1 Decompose D (i) into di := #{p ∈ K | ρp (D (i) ) > 0} simple divisors.
(i)
i.2 For each
P j = 1, . . . , di , consider qj ∈ {p ∈ K | ρp (D ) > 0} and assume
Bqj = p∈K ep E p .

i.j.1 Stop at the maximal ideal:

If Bqj = BO := Div(π ∗ m), then set HBqj = (fi0 , fi1 ) for i0 , i1 ∈ I such
that they are smooth and transverse at O, then stop. Otherwise, proceed
from i.j.2.
i.j.2 Compute the adjacent ideal to HBqj :
bj .
Perform unloading on the divisor Bqj + EO to get its antinef closure D

i.j.3 Select a maximal contact element in HBqj \ HDb j :

Let p ∈ K be the last free point such that ep 6= 0. Take τj ∈ I such that
ep (fτj ) = 1 and eO (fτj ) ≤ eO . Define the integer nj := eO /eO (fτj ).
i.j.4 Recursive step:
Assume that HDb j has been computed after performing step (i + 1) with
bj .
D (i+1) := D

i.j.5 Set:

 
HBqj = fτnjj + HDb j .

MONOMIAL GENERATORS OF COMPLETE PLANAR IDEALS

11

i.3 Apply Zariski’s factorization theorem:
Qi
ρq
HBqj , giving generators h1 , . . . , hsi .
Compute the product HD(i) = dj=1
j

i.4 Set, using Nakayama’s lemma:



HD(i) = hk π ∗ hk 6∈ OX ′ − D (i) − E O , k = 1, . . . , si OX,O .
3. Return: HD = HD(0) .
Remark 3.6. In order to clarify some steps of the algorithm we point out the following:
· At step 1 of Algorithm 3.5 a set of maximal contact elements {fi }i∈I of π is fixed.
The specific choice of the germs fi = 0 nor of the equations fi do not affect the
output of the algorithm: the monomial expression remains the same for whatever
choice, since the algorithm only uses the information of the equisingularity types of
the maximal contact elements. However, as mentioned in Section 2.4, the set {fi }
can be chosen in an intrinsic unique way, which is convenient for computational
issues.
· At steps (i.j.1) and (i.j.3) of Algorithm 3.5 we have to choose maximal contact
elements. These choices are not necessary unique as several maximal contact
elements may fulfill the required conditions. However, by simply keeping record of
the dead-ends indexing these chosen maximal contact elements, our algorithm is
able to produce the same output whatever the input equations of the ideal are, and
whenever it is used with two different input ideals which have the same integral
closure.
· Since the sheaf ideals OX ′ (−D), with D ∈ Div(X ′ ), are defined by valuations,
testing whether the pullback of an element f belongs to OX ′ (−D) or not is only
a matter of comparing the values vp (Div(π ∗ f )) and vp (D) for all p ∈ K.
· It is clear from Nakayama’s lemma and Lemma 3.4 that a set of elements of OX,O
is a system of generators of HD if and only if its classes modulo HD+E O are a
system of generators of HD /HD+E O as C-vector space. Equivalently, any element
of HD+E O is redundant in a system of generators of HD .
Example 3.7. We will compute HD for the divisor D and the morphism π from Example
2.8. Let us fix the set of maximal contact elements f0 = x, f1 = y, f2 = y 2 − x3 . The steps
of Algorithm 3.5 applied to D = 4EO + 6Ep1 + 12Ep2 + 13Ep3 + 26Ep4 will be illustrated
by means of the tree-shaped graph in Figure 1.
Each vertex of the tree contains an antinef divisor. In this example, we use dual-graphs
to represent them. The root node contains the initial divisor D. Dashed arrows connect
b j from step (i.j.2). The maximal
simple divisors Bqj with its corresponding adjacent D
contact elements from step (i.j.3) that belong to HBqj but not to HDb j are represented
b j =: D (i+1) with each of its irreducible
next to dashed arrows. Solid arrows connect D
(i)
b (i) is
components Bp , with p ∈ K. Finally, the weight ρp of each divisor Bp , p ∈ K, in D
written next to the solid arrows.
We have added some extra indices to the divisors appearing in the algorithm to highlight
at which step we encounter them. Hopefully it does not create any confusion since its

