# Multiplier ideals of meromorphic functions in dimension two

Maria Alberich-Carramiñana  Departament de Matemàtiques and Institut de
Matemàtiques de la UPC-BarcelonaTech (IMTech)
Universitat Politècnica de Catalunya
Av. Diagonal 647, Barcelona 08028; and Institut de Robòtica i Informàtica
Industrial
CSIC-UPC
Llorens i Artigues 4-6, Barcelona 08028, Spain<EMAIL_ADDRESS>,
Josep Àlvarez Montaner  Departament de Matemàtiques and Institut de
Matemàtiques de la UPC-BarcelonaTech (IMTech)
Universitat Politècnica de Catalunya
Av. Diagonal 647, Barcelona 08028; and Centre de Recerca Matemàtica (CRM),
Bellaterra, Barcelona 08193<EMAIL_ADDRESS>and  Roger Gómez-López
Departament de Matemàtiques
Universitat Politècnica de Catalunya
Av. Diagonal 647, Barcelona 08028<EMAIL_ADDRESS>
###### Abstract.

We provide an effective method to compute multiplier ideals of meromorphic
functions in dimension two. We also prove that meromorphic functions only have
integer jumping numbers after reaching some threshold.

###### Key words and phrases:

Multiplier ideals, jumping numbers, meromorphic functions.

###### 2000 Mathematics Subject Classification:

Primary 13D45, 13N10

All three authors are partially supported by grant PID2019-103849GB-I00 funded
by MICIU/AEI/ 10.13039/501100011033 and AGAUR grant 2021 SGR 00603. JAM is
also supported by Spanish State Research Agency, through the Severo Ochoa and
María de Maeztu Program for Centers and Units of Excellence in R$\&$D (project
CEX2020-001084-M). RGL gratefully acknowledges Secretaria d’Universitats i
Recerca del Departament d’Empresa i Coneixement de la Generalitat de Catalunya
and Fons Social Europeu Plus for the financial support of his FI Joan Oró
predoctoral grant.

##  1\. Introduction

Let $X$ be a $n$-dimensional complex smooth algebraic variety and
${\mathcal{O}}_{X,O}$ the local ring of a point $O\in X$. Let
$f,g:(X,0)\longrightarrow(\mathbb{C},0)$ be two germs of holomorphic functions
and consider the germ of meromorphic function
$f/g:(X,0)\longrightarrow(\mathbb{C},0).$ Taking local coordinates we assume
$f,g\in{\mathcal{O}}_{X,O}=\mathbb{C}\\{x_{1},\dots,x_{n}\\}$. Notice also
that $f/g$ and $f^{\prime}/g^{\prime}$ define the same germ if
$fg^{\prime}=f^{\prime}g$.

In order to study the singularities of the meromorphic germ $f/g$, the authors
in [AGLN21] introduced a theory of multiplier ideals mimicking the classical
one for holomorphic functions (see also [Tak23]). However this variant for
meromorphic functions has several features that do not behave quite as in the
classical way.

The aim of this note is to provide an effective method to compute multiplier
ideals of meromorphic functions in dimension two. In Section 3 we present an
algorithm to compute sequentially the set of jumping numbers and the
corresponding multiplier ideals. This algorithm is a mild modification of the
one given by the first two authors together with Dachs-Cadefau in [AAD16]
combined with the methods developed in [AAB19, AAB21]. For completeness we
will describe the differences between both cases.

In Section 4 we pay attention to the version of Skoda’s theorem for
meromorphic functions considered in [AGLN21] which provides some interesting
phenomena. For instance, we prove in Theorem 4.4 that after reaching some
threshold the only jumping numbers of $f/g$ are integer numbers.

In Section 5 we present some examples that illustrate the effectiveness of our
methods and the properties proved in Theorem 4.4. In particular, the examples
show how the jumping numbers of $f/g$ depend on the contact between the two
plane curves $f$ and $g$.

##  2\. Multiplier ideals of meromorphic functions

Let $X$ be a $n$-dimensional complex smooth algebraic variety and let $U$ be a
neighbourhood of $O\in X$. Let $\pi:Y\longrightarrow U$ be a log resolution of
a meromorphic germ $f/g:(X,0)\longrightarrow(\mathbb{C},0).$ Namely, $\pi$ is
a proper birational map such that:

  * •

$\pi$ is a log resolution of the hypersurface
$H=\\{f_{|_{U}}=0\\}\cup\\{g_{|_{U}}=0\\}$, i.e., $\pi$ is an isomorphism
outside a proper analytic subspace in $U$;

  * •

there is a normal crossing divisor $F$ on $Y$ such that
$\pi^{-1}(H)=\mathcal{O}_{Y}(-F)$;

  * •

the lifting $\tilde{f}/\tilde{g}=(f/g)\circ\pi=\frac{f\circ\pi}{g\circ\pi}$
defines a holomorphic map $\tilde{f}/\tilde{g}:Y\to\mathbb{P}^{1}$.

Let $\\{E_{i}\\}_{i\in I}$ be the irreducible components of $F$. Some of the
ingredients that we will use in the sequel are the relative canonical divisor,
defined by the Jacobian determinant of $\pi$, that we denote

$K_{\pi}=\sum k_{i}E_{i},$

and the total transforms of $f$ and $g$ that we denote by

$\tilde{f}=\pi^{*}f=\sum N_{f,i}E_{i},\quad\quad\tilde{g}=\pi^{*}g=\sum
N_{g,i}E_{i}.$

We point out that these divisors can be decomposed into their exceptional and
affine parts, depending on whether the divisor $E_{i}$ has exceptional support
or it corresponds to the components of the strict transform, which is the
closure of $\pi^{-1}(C-\\{O\\})$ with $C$ being the hypersurface defined by
$f=0$ or $g=0$ respectively. We call $v_{i}(f):=N_{f,i}$ the value of $f$ at
$E_{i}$.

