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 protected] , 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 protected] and Roger Gómez-López Departament de Matemàtiques
Universitat Politècnica de Catalunya
Av. Diagonal 647, Barcelona 08028 [email protected]

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)

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

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)

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)

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

$\cdot$

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

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

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 $1)$	$4$	$4$	$4$	$4$	$4$
⋮	⋮	⋮	⋮	⋮	⋮

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Multiplier ideals of meromorphic functions in dimension two

Abstract.

Key words and phrases:

2000 Mathematics Subject Classification:

1. Introduction

2. Multiplier ideals of meromorphic functions

Definition 2.1.

Remark 2.2.

3. Explicit computation in dimension two

Algorithm 3.1 (Jumping Numbers and Multiplier Ideals).

3.1. Computing the minimal log resolution of meromorphic functions

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

3.2. Computing the integral closure of an ideal

Algorithm 3.3 (Unloading procedure).

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

4. Skoda’s theorem version for meromorphic functions

Proposition 4.1.

Lemma 4.2.

Lemma 4.3.

Proof.

Theorem 4.4.

Proof.

Corollary 4.5.

5. Examples

Example 5.1.

Example 5.2.

Example 5.3.

References

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