12

M. ALBERICH-CARRAMIÑANA, J. ÀLVAREZ MONTANER, AND G. BLANCO

13
26

p3
p4

4

p2

O

12

6

(y 2 − x3 )2

p1

13
26

5

BO

1

2

D =: D (0) = Bp4

13

7

1

2

b 1(0) =: D (1)
D

Bp1

1

Bp2

2

3

6

4

6

12

1

1

3

2

2

y

6

y2
4

b 1(1) =: D1(2)
D

6

b 2(1) =: D2(2)
D

8

2

4

2

12

3

2

6

3

3
2

BO

1

3

2

BO

4

2

1

1

Figure 1. Tree of divisors from Algorithm 3.5 in Example 3.7.

4

2

1

MONOMIAL GENERATORS OF COMPLETE PLANAR IDEALS

13

meaning should be clear from the context. The generators of the ideals associated to the
divisors in each intermediate step are then:
• HBO = m = (x, y).
• HBp1 = (y) + HD(2) = (y) + m2 = (y, x2 , xy, y 2).
1

2

• HBp2 = (y ) + HD(2) = (y 2 ) + m3 = (y 2, x3 , x2 y, xy 2, y 3 ).
2

• HD(1) =

BO2

· Bp1 · Bp2 = (x, y)2 · (y, x2, xy, y 2) · (y 2, x3 , x2 y, xy 2, y 3)

= (x7 , ✚
x6✚
y, ✟
x5✟
y✟2 , ✟
x4✟
y✟3 , . . . , x5 y, x4y 2 , . . . , ✟
x3✟
y✟3 , ✟
x2✟
y✟4 , x2 y 3, xy 4 , y 5)
= (x7 , x5 y, x4 y 2 , x2 y 3 , xy 4 , y 5).


• Bp4 = (y 2 − x3 )2 + HD(1) = (y 2 − x3 )2 , x7 , x5 y, x4 y 2 , x2 y 3 , xy 4 , y 5 .

• HD := HD(0) = Bp4 = (y 2 − x3 )2 , x7 , x5 y, x4y 2 , x2 y 3 , xy 4 , y✓✓5 .
The crossed out elements are those that are redundant by step (i.4) and Remark 3.6.
Observe that, although many crossed out elements are actually multiple of other elements,
step (i.4) and Remark 3.6 allows us to remove y 5 which is not multiple of any other
element.
Remark 3.8. As an outcome of the algorithm, we see that HD admits the following monomial expression

HD = f22 , f07 , f05 f1 , f04 f12 , f02 f13 , f0 f14 ,

in the set of maximal contact elements f0 = x, f1 = y, f2 = y 2 − x3 associated to π that
we fixed in the beginning. It is worth remarking that we would get this same monomial
expression for any other set of maximal contact elements chosen in the beginning.

However, we might get a different monomial expression depending on the maximal
contact elements (or powers of) that we choose in step (i.j.3) of Algorithm 3.5. In this
example, when choosing an element in HBp3 that does not belong to HDb (1) we took f12 = y 2,
2
but we could also had chosen f2 = y 2 − x3 . In this later case the final system of generators
is

HD = (y 2 − x3 )2 , x2 y(y 2 − x3 ), xy 2(y 2 − x3 ), x7 , x5 y, x4(y 2 − x3 ), x4 y 2 ,
so we get the monomial expression