Associated to the meromorphic germ $f/g$ we consider the divisor

$\tilde{F}=\sum_{i\in I}N_{i}E_{i},$

with $N_{i}:=N_{f,i}-N_{g,i}.$ We define

$\tilde{F}_{0}=\sum_{N_{i}>0}N_{i}E_{i},\quad\tilde{F}_{\infty}=\sum_{N_{i}<0}N_{i}E_{i}.$

######  Definition 2.1.

Let $\pi:Y\longrightarrow U$ be a log resolution of a non-constant meromorphic
germ $f/g:(X,0)\longrightarrow(\mathbb{C},0).$ We define the $0$-multiplier
ideal of $f/g$ at $\lambda$ as the stalk at the origin of

${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda}\right)=\pi_{*}\mathcal{O}_{Y}(\lceil
K_{\pi}-\lambda\cdot\tilde{F}_{0}\rceil),$

where $\lceil\cdot\rceil$ denotes the upper integer part of a real number or
$\mathbb{R}$-divisor. If no confusion arises we denote the stalk at the origin
in the same way and thus
${\mathcal{J}}((\frac{f}{g})^{\lambda})\subseteq{\mathcal{O}}_{X,O}.$

A useful remark mentioned in [AGLN21] shows that we can describe the
meromorphic multiplier ideal from the theory of mixed multiplier ideals
${\mathcal{J}}(f^{\lambda_{1}}g^{\lambda_{2}})$ associated to the pair of
functions. Namely, pick $t\in\mathbb{N}$ such that $t\geq\lambda$. Then,

${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda}\right)={\mathcal{J}}\left(f^{\lambda}g^{t-\lambda}\right):g^{t}=\left\\{h\in{\mathcal{O}}_{X,O}\;|\;hg^{t}\in{\mathcal{J}}\left(f^{\lambda}g^{t-\lambda}\right)\right\\}.$

######  Remark 2.2.

Analogously, we define the $\infty$-multiplier ideal of $f/g$ at $\lambda$ as
the stalk at the origin of

${\mathcal{J}}^{\infty}\left(\left(\frac{f}{g}\right)^{\lambda}\right)=\pi_{*}\mathcal{O}_{Y}(\lceil
K_{\pi}+\lambda\cdot\tilde{F}_{\infty}\rceil)={\mathcal{J}}\left(\left(\frac{g}{f}\right)^{\lambda}\right).$

Many of the properties of the classical multiplier ideals are still satisfied
in the meromorphic case. For instance, the definition is independent of the
chosen log resolution $\pi$ and there exists a discrete strictly increasing
sequence of rational numbers $\lambda_{1}<\lambda_{2}<\cdots$ such that

${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda_{i+1}}\right)\subsetneq{\mathcal{J}}\left(\left(\frac{f}{g}\right)^{c}\right)={\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda_{i}}\right)$

for $c\in[\lambda_{i},\lambda_{i+1})$, and all $i$. These rational numbers are
called the $0$-jumping numbers of the meromorphic function $f/g$. If no
confusion may arise, we shall refer to them and to their corresponding
$0$-multiplier ideals simply as jumping numbers and multiplier ideals.

##  3\. Explicit computation in dimension two

The classical case of multiplier ideals of holomorphic functions has been
extensively studied when the complex smooth algebraic variety $X$ has
dimension two. For simple complete ideals or irreducible plane curves,
Järviletho [Jär11] and Naie [Nai09] provided a closed formula for the set of
jumping numbers. For the case of any ideal or any plane curve we must refer to
the work of Tucker [Tuc10] or the papers by the first two authors with Dachs-
Cadefau and Alonso-González [AAD16, AADG17]. Hyry and Järviletho [HJ11, HJ18]
also worked on this general setting in dimension two.

The method presented in [AAD16] is an algorithm that computes sequentially at
each step a jumping number and its associated multiplier ideal and thus it
provides the ordered sequence of multiplier ideals in any desired range of the
real line. Combined with the effective methods developed by the first two
authors with Blanco [AAB19, AAB21] one may start the algorithm with the
equation of a plane curve or generators of any ideal and get a set of
generators of each multiplier ideal as an output (see also [BD18]). In this
section we will adjust these methods to the case of meromorphic functions. We
present first the algorithm and then we will explain each step with some
detail highlighting the main differences with the classical case.

######  Algorithm 3.1 (Jumping Numbers and Multiplier Ideals).

Input: A pair of holomorphic functions $f,g\in\mathcal{O}_{X,O}$.

Output: List of Jumping Numbers $\lambda_{1}<\lambda_{2}<\dots$ of $f/g$ and a
set of generators of their corresponding multiplier ideals.

  * (Step 1)

Compute the minimal log resolution of $f/g$ and set
$\tilde{F}_{0}=\sum_{N_{i}>0}N_{i}E_{i}$.

  * (Step 2)

Set $\lambda_{0}=0$, $e_{i}^{\lambda_{0}}=0$. From $j=1$, incrementing by $1$

    1. (Step 2.$j$)

      1. $\cdot$

Jumping number: Compute

$\lambda_{j}=\min_{N_{i}>0}\left\\{\frac{k_{i}+1+e_{i}^{\lambda_{j-1}}}{N_{i}}\right\\}.$

      2. $\cdot$

Multiplier ideal: Compute the antinef closure $D_{\lambda_{j}}=\sum
e_{i}^{\lambda_{j}}E_{i}$ of $\lfloor\lambda_{j}\tilde{F}_{0}-K_{\pi}\rfloor$
using the unloading procedure.
${\mathcal{J}}((\frac{f}{g})^{\lambda_{j}})=\pi_{*}\mathcal{O}_{Y}(-D_{\lambda_{j}})$.

      3. $\cdot$

Generators: Give a system of generators of the complete ideal associated to
$D_{\lambda_{j}}$.

###  3.1. Computing the minimal log resolution of meromorphic functions

Computing the log resolution of reduced plane curves is an already well-known
procedure see, for instance [CA00]. To obtain a log resolution of $f/g$, one
can begin with a log resolution of the curve defined by $f\cdot g$ and then
perform additional blow-ups to ensure a log resolution of the generic fibers
of $f/g$, which is achieved by blowing up the base points of the ideal
$(f,g)$. These latter blow-ups also guarantee that each dicritical component
$E$ of $f/g$ has appeared, i.e. an exceptional divisor $E$ for which the
restricted map $(\tilde{f}/\tilde{g})_{|_{E}}:Y\to\mathbb{P}^{1}$ is
surjective (non-constant). Notice that if $E_{i}$ is a dicritical component
then $N_{i}=0$.