HD = f22 , f02 f1 f2 , f0 f12 f2 , f07 , f05 f1 , f04 f2 , f04f12 .
4. Correctness of the algorithm
In this section we will prove that Algorithm 3.5 developed in Section 3 is correct. First,
we need to check that it ends after a finite number of steps. The key point is to prove that
b j defining the adjacent ideal to the simple ideal HBq has smaller support in
the divisor D
j
the total transform basis than Bqj .
Lemma 4.1. Using the notations in Algorithm 3.5, assume that Bqj is a simple divisor
b j be the antinef closure of Bq + EO computed in
different from BO = Div(π ∗ m). Let D
j
b
step (i.j.2). Then, Dj has smaller support than Bqj in the total transform basis. That is,
b j )|.
|SuppE (Bqj )| > |SuppE (D

14

M. ALBERICH-CARRAMIÑANA, J. ÀLVAREZ MONTANER, AND G. BLANCO

Proof. Since Bqj =
6 BO , the excess of Bqj at O is ρO (Bqj ) = 0. Then, according to
Proposition 3.1, the ideal defined by Bqj + EO is an adjacent ideal below HBqj . Let
bj
T ⊆ K be the points on which unloading is performed to obtain the antinef closure D
from Bqj + EO . By Proposition 3.1, T are the points p ∈ K whose
P associated exceptional
divisor Ep belongs to the same connected component as EO in O≤p<qj Ep . Observe that
P
0≤p<qj Ep has either one or two components, according if qj is either free or satellite.
In both cases, qj is proximate to the point p ∈ T whose exceptional divisor cuts Eqj , i.e.
b j , after
Ep · Eqj = 1. Hence, invoking Proposition 3.1 again, the multiplicity at qj of D
performing unloading on Bqj + EO , decreases by one. Since Bqj is simple, the multiplicity
b j at qj is zero, giving the desired result. 
of Bqj at qj is one. Hence, the multiplicity of D
In the next proposition we prove that Algorithm 3.5 ends after a finite number of steps.
To emphasize the dependence of the divisors on a specific step (i) of the algorithm we
(i)
b (i) .
will use the notation Bqj and D
j

Proposition 4.2. Algorithm 3.5 ends after a finite number of steps.

Proof. As noted earlier, the points q ∈ K such that eq (Bp ) 6= 0 are totally ordered. Then,
using Equation (2.4), the sequence of multiplicities of Bp decrease along those points.
Hence, we have |Bp |E = 1 for some p ∈ K if and only if p = O and then, Bp = Div(π ∗ m).
Using the notations in Algorithm 3.5, assume that we are in step (i) and we have
a simple divisor Bqj in step (i.j.1). If qj = O, then BO = Div(π ∗ m) and we are
b (i) )| by Lemma
done. Otherwise, since qj 6= O, we have that |SuppE (Bqj )| > |SuppE (D
j
(i+1)
b (i) admits a decomposition D (i+1) = P
4.1. Since D (i+1) := D
ρ
Bp , we have
p
j
p∈K
(i+1)
|SuppE (D
)| ≥ |SuppE (Bp )| for all Bp with ρ(i + 1)p > 0. Hence, |SuppE (Bqj )| >
(i+1)

|SuppE (Bp )|, for all p with ρp

> 0, and by induction we obtain the desired result.



Lemma 4.3. Let Bq be a branch basis divisor associated to a satellite point q ∈ K. Let
Σ(ξ) = hn, m̌1 , . . . , m̌r i be the semigroup of ξ : fq = 0 and take γ : fr = 0 such that
[γ.ξ]O = m̌r . Then, Bq2 = [ζ.ξ]O with ζ : frnr−1 = 0. Furthermore, vp (frnr−1 ) ≥ vp (Bq ), for
all p ≤ q.
P
Proof. Assume that Bq = p∈K ep E p . The first claim follows from the following computation:
r−1
r
X
X
X
2
2
(ni−1 −ni )mi +nr−1 mr = nr−1 m̌r = nr−1 [γ.ξ]O = [ζ.ξ]O .
(mi −mi−1 )ni−1 =
Bq =
ep =
p∈K

i=1

i=1

where the second equality is true since q ∈ K is a satellite point (see [9, §5.10]) and the
fourth equality comes from (2.7).
To prove the second claim, in virtue of Remark 2.4, it suffices to check the inequalities
[ζ.ηp ]O ≥ [ξ.ηp ]O for any p ≤ q with ηp : fp = 0. We will use known properties of the
ultrametric dC distance (introduced in [25]) defined over the space C of plane branches
[C.D]O
1
as dC (C,D)
= eO (C)e
for any C, D ∈ C. Hence, the inequalities above are equivalent to
O (D)
dC (ξ, ηp ) ≥ dC (γ, ηp ) for any p ≤ q, since in our case
[ζ.ηp ]O
ng−1 [γ.ηp ]O
1
=
=
.
eO (ζ)eO (ηp )
ng−1 eO (γ)eO (ηp )
dC (γ, ηp )

MONOMIAL GENERATORS OF COMPLETE PLANAR IDEALS

15

Notice that fq = fqg for some point qg , O ≤ qg ≤ q, which corresponds to a dead-end
in the dual graph of Bq . Now we summarize the results on the ultrametric space of plane
branches of [1, 3.1, 3.2 and 3.4] adapted to our setting:
· dC (ξ, ηp) = dC (γ, ηp ), if O ≤ p < qg ;
· dC (ξ, ηp) = dC (γ, ηp ), if qg < p ≤ q and in the dual graph of Bq the vertex of p lies
on the segment joining the vertexes q and qg ;
· dC (ξ, ηp) > dC (γ, ηp ), otherwise.
Hence, the second claim follows.

Proposition 4.4. Using the notations in Algorithm 3.5, at any step (i) of the algorithm,
n
there exists a power of a maximal contact element fτjj ∈ OX,O as required at step (i.j.3)
and such element belongs to HBqj but not to HDb j .
Proof. We are going to break the proof of the first statement in two cases depending on
whether the point qj ∈ K is free or satellite. With the notations from step (i.j.3), p ∈ K
will be the last free point such that ep (Bqj ) 6= 0.
Assume first that qj is free, i.e. p = qj . If, in addition, the vertex of qj is a dead-end of
the dual graph of F we are done, since fτj = fqj and fqj ∈ HBqj . If qj is not a dead-end of
the dual graph, there is a dead-end q ∈ K and a totally ordered sequence qj ≤ p1 ≤ · · · ≤
pr ≤ q of free points such that eqj (Bq ) = ep1 (Bq ) = · · · = epr (Bq ) = eq (Bq ) = 1, by (2.4).
Therefore, fτj = fq with eqj (fq ) = 1 and Bq = Bp + E p1 + · · · + E pr + E q , which implies,
by (2.1), that fq ∈ HBq HBqj . In either case we have that nj = eO (Bqj )/eO (fτj ) = 1.
Now, assume that qj is satellite and hence p < qj . Let Σ(ξ) = hn, m̌1 , . . . , m̌g i the
semigroup of ξ : fqj = 0. By [9, §5.8], p has the property that any γ : fp = 0, fp ∈ OX,O
satisfies [γ.ξ]O = Bp · Bqj = m̌g . However, it may happen that p ∈ K is not a dead-end.
In this case, using the same argument as before, there is a dead-end q ∈ K and a totally
ordered sequence p ≤ p1 ≤ · · · ≤ pr ≤ q of free points and Bq = Bp + E p1 + · · · + E pr + E q .
Since p is the last free point of Bqj , Bqj · E pi = 0 for i = 1, . . . , r and also Bqj · E q = 0.
Hence, [e
γ .ξ]O = Bq · Bqj = m̌g with γe : fq = 0, i.e. we can take fτj = fq . We can then
n
apply Lemma 4.3 to Bqj with fg = fτj yielding that fτjj ∈ HBqj with nj = eO (Bqj )/eO (fτj ).
n