In [AC04], an algorithmic procedure was described to compute the base points
of $(f,g)$; see also [AAB19]. To compute the log resolution of $f/g$, we may
adapt Algorithm 4.6 from [AAB19], which is more convenient since it begins
with the log resolution of $f\cdot g$, by omitting the last part (steps v to
viii). The finiteness and correctness of this procedure are proved there.

######  Algorithm 3.2 (Minimal log resolution of $f/g$).

Input: A pair of holomorphic functions $f,g\in\mathbb{C}\\{x,y\\}$.

Output: Divisor $F=\sum_{i\in I}E_{i}$ of the minimal log resolution of $f/g$
described by means of the proximity matrix, and the values
$(N_{f,i},N_{g,i})_{i\in I}$ of $f$ and $g$.

  1. (1)

Find $b=\gcd(f,g)$ and set $f=ba_{1}$, $g=ba_{2}$. Compute $h=ba_{1}a_{2}$.

  2. (2)

Find the exceptional divisor $F^{\prime}=\sum_{i\in I^{\prime}}E_{i}$ of the
minimal log resolution of the reduced germ $\xi_{\textrm{red}}$, where
$\xi:h=0$, and the values $v_{i}(a_{k})$ and $v_{i}(b)$ at each $E_{i}$.
Compute $v_{i}=\min_{i}\\{v_{i}(a_{1}),v_{i}(a_{2})\\}$ for $i\in I^{\prime}$.

  3. (3)

Define $F^{\prime\prime}=\sum_{i\in I^{\prime\prime}}E_{i}\geq F^{\prime}$ by
adding, if necessary, exceptional components of blow-ups of free points using
the following criterion and set, for each new $E_{i}$,
$v_{i}=\min\\{v_{i}(a_{1}),v_{i}(a_{2})\\}$.

Repeat for $k\in\\{1,2\\}$:

    1. $\cdot$

Blow-up the free point $q\in E_{i}$ on the strict transform of $a_{k}$,

while $v_{i}(a_{k})=v_{i}$.

  4. (4)

Define $F=\sum_{i\in I}E_{i}\geq F^{\prime\prime}$ by adding, if necessary,
exceptional components of blow-ups of satellite points using the following
criterion and set, for each new $E_{i}$,
$v_{i}=\min\\{v_{i}(a_{1}),v_{i}(a_{2})\\}$.

Repeat:

    1. $\cdot$

Blow-up the satellite point $q\in E_{i}\cap E_{j}$,

while $v_{i}(a_{1})>v_{i}$ and $v_{j}(a_{2})>v_{j}$.

  5. (5)

Return $F$ and
$\left(N_{f,i}=v_{i}(b)+v_{i}(a_{1}),N_{g,i}=v_{i}(b)+v_{i}(a_{2})\right)_{i\in
I}$.

Observe that the strategy of performing further blow-ups from the log
resolution of $f\cdot g$ until the irreducible components of the strict
transform in $f$ and $g$ are separated by a dicritical component is rather
vague. In contrast, Algorithm 3.2 determines precisely the minimal blow-ups
needed to achieve that.

###  3.2. Computing the integral closure of an ideal

Let $\pi:Y\longrightarrow X$ be a proper birational morphism obtained after a
sequence of point blowings-up. An effective divisor with integral coefficients
$D\in{\rm Div}(Y)$ is called _antinef_ if $-D\cdot E_{i}\geqslant 0$, for
every exceptional prime divisor $E_{i}$. Lipman [Lip69] established a one to
one correspondence between antinef divisors in ${\rm Div}(Y)$ and complete
ideals in $\mathcal{O}_{X,O}$. Namely, given an effective divisor $D=\sum
d_{i}E_{i}\in\textrm{Div}_{\mathbb{Q}}(Y)$, we may consider its associated
sheaf ideal $\pi_{*}\mathcal{O}_{X^{\prime}}(-D)$ whose stalk at $O$ is

$H_{D}=\\{h\in\mathcal{O}_{X,O}\ |\ v_{i}(h)\geqslant\lceil d_{i}\rceil\
\textrm{for all}\ E_{i}\leqslant D\\},$

where $v_{i}(h)$ is the value of $h$ at $E_{i}$. These ideals are _complete_ ,
see [Zar38], and $\mathfrak{m}$-primary whenever $D$ has exceptional support.

The divisors $\lfloor\lambda_{j}\tilde{F}_{0}-K_{\pi}\rfloor$ used in the
definition of multiplier ideals are not antinef. Given a non-antinef divisor
$D$, one can compute an antinef divisor defining the same ideal, called the
antinef closure, via the so called unloading procedure (see [AAD16, §2.2] or
[CA00, §4.6]). The unloading procedure incrementally expands the exceptional
part $D_{\rm exc}$ of $D$ to obtain the minimal antinef divisor $D^{\prime}$
containing $D$. This is one of the key ingredients of the algorithm in
[AAD16]. The finiteness and correctness of the unloading procedure it is
proved there.

######  Algorithm 3.3 (Unloading procedure).

Input: A divisor $D$.

Output: The divisor $D^{\prime}$ that is the antinef closure of $D$.

Repeat

  1. $\cdot$

Define $\Theta:=\\{E_{i}\leqslant D_{\rm exc}\hskip 5.69054pt|\hskip
5.69054pt\rho_{i}=-\lceil D\rceil\cdot E_{i}<0\\}.$

  2. $\cdot$

Let $n_{i}=\left\lceil\frac{\rho_{i}}{E_{i}^{2}}\right\rceil$ for
$i\in\Theta$. Notice that $(\lceil D\rceil+n_{i}E_{i})\cdot E_{i}\leqslant 0$.

  3. $\cdot$

Define a new divisor as $D^{\prime}=\lceil
D\rceil+\sum_{E_{i}\in\Theta}n_{i}E_{i}$.