n

Finally, if fτjj ∈ OX,O fulfills the requirements of step (i.j.3), then eO (fτjj ) = eO (Bqj ),
b j ) > eO (Bq ) by Proposition 3.1, therefore we have that fτnj 6∈ H b .
but eO (D

j
Dj

Theorem 4.5. Let π : X ′ → X be a proper birational morphism and let D ∈ Λπ . Then,
Algorithm 3.5 computes a set of generators for HD that are monomial in any given set of
maximal contact elements of π.
Proof. Let us prove that the i-th step of the algorithm returns a system of generators
of HD(i) which has the desired properties. By Zariski’s factorization Theorem 2.6, it is
enough to focus on computing generators for each simple ideal HBqj , j = 1, . . . , di , in the
decomposition of D (i) . Fixing Bqj at step (i.2), we will make induction on the order of the
neighbourhood that qj ∈ K belongs to, and we will show that Algorithm 3.5 computes
generators for HBqj which are monomials in the set of maximal contact elements.
If qj = O, then BO = Div(π ∗ m) and step (i.j.1) returns HD(i) = m, since a pair of
smooth transverse elements generate m. By construction, any set of maximal contact
elements contain such a pair of elements.

16

M. ALBERICH-CARRAMIÑANA, J. ÀLVAREZ MONTANER, AND G. BLANCO

Assume now that qj 6= O and that the algorithm computes the generators of the ideals
associated to Bp for p < qj . By Proposition 3.1, HDb j
HBqj are adjacent ideals. Since
P
(i)
bj =
D
ρp Bp , we can apply the induction hypothesis to the simple divisors Bp , p < qj

such

p<qj
(i)
that ρp

6= 0 and apply Theorem 2.6 to get
Y ρ(i)
HDb j =
HBpp

HD(i) .

p<qj

At this point is it enough to add any element that belongs to HBqj but not to HDb j to
get a system of generators of HBqj . By Proposition 4.4, the element chosen at step (i.j.3)
has the desired properties, namely, it is a power of a maximal contact element. Finally,
we can remove unnecessary elements from the system of generators of HD(i) using Lemma
3.4.
The dependency on the set of maximal contact elements {fi }i∈I is only used in step
(i.j.3). The conditions required to {fp }i∈I depend only on a finite number of valuations
associated to the exceptional divisors of π. These conditions are fulfilled by an infinite
number of elements which can be part of a set of maximal contact elements.

Remark 4.6. We would like to stress theQgenerality of the monomial generators in Theorem
4.5. Consider the monomials zα = i∈I ziαi , α = (αi )i∈I in the variables zi , i ∈ I.
Each variable zi formally represents all possible elements fp for a fixed dead-end of the
dual graph of π. They all have the same value for the valuations associated to the
exceptional divisors.
Take now any set of maximal contact elements f = {fpi }i∈I and
Q
αi
denote z α
=
f
f
i∈I pi the specialization zi 7→ fpi .
The result of Theorem 4.5 is that Algorithm 3.5 returns formally (z α1 , . . . , z αr ) and
that for any two sets of maximal contact elements f = {fpi }i∈I and g = {gpi }i∈I , both
αr
α1
αr
1
specializations HD,f = (z α
f , . . . , z f ) and HD,g = (z g , . . . , z g ) are equal, HD = HD,f =
HD,g , and are wanted the complete ideal HD .
Corollary 4.7. The set of monomial expressions returned by Algorithm 3.5 is a topological
invariant of D, i.e. it is an invariant of the weighted dual graph of D.
Proof. From the proof of Theorem 4.5 it follows that the algorithm only uses the information of the equisingularity types of the maximal contact elements and of the dual graph
of D weighted by the natural numbers vp (D) for p ∈ K.

5. Application to some families of complete ideals
Let (X, O) be a germ of smooth complex surface and let π : X ′ −→ X be a proper
birational transform. The sheaf ideals OX ′ (D) and its pushforward, for some D ∈ Λπ ,
arise in many different contexts. The goal of this sections is two show how Algorithm 3.5
and Theorem 4.5 applies to different problems.
Our approach is specially useful when studying families of divisor {Di } in Λπ since all
the generators for all the ideals HDi will be given as monomials in any set of maximal
contact elements.
5.1. Integral closure. Let a ⊆ OX,O be an ideal which can be assumed to be m-primary
after considering the decomposition a = (a) · a′ with a = gcd(a). Let π : X ′ → X be a

MONOMIAL GENERATORS OF COMPLETE PLANAR IDEALS

17

log-resolution of the ideal a, i.e., a proper birational morphism such that there exists F
an effective Cartier divisor such that a · OX ′ = OX ′ (−F ). Then, the integral closure a of
a is just the ideal HF , see [9, §8.3].
Therefore, we have a very simple method to compute the integral closure of any planar
ideal that boils down to the following steps:
· Compute the divisor F associated to the log-resolution of a by using the algorithm
from [2].
· Compute a set of generators for the ideal HF using Algorithm 3.5.
Let us illustrate this situation with an small example.
Example 5.1. Let a = ((y 2 − x3 )2 , x2 y 3 ) ⊆ OX,O an ideal. One can compute the logresolution divisor of a using the algorithm from [2]. The log-resolution and its associated
divisor F of a are precisely the proper birational morphism π and the divisor D from
Example 2.8.
Namely, F = 4EO + 6Ep1 + 12Ep2 + 13Ep3 + 26Ep4 and we can take f0 = x, f1 = y, f2 =
y 2 − x3 as a set of maximal contact elements of the log-resolution π. Thus, from the
computation in Example 3.7, one deduces that
a = HF = π∗ OX ′ (F ) = ((y 2 − x3 )2 , x7 , x5 y, x4 y 2, x2 y 3 , xy 4).
The following results follows directly from Theorem 4.5 and its corollaries.
Theorem 5.2. Let a ⊆ OX,O be an ideal. There exists a set of generators of the integral
closure a that are monomial in any given set of maximal contact elements of the logresolution of a.
Corollary 5.3. Let π : X ′ −→X be a proper birational morphism. Any complete ideal
a ⊆ OX,O whose log-resolution is dominated by π admits a system of generators given by
monomials in any set of maximal contact elements associated to π.
Corollary 5.4. The set of monomial expressions returned by Algorithm 3.5 for the integral
closure of an ideal a is an equisingular invariant of a.
5.2. Multiplier ideals. Let π : X ′ −→ X be a log-resolution of an ideal a ⊆ OX and let
′
F be the divisor such that a · OX
= OX ′ (−F ). The multiplier ideal associated to a and
some rational number λ ∈ Q≥0 is defined as
J (aλ ) = π∗ OX ′ (⌈Kπ − λF ⌉),
where Kπ is the so-called relative canonical divisor (see [20] for details). Multiplier ideals
form a discrete nested sequence of ideals
OX ) J (aλ0 ) ) J (aλ1 ) ) J (aλ2 ) ) · · · ) J (aλi ) ) · · ·
and the rational numbers 0 < λ0 < λ1 < · · · where an strict inclusion of ideals is achived
are called the jumping numbers associated to a.
There are general algorithms as those developed by Shibuta [26] and Berkersch and
Leykin [6] that, given a set of generators of a, return the list of jumping numbers and a
minimal set of generators of the corresponding multiplier ideals. These algorithms use the
theory of Bernstein-Sato polynomials and have been implemented in Macaulay2. However,
it is difficult to compute large examples due to the complexity of these algorithms.