Until the resulting divisor $D^{\prime}$ is antinef.

###  3.3. Computing the generators of an ideal associated to an antinef
divisor

Once we have the antinef divisor that describes the corresponding multiplier
ideal of the meromorphic function we may use the algorithm developed in
[AAB21] to compute a set of generators of the multiplier ideal. This step of
our method does not require any adjustment for the case of meromorphic
functions and we briefly recall it here.

First, start with a divisor $D$, which we assume to be antinef. The divisor
$D$ is decomposed into simple divisors $D_{1},\dots,D_{r}$ by using Zariski’s
decomposition theorem [Zar38, CA00]. This decomposition is precisely

$D=\sum\rho_{i}D_{i},\quad\textrm{where}\quad\rho_{i}=-D\cdot E_{i}\geqslant
0,$

and $H_{D_{i}}$ is a simple ideal appearing in the factorization of $H_{D}$
with multiplicity $\rho_{i}$.

Now, for each simple divisor $D_{i}$, compute $D^{\prime}_{i}$ an antinef
divisor defining an adjacent ideal $H_{D^{\prime}_{i}}\subset H_{D_{i}}$, such
that $D^{\prime}_{i}$ is the antinef closure of $D_{i}+E_{O}$ ($E_{O}$ is the
exceptional divisor obtained blowing-up the origin $O$). Next, find an element
$f\in\mathcal{O}_{X,O}$ belonging to $H_{D_{i}}$ but not to
$H_{D^{\prime}_{i}}$. Now, $D^{\prime}_{i}$ is no longer simple but has
smaller support than $D_{i}$. This part is repeated with $D:=D^{\prime}_{i}$
until $H_{D}$ is the maximal ideal $\mathfrak{m}\subseteq\mathcal{O}_{X,O}$.

This first part of the algorithm generates a tree where each vertex is an
antinef divisor and where the leafs of the tree are all $\mathfrak{m}$. The
second part traverses the tree bottom-up computing in each node the ideal
associated to the divisor. Using the notations from the above paragraph, given
any node in the tree with divisor $D$, the ideal $H_{D}$ is computed
multiplying the ideals in child nodes $H_{D^{\prime}_{1}}\cdots
H_{D^{\prime}_{r}}$ and adding the element $f$ to the resulting generators.

##  4\. Skoda’s theorem version for meromorphic functions

Let $f\in{\mathcal{O}}_{X,O}=\mathbb{C}\\{x_{1},\dots,x_{n}\\}$ be an
holomorphic function and consider the classical multiplier ideals
${\mathcal{J}}(f^{\lambda})$. A version of Skoda’s theorem for hypersurfaces
[Laz04, §9] states that

${\mathcal{J}}\left({f}^{\lambda+\ell}\right)={f^{\ell}}{\mathcal{J}}\left(f^{\lambda}\right)$

for every $\ell\in\mathbb{N}.$ This implies a peridiocity of the set of
jumping numbers, indeed any jumping number is of the form $\lambda+\ell$ with
$\lambda$ being a jumping number in the interval $[0,1]$ and
$\ell\in\mathbb{N}.$

For the case of meromorphic functions we only have a weaker version of Skoda’s
theorem.

######  Proposition 4.1.

[AGLN21, Proposition 6.5] Let $f,g\in{\mathcal{O}}_{X,O}$ be a nonzero
elements and $\lambda\in\mathbb{R}_{\geq 0}$. Then,

${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda+\ell}\right)=\left(\frac{f^{\ell}}{g^{\ell}}\;{\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda}\right)\right)\cap{\mathcal{O}}_{X,O}=f^{\ell}\left({\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\lambda}\right):g^{\ell}\right)$

for every $\ell\in\mathbb{N}.$ In particular, if $\lambda+1$ is a jumping
number, then $\lambda$ is a jumping number.

We also have the following property.

######  Lemma 4.2.

[AGLN21, Lemma 7.5] For any $\lambda\in\mathbb{R}_{>0}$ we have
${\mathcal{J}}(f^{\lambda})\subseteq{\mathcal{J}}((\frac{f}{g})^{\lambda})$.
In addition, if $f$ and $g$ have no common factors, it holds
${\mathcal{J}}((\frac{f}{g})^{n})=(f^{n})$ for any $n\in\mathbb{Z}_{>0}$.

Additionally,

######  Lemma 4.3.

Let $f,g\in\mathbb{C}\\{x,y\\}$. If $f$ vanishes at the origin and has no
common factor with $g$, then $\mathbb{Z}_{>0}$ are jumping numbers of the
meromorphic function $f/g$.

###### Proof.

Let $E_{i}$ be an affine component of $f$. Then $N_{i}>0$, $k_{i}=0$, so
$\lfloor\lambda_{j}N_{i}-k_{i}\rfloor=\lfloor\lambda_{j}\rfloor
N_{i}=e_{i}^{\lambda_{j}}$, where the last equality is due to the unloading
procedure not modifying the multiplicity of affine components. Following
Algorithm 3.1, if $\lambda_{j}<n\in\mathbb{Z}_{>0}$ then
$e_{i}^{\lambda_{j}}=\lfloor\lambda_{j}\rfloor N_{i}\neq
nN_{i}=e_{i}^{n}\implies D_{\lambda_{j}}\neq D_{n}$. Therefore, $n$ is a
jumping number. ∎

These results lead to some behaviours of the set of jumping numbers that
contrast with the classical case. For the rest of this section we will only
consider the case of dimension two, that is
${\mathcal{O}}_{X,O}=\mathbb{C}\\{x,y\\}$, and we will use Algorithm 3.1 to
compute consecutive jumping numbers. In the sequel, we fix the following
notation:

Let $\pi:Y\longrightarrow U$ be a log resolution of the meromorphic function
$f/g$ and consider the total transforms of $f$ and $g$.

$\tilde{f}=\pi^{*}f=\sum N_{f,i}E_{i},\quad\quad\tilde{g}=\pi^{*}g=\sum
N_{g,i}E_{i}.$

Consider also the relative canonical divisor $K_{\pi}=\sum k_{i}E_{i}$.