18

M. ALBERICH-CARRAMIÑANA, J. ÀLVAREZ MONTANER, AND G. BLANCO

In the case of planar ideals, there are methods given by Järviletho [19], Naie [24]
and Tucker [29] to compute the set of jumping numbers. The first two authors of this
manuscript and Dachs-Cadefau [5] gave an algorithm that computes sequentially the list of
jumping numbers of a planar ideal and the antinef divisor associated to the corresponding
multiplier ideal.
Combining the algorithms developed in [2] and [5] with Algorithm 3.5 we may provide
a method that, given a set of generators of a planar ideal a, returns the set of jumping
numbers and a set of generators of the corresponding multiplier ideals. Namely, we have
to perform the following steps:
· Compute the divisor F associated to the log-resolution of a by using the algorithm
from [2].
· Compute the sequence of jumping numbers {λj }j∈Z≥0 and the divisor corresponding to
the associated multiplier ideals {J (aλj )}j∈Z≥0 , i.e. the antinef closures Dλj of ⌊λj F −Kπ ⌋,
using the main algorithm of [5].
· Compute a set of generators for the ideals HDλj using Algorithm 3.5.
This method is illustrated with the following:
Example 5.5. Consider the ideal a = ((y 2 − x3 )3 , x3 (y 2 − x3 )2 , x6 y 3 ) ⊆ OX,O . The logresolution of a can be computed using the algorithm from [2] and is represented by means
of the following dual-graph:

6

21

p4

42

p5

20

p3

18

9

O

p2

p1

′
The divisor F such that a·OX
= OX ′ (−F ) is F = 6EO +9E1 +18E2 +20E3 +21E4 +42E5 .
A set of maximal contact elements for the log-resolution of a is, for instance, f0 = x, f1 =
y, f2 = y 2 − x3 .

The jumping numbers smaller than 1 computed using the algorithm from [5] and generators for the associated multiplier ideals computed using Algorithm 3.5 can be found in
Table 1.
It is a known result that the multiplier ideals J (aλ ) associated to a are the same when
taking the completion a of a, i.e. J (aλ ) = J (aλ ). As a corollary of Theorem 4.5 and
Corollary 5.3 we obtain the following result which resembles Howald’s theorem [18] on
the fact that multiplier ideals of monomial ideals are also monomial.
Theorem 5.6. Let a ⊆ OX,O be an ideal and consider its completion a, that can be generated by monomials in any given set of maximal contact elements. Then, the multiplier
ideals J (aλ ) are also generated by monomials in the same set of maximal contact elements
of the log-resolution of a.

MONOMIAL GENERATORS OF COMPLETE PLANAR IDEALS

λi

J (aλi )

5
18
7
18
4
9
1
2
23
42
25
42
11
18
9
14
29
42
13
18
31
42
7
9
11
14
5
6
37
42
8
9
13
14
17
18
41
42

x, y

19

y, x2
x2 , xy, y 2
xy, y 2, x3
y 2, x3 , x2 y
y 2 − x3 , x2 y, xy 2, x4
x2 y, xy 2, y 3, x4
xy 2 , y 3, x4 , x3 y
x(y 2 − x3 ), y(y 2 − x3 ), x3 y, x2 y 2 , x5
y 3 , x3 y, x2 y 2 , x5
y(y 2 − x3 ), x2 y 2, xy 3 , x2 (y 2 − x3 ), x4 y
x2 y 2 , xy 3 , y 4, x5 , x4 y
x2 (y 2 − x3 ), xy(y 2 − x3 ), y 2(y 2 − x3 ), x4 y, x3 y 2 , x6
xy(y 2 − x3 ), y 2(y 2 − x3 ), x3 (y 2 − x3 ), x3 y 2 , x2 y 3 , x5 y
xy(y 2 − x3 ), y 2(y 2 − x3 ), x3 (y 2 − x3 ), x2 y 3 , xy 4, x5 y, x4 y 2, x7
y 2 (y 2 − x3 ), x5 y, x3 (y 2 − x3 ), x2 y 3, x2 y(y 2 − x3 ), xy 4 , x4 y 2 , x7
y 2 (y 2 − x3 ), x3 (y 2 − x3 ), x2 y(y 2 − x3 ), xy 4 , y 5, x4 y 2, x3 y 3, x7 , x6 y
x2 y(y 2 − x3 ), xy 2 (y 2 − x3 ), y 3(y 2 − x3 ), x4 y 2, x4 (y 2 − x3 ), x3 y 3 , x6 y
x2 y(y 2 − x3 ), xy 2 (y 2 − x3 ), y 3(y 2 − x3 ), x4 (y 2 − x3 ), x3 y 3 , x2 y 4, x6 y, x5y 2 , x8
Table 1. The jumping numbers smaller than 1 and generators of the associated multiplier ideal for a = ((y 2 − x3 )3 , x3 (y 2 − x3 )2 , x6 y 3 ).