######  Theorem 4.4.

Let $f,g\in\mathbb{C}\\{x,y\\}$ be reduced holomorphic functions vanishing at
the origin with no common factor. Then, there exist $n\in\mathbb{Z}_{>0}$ such
that the only jumping numbers of the meromorphic function $f/g$ larger than
$n$ are the integer numbers.

###### Proof.

Since $f$ is reduced, for any affine component $E_{i}$ of $f$ we have
$N_{f,i}=1$. Moreover, since $f$ and $g$ do not have common factor, for these
affine components we have $N_{g,i}=0$ and thus $N_{i}=N_{f,i}-N_{g,i}=1$. The
relative canonical divisor only has exceptional support so we also have
$k_{i}=0$ on affine components.

Now consider an integer jumping number $\lambda_{j-1}=n\in\mathbb{Z}_{>0}$ and
recall that, by Lemma 4.2,

${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{n}\right)=\left(f^{n}\right)=\pi_{*}\mathcal{O}_{Y}(-n\tilde{f}),$

where $n\tilde{f}=\sum nN_{f,i}\,E_{i}$ is an antinef divisor. Then, following
Algorithm 3.1, the next jumping number is

$\lambda_{j}=\min_{N_{i}>0}\left\\{\frac{k_{i}+1+nN_{f,i}}{N_{i}}\right\\}.$

We have that this next jumping number is $n+1$ if and only if for all $i$ such
that $N_{i}>0$ we have:

$\displaystyle n+1\leqslant\frac{k_{i}+1+nN_{f,i}}{N_{i}}$
$\displaystyle\iff(n+1)(N_{f,i}-N_{g,i})\leqslant k_{i}+1+nN_{f,i}$
$\displaystyle\iff N_{f,i}\leqslant(n+1)N_{g,i}+k_{i}+1.$

For the affine components of $f$ we have $N_{f,i}=1$ (and $N_{g,i}=0$,
$k_{i}=0$) so the inequality is satisfied. We don’t consider the affine
components of $g$ since $N_{i}<0$. Since $g$ vanishes at the origin, for all
exceptional components we have $N_{g,i}\geqslant 1$. Therefore, if $n$ is
large enough, the inequality holds for all $i$ such that $N_{i}>0$, and thus
$\lambda_{j}=\lambda_{j-1}+1=n+1$.

Notice that, given a large enough integer jumping number
$\lambda_{j-1}=n\in\mathbb{Z}_{>0}$ (which exists by Lemma 4.3), all the
consecutive jumping numbers satisfy
$\lambda_{i}=\lambda_{i-1}+1\in\mathbb{Z}_{>0}$ for all $i\geqslant j$. ∎

Using the notations we fixed in this section we also obtain the following:

######  Corollary 4.5.

Let $f,g\in\mathbb{C}\\{x,y\\}$ be reduced holomorphic functions vanishing at
the origin, with no common factor. Then, the jumping numbers of $f/g$ are
precisely the set of integers $\mathbb{Z}_{>0}$ if and only if
$N_{f,i}\leqslant N_{g,i}+k_{i}+1$ for all $i$.

##  5\. Examples

In this section we provide examples where we compute the sequence of
multiplier ideals of meromorphic functions using the algorithm developed in
Section 2. In particular, we will illustrate the phenomenon described in
Section 4. The algorithm has been implemented with the mathematical software
Magma [BCP97] and is available at:

https://github.com/rogolop

######  Example 5.1.

Consider the holomorphic functions:

$\displaystyle f=(y^{2}-x^{3})^{4}+x^{8}y^{5},$ $\displaystyle g=y^{2}-x^{3}.$

The values of $f$ and $g$ are collected in the following vectors:

$\displaystyle
N_{f}=(8,12,24,28,31,31,60,92,124,125,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31),$
$\displaystyle
N_{g}=(2,3,6,7,8,9,15,23,31,31,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31).$

Therefore, we get the divisor $\tilde{F}_{0}$ given by the values

$N=(6,9,18,21,23,22,45,69,93,94,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0).$

The jumping numbers of the meromorphic function $f/g$ and the generators of
the corresponding multiplier ideals are:

$\begin{array}[]{c c
c}\begin{array}[t]{|l|l|}\hline\cr\bm{\lambda_{j}}&\bm{{\mathcal{J}}((f/g)^{\lambda_{j}})}\\\\[0.0pt]
\hline\cr\hline\cr 5/18&x,y\\\ \hline\cr 35/93&x^{2},y\\\ \hline\cr
13/31&x^{2},xy,y^{2}\\\ \hline\cr 43/93&x^{3},xy,y^{2}\\\ \hline\cr
47/93&g,x^{2}y,x^{3}\\\ \hline\cr 17/31&g,x^{4},xy^{2},x^{2}y\\\ \hline\cr
55/93&g,x^{4},x^{3}y\\\ \hline\cr\bm{11/18}&xg,x^{4},x^{3}y,yg\\\ \hline\cr
59/93&x^{2}y^{2},x^{5},xg,x^{3}y,yg\\\ \hline\cr 21/31&x^{5},x^{4}y,xg,yg\\\
\hline\cr\bm{22/31}&x^{2}y^{2},x^{2}g,x^{4}y,yg,xy^{3}\\\ \hline\cr
67/93&x^{3}y^{2},x^{2}g,x^{4}y,x^{6},yg\\\
\hline\cr\bm{70/93}&x^{2}g,x^{3}y^{2},x^{6},y^{2}g,xyg,x^{4}y\\\ \hline\cr
71/93&x^{2}g,x^{5}y,x^{6},y^{2}g,xyg\\\
\hline\cr\bm{74/93}&x^{3}y^{2},x^{2}y^{3},x^{3}g,y^{2}g,xyg,x^{5}y\\\
\hline\cr 25/31&x^{7},x^{3}g,y^{2}g,xyg,x^{5}y,x^{4}y^{2}\\\
\hline\cr\bm{26/31}&g^{2},x^{7},x^{2}yg,x^{3}g,x^{5}y,x^{4}y^{2}\\\ \hline\cr
79/93&g^{2},x^{7},x^{2}yg,x^{3}g,x^{6}y\\\
\hline\cr\bm{82/93}&g^{2},xy^{2}g,x^{4}g,x^{2}yg,x^{3}y^{3},x^{6}y,x^{4}y^{2}\\\
\hline\cr 83/93&g^{2},xy^{2}g,x^{8},x^{4}g,x^{2}yg,x^{5}y^{2},x^{6}y\\\
\hline\cr\bm{86/93}&g^{2},x^{6}y,x^{5}y^{2},x^{8},x^{3}yg,x^{4}g\\\ \hline\cr
29/31&g^{2},x^{7}y,x^{3}yg,x^{8},x^{4}g\\\
\hline\cr\bm{17/18}&x^{7}y,xg^{2},x^{8},yg^{2},x^{3}yg,x^{4}g\\\
\hline\cr\bm{30/31}&x^{7}y,x^{5}y^{2},xg^{2},x^{5}g,yg^{2},x^{3}yg,x^{4}y^{3},x^{2}y^{2}g\\\
\hline\cr
91/93&x^{7}y,x^{9},xg^{2},x^{5}g,yg^{2},x^{3}yg,x^{6}y^{2},x^{2}y^{2}g\\\
\hline\cr 1&f\\\
\hline\cr\end{array}&\quad\begin{array}[t]{|l|l|}\hline\cr\bm{\lambda_{j}}&\bm{{\mathcal{J}}((f/g)^{\lambda_{j}})}\\\\[0.0pt]
\hline\cr\hline\cr\bm{29/18}&f\cdot(x,y)\\\
\hline\cr\bm{53/31}&f\cdot(y,x^{2})\\\
\hline\cr\bm{163/93}&f\cdot(x^{2},xy,y^{2})\\\
\hline\cr\bm{167/93}&f\cdot(x^{3},xy,y^{2})\\\
\hline\cr\bm{57/31}&f\cdot(g,x^{2}y,x^{3})\\\
\hline\cr\bm{175/93}&f\cdot(g,x^{4},xy^{2},x^{2}y)\\\
\hline\cr\bm{179/93}&f\cdot(g,x^{4},x^{3}y)\\\
\hline\cr\bm{35/18}&f\cdot(xg,x^{4},x^{3}y,yg)\\\
\hline\cr\bm{61/31}&f\cdot(x^{2}y^{2},x^{5},xg,x^{3}y,yg)\\\ \hline\cr
2&f^{2}\\\ \hline\cr\bm{53/18}&f^{2}\cdot(x,y)\\\ \hline\cr 3&f^{3}\\\
\hline\cr 4&f^{4}\\\ \hline\cr 5&f^{5}\\\ \hline\cr 6&f^{6}\\\ \hline\cr
7&f^{7}\\\ \hline\cr 8&f^{8}\\\ \hline\cr 9&f^{9}\\\
\hline\cr\lx@intercol\hfil\vdots\hfil\lx@intercol\end{array}\end{array}$

Notice that the jumping numbers larger than $3$ are only integer numbers which
illustrates Theorem 4.4. We also point out that the version of Skoda’s theorem
4.1 gives us

${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\frac{29}{18}}\right)={\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\frac{11}{18}+1}\right)=f\cdot\left({\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\frac{11}{18}}\right):g\right)=f\cdot{\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\frac{5}{18}}\right).$

In this case
${\mathcal{J}}\left(\left(\frac{f}{g}\right)^{\frac{11}{18}}\right)$ is the
first ideal that does not contain $g$. A similar phenomenon happens for the
jumping numbers stated in bold at the table.

######  Example 5.2.

Consider the holomorphic funcions:

$\displaystyle f=(y^{2}-x^{3})^{4}+x^{8}y^{5},$ $\displaystyle
g^{\prime}=y^{2}+x^{3}.$

The values of $f$ and $g$ are collected in the following vectors:

$\displaystyle
N_{f}=(8,12,24,28,31,60,92,124,125,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24),$
$\displaystyle
N_{g^{\prime}}=(2,3,6,6,6,12,18,24,24,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24).$

Therefore, we get the divisor $\tilde{F}_{0}$ given by the values

$N=(6,9,18,22,25,48,74,100,101,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0).$

Comparing the jumping numbers of $f/g^{\prime}$ with the previous example (in
which $g=y^{2}-x^{3}$ and thus the log resolution of $f/g$ and $f/g^{\prime}$
differ):

$\bm{g^{\prime}=y^{2}+x^{3}}$ |  $\bm{g=y^{2}-x^{3}}$
---|---
$\frac{27}{100}$, $\frac{7}{20}$, $\frac{39}{100}$, $\frac{43}{100}$, $\frac{47}{100}$, $\frac{51}{100}$, $\frac{11}{20}$, $\frac{29}{50}$, $\frac{59}{100}$, $\frac{63}{100}$, $\frac{33}{50}$, $\frac{67}{100}$, $\frac{7}{10}$, $\frac{71}{100}$, $\frac{37}{50}$, $\frac{3}{4}$, $\frac{39}{50}$, $\frac{79}{100}$, $\frac{41}{50}$, $\frac{83}{100}$, $\frac{43}{50}$, $\frac{87}{100}$, $\frac{89}{100}$, $\frac{9}{10}$, $\frac{91}{100}$, $\frac{47}{50}$, $\frac{19}{20}$, $\frac{97}{100}$, $\frac{49}{50}$, $\frac{99}{100}$ |  $\frac{5}{18}$, $\frac{35}{93}$, $\frac{13}{31}$, $\frac{43}{93}$, $\frac{47}{93}$, $\frac{17}{31}$, $\frac{55}{93}$, $\frac{11}{18}$, $\frac{59}{93}$, $\frac{21}{31}$, $\frac{22}{31}$, $\frac{67}{93}$, $\frac{70}{93}$, $\frac{71}{93}$, $\frac{74}{93}$, $\frac{25}{31}$, $\frac{26}{31}$, $\frac{79}{93}$, $\frac{82}{93}$, $\frac{83}{93}$, $\frac{86}{93}$, $\frac{29}{31}$, $\frac{17}{18}$, $\frac{30}{31}$, $\frac{91}{93}$
$1$, $\frac{151}{100}$, $\frac{159}{100}$, $\frac{163}{100}$, $\frac{167}{100}$, $\frac{171}{100}$, $\frac{7}{4}$, $\frac{179}{100}$, $\frac{91}{50}$, $\frac{183}{100}$, $\frac{187}{100}$, $\frac{19}{10}$, $\frac{191}{100}$, $\frac{97}{50}$, $\frac{39}{20}$, $\frac{99}{50}$, $\frac{199}{100}$ |  $1$, $\frac{29}{18}$, $\frac{53}{31}$, $\frac{163}{93}$, $\frac{167}{93}$, $\frac{57}{31}$, $\frac{175}{93}$, $\frac{179}{93}$, $\frac{35}{18}$, $\frac{61}{31}$
$2$, $\frac{11}{4}$, $\frac{283}{100}$, $\frac{287}{100}$, $\frac{291}{100}$, $\frac{59}{20}$, $\frac{299}{100}$ |  $2$, $\frac{53}{18}$
$3$, $\frac{399}{100}$ |  $3$
$4$ |  $4$
⋮ | ⋮