5.3. Valuation filtration. Throughout this subsection we will use the notation from
Section 2.4. Let ξ : f = 0, f ∈ OX,O be a plane branch and let vξ be the valuation
induced by the intersection multiplicity. Let Vi denote the ideal of all the elements in
OX,O with valuation greater or equal to i:
Vi := {ζ ∈ OX,O | vξ (ζ) ≥ i}.
These ideals form a filtration in OX,O :
(5.1)

OX,O = V0 ⊇ V1 ⊇ · · · ⊇ Vi ⊇ Vi+1 ⊇ · · ·

such that Vi · Vj ⊂ Vi+j and ∩i∈Z≥0 Vi = (f ). This type of filtration was considered by
Teissier in [28]. Since, the ideals Vi are defined by valuations, they are complete, and
hence, have the form π∗ OXi′ (Di ) for some effective divisor Di in some surface Xi′ .
Example 5.7. Consider the germ of plane curve ξ : f = 0, f = (y 2 − x3 )2 − x5 y ∈ OX,O .
A resolution morphism π : X ′ −→ X of ξ is the same as the one given for the ideal in

20

M. ALBERICH-CARRAMIÑANA, J. ÀLVAREZ MONTANER, AND G. BLANCO

Example 2.8. Using Algorithm 3.5 we obtain the generators of the filtration (5.1) until
the last Vi dominated by the log-resolution of (f ), which we collect in Table 2.
i

Vi

1, 2, 3, 4

x, y

5, 6

y, x2

7, 8

xy, x2 , y 2

9, 10

xy, y 2, x3

11, 12

x2 y, y 2, x3

13

y 2 − x3 , x2 y, xy 2, x4

14

x2 y, xy 2, y 3, x4

15, 16

xy 2 , y 3, x4 , x3 y

17

x(y 2 − x3 ), y(y 2 − x3 ), x3 y, x2 y 2, x5

18

y 3 , x3 y, x2 y 2 , x5

19

y(y 2 − x3 ), x2 (y 2 − x3 ), x4 y, x2 y 2 , xy 3

20

x4 y, x2 y 2, xy 3 , y 4, x5

21

x2 (y 2 − x3 ), xy(y 2 − x3 ), y 2 (y 2 − x3 ), x4 y, x3 y 2 , x6

22

xy 3 , y 4 , x4 y, x3 y 2, x6

23

xy(y 2 − x3 ), y 2(y 2 − x3 ), x3 y 2 , x3 (y 2 − x3 ), x2 y 3 , x5 y

24

y 4, x3 y 2, x2 y 3, x6 , x5 y

25

(y 2 − x3 )2 , x3 (y 2 − x3 ), x2 y(y 2 − x3 ), x5 y, x4 y 2 , x7

26

(y 2 − x3 )2 , x2 y 3, xy 4 , x5 y, x4y 2 , x7

Table 2. The ideals Vi of the filtration associated to the plane branch
f = (y 2 − x3 )2 − x5 y for i = 1, 2, . . . , 26.

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[34] O. Zariski, Le problème des modules pour les branches planes. Hermann, Paris, 1986. 7, 21
Departament de Matemàtiques, Univ. Politècnica de Catalunya, Av. Diagonal 647,
Barcelona 08028, Spain
E-mail address: Maria.Alberich@upc.edu, Josep.Alvarez@upc.edu, Guillem.Blanco@upc.edu