The curves $g=y^{2}-x^{3}$ and $g^{\prime}=y^{2}+x^{3}$ are analytically
isomorphic. The difference in the jumping numbers is due to the different
contact with the curve defined by $f$. The higher contact results in less
jumping numbers.

######  Example 5.3.

Consider the holomorphic functions

$\displaystyle f=(y^{2}-x^{3})^{5}+x^{18},$ $\displaystyle
g_{k}=(y^{2}-x^{3})^{k},$

with $k\in\mathbb{Z}_{\geqslant 0}$. The contact of $g_{k}$ with $f$ increases
with the exponent $k$, thus we get less jumping numbers.

$\bm{k=0}$ |  $\bm{k=1}$ |  $\bm{k=2}$ |  $\bm{k=3}$ |  $\bm{k=4}$ |  $\bm{k\geqslant 5}$
---|---|---|---|---|---
$\frac{1}{6}$, $\frac{41}{180}$, $\frac{23}{90}$, $\frac{17}{60}$, $\frac{14}{45}$, $\frac{61}{180}$, $\frac{11}{30}$, $\frac{71}{180}$, $\frac{19}{45}$, $\frac{77}{180}$, $\frac{9}{20}$, $\frac{41}{90}$, $\frac{43}{90}$, $\frac{29}{60}$, $\frac{91}{180}$, $\frac{23}{45}$, $\frac{8}{15}$, $\frac{97}{180}$, $\frac{101}{180}$, $\frac{17}{30}$, $\frac{53}{90}$, $\frac{107}{180}$, $\frac{37}{60}$, $\frac{28}{45}$, $\frac{113}{180}$, $\frac{29}{45}$, $\frac{13}{20}$, $\frac{59}{90}$, $\frac{121}{180}$, $\frac{61}{90}$, $\frac{41}{60}$, $\frac{7}{10}$, $\frac{127}{180}$, $\frac{32}{45}$, $\frac{131}{180}$, $\frac{11}{15}$, $\frac{133}{180}$, $\frac{34}{45}$, $\frac{137}{180}$, $\frac{23}{30}$, $\frac{47}{60}$, $\frac{71}{90}$, $\frac{143}{180}$, $\frac{73}{90}$, $\frac{49}{60}$, $\frac{37}{45}$, $\frac{149}{180}$, $\frac{151}{180}$, $\frac{38}{45}$, $\frac{17}{20}$, $\frac{77}{90}$, $\frac{13}{15}$, $\frac{157}{180}$, $\frac{79}{90}$, $\frac{53}{60}$, $\frac{161}{180}$, $\frac{9}{10}$, $\frac{163}{180}$, $\frac{41}{45}$, $\frac{83}{90}$, $\frac{167}{180}$, $\frac{14}{15}$, $\frac{169}{180}$, $\frac{19}{20}$, $\frac{43}{45}$, $\frac{173}{180}$, $\frac{29}{30}$, $\frac{44}{45}$, $\frac{59}{60}$, $\frac{89}{90}$, $\frac{179}{180}$ |  $\frac{5}{24}$, $\frac{41}{144}$, $\frac{23}{72}$, $\frac{17}{48}$, $\frac{7}{18}$, $\frac{61}{144}$, $\frac{11}{24}$, $\frac{71}{144}$, $\frac{19}{36}$, $\frac{77}{144}$, $\frac{9}{16}$, $\frac{41}{72}$, $\frac{43}{72}$, $\frac{29}{48}$, $\frac{91}{144}$, $\frac{23}{36}$, $\frac{2}{3}$, $\frac{97}{144}$, $\frac{101}{144}$, $\frac{17}{24}$, $\frac{53}{72}$, $\frac{107}{144}$, $\frac{37}{48}$, $\frac{7}{9}$, $\frac{113}{144}$, $\frac{29}{36}$, $\frac{13}{16}$, $\frac{59}{72}$, $\frac{121}{144}$, $\frac{61}{72}$, $\frac{41}{48}$, $\frac{7}{8}$, $\frac{127}{144}$, $\frac{8}{9}$, $\frac{131}{144}$, $\frac{11}{12}$, $\frac{133}{144}$, $\frac{17}{18}$, $\frac{137}{144}$, $\frac{23}{24}$, $\frac{47}{48}$, $\frac{71}{72}$, $\frac{143}{144}$ |  $\frac{5}{18}$, $\frac{41}{108}$, $\frac{23}{54}$, $\frac{17}{36}$, $\frac{14}{27}$, $\frac{61}{108}$, $\frac{11}{18}$, $\frac{71}{108}$, $\frac{19}{27}$, $\frac{77}{108}$, $\frac{3}{4}$, $\frac{41}{54}$, $\frac{43}{54}$, $\frac{29}{36}$, $\frac{91}{108}$, $\frac{23}{27}$, $\frac{8}{9}$, $\frac{97}{108}$, $\frac{101}{108}$, $\frac{17}{18}$, $\frac{53}{54}$, $\frac{107}{108}$ |  $\frac{5}{12}$, $\frac{41}{72}$, $\frac{23}{36}$, $\frac{17}{24}$, $\frac{7}{9}$, $\frac{61}{72}$, $\frac{11}{12}$, $\frac{71}{72}$ |  $\frac{5}{6}$ |
$1+($jumping numbers
between $0$ and $1)$ |  $1$, $\frac{35}{24}$, $\frac{221}{144}$, $\frac{113}{72}$, $\frac{77}{48}$, $\frac{59}{36}$, $\frac{241}{144}$, $\frac{41}{24}$, $\frac{251}{144}$, $\frac{16}{9}$, $\frac{257}{144}$, $\frac{29}{16}$, $\frac{131}{72}$, $\frac{133}{72}$, $\frac{89}{48}$, $\frac{271}{144}$, $\frac{17}{9}$, $\frac{23}{12}$, $\frac{277}{144}$, $\frac{281}{144}$, $\frac{47}{24}$, $\frac{143}{72}$, $\frac{287}{144}$ |  $1$, $\frac{35}{18}$ |  $1$ |  $1$ |  $1$
$2+($jumping numbers
between $0$ and $1)$ |  $2$, $\frac{65}{24}$, $\frac{401}{144}$, $\frac{203}{72}$, $\frac{137}{48}$, $\frac{26}{9}$, $\frac{421}{144}$, $\frac{71}{24}$, $\frac{431}{144}$ |  $2$ |  $2$ |  $2$ |  $2$
$3+($jumping numbers
between $0$ and $1)$ |  $3$, $\frac{95}{24}$ |  $3$ |  $3$ |  $3$ |  $3$
$4+($jumping numbers
between $0$ and $1)$ |  $4$ |  $4$ |  $4$ |  $4$ |  $4$
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮

## References

  * [AC04] Maria Alberich-Carramiñana, _An algorithm for computing the singularity of the generic germ of a pencil of plane curves_ , Comm. Algebra 32 (2004), no. 4, 1637–1646. MR 2100381
  * [AAB19] Maria Alberich-Carramiñana, Josep Àlvarez Montaner, and Guillem Blanco, _Effective computation of base points of ideals in two-dimensional local rings_ , J. Symbolic Comput. 92 (2019), 93–109. MR 3907349
  * [AAB21] by same author, _Monomial generators of complete planar ideals_ , J. Algebra Appl. 20 (2021), no. 3, Paper No. 2150032, 24. MR 4242204
  * [AAD16] Maria Alberich-Carramiñana, Josep Àlvarez Montaner, and Ferran Dachs-Cadefau, _Multiplier ideals in two-dimensional local rings with rational singularities_ , Michigan Math. J. 65 (2016), no. 2, 287–320. MR 3510908
  * [AADG17] Maria Alberich-Carramiñana, Josep Àlvarez Montaner, Ferran Dachs-Cadefau, and Víctor González-Alonso, _Poincaré series of multiplier ideals in two-dimensional local rings with rational singularities_ , Adv. Math. 304 (2017), 769–792. MR 3558221
  * [AGLN21] Josep Àlvarez Montaner, Manuel González-Villa, Edwin León-Cardenal, and Luis Núñez Betancourt, _Bernstein-Sato polynomial and related invariants for meromorphic functions_ , arXiv:2112.08492 (2021).
  * [BD18] Guillem Blanco and Ferran Dachs Cadefau, _Computing multiplier ideals in smooth surfaces_ , Extended abstracts February 2016—positivity and valuations, Trends Math. Res. Perspect. CRM Barc., vol. 9, Birkhäuser/Springer, Cham, 2018, pp. 57–63. MR 3946254
  * [BCP97] Wieb Bosma, John Cannon, and Catherine Playoust, _The Magma algebra system. I. The user language_ , J. Symbolic Comput. 24 (1997), no. 3-4, 235–265, Computational algebra and number theory (London, 1993). MR MR1484478
  * [CA00] Eduardo Casas-Alvero, _Singularities of plane curves_ , London Mathematical Society Lecture Note Series, vol. 276, Cambridge University Press, Cambridge, 2000\. MR 1782072
  * [HJ11] Eero Hyry and Tarmo Järvilehto, _Jumping numbers and ordered tree structures on the dual graph_ , Manuscripta Math. 136 (2011), no. 3-4, 411–437. MR 2844818
  * [HJ18] by same author, _A formula for jumping numbers in a two-dimensional regular local ring_ , J. Algebra 516 (2018), 437–470. MR 3863487
  * [Jär11] Tarmo Järvilehto, _Jumping numbers of a simple complete ideal in a two-dimensional regular local ring_ , Mem. Amer. Math. Soc. 214 (2011), no. 1009, viii+78. MR 2856648
  * [Laz04] Robert Lazarsfeld, _Positivity in algebraic geometry. II_ , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 49, Springer-Verlag, Berlin, 2004.
  * [Lip69] Joseph Lipman, _Rational singularities, with applications to algebraic surfaces and unique factorization_ , Inst. Hautes Études Sci. Publ. Math. (1969), no. 36, 195–279. MR 276239
  * [Nai09] Daniel Naie, _Jumping numbers of a unibranch curve on a smooth surface_ , Manuscripta Math. 128 (2009), no. 1, 33–49. MR 2470185
  * [Tak23] Kiyoshi Takeuchi, _On a Bernstein-Sato polynomial of a meromorphic function_ , Nagoya Math. J. 251 (2023), 715–733. MR 4637148
  * [Tuc10] Kevin Tucker, _Jumping numbers on algebraic surfaces with rational singularities_ , Trans. Amer. Math. Soc. 362 (2010), no. 6, 3223–3241. MR 2592954
  * [Zar38] Oscar Zariski, _Polynomial ideals defined by infinitely near base points_ , Amer. J. Math. 60 (1938), no. 1, 151–204.