Motivic Milnor fibers of plane curve singularities

TL;DR

This paper presents a combinatorial method to compute motivic Milnor fibers of plane curve singularities using resolution graphs, enabling analysis of their Hodge-Steenbrink spectrum through quasi-homogeneous singularities.

Contribution

Introduces an inductive, combinatorial approach to compute motivic Milnor fibers using extended simplified resolution graphs, linking to Hodge-Steenbrink spectrum analysis.

Findings

Method allows computation of motivic Milnor fibers for plane curve singularities.

Connects the spectrum of such singularities to that of quasi-homogeneous ones.

Facilitates spectral study via combinatorial resolution data.

Abstract

We compute the motivic Milnor fiber of a complex plane curve singularity in an inductive and combinatoric way using the extended simplified resolution graph. The method introduced in this article has a consequence that one can study the Hodge-Steenbrink spectrum of such a singularity in terms of that of a quasi-homogeneous singularity.

Equations

$$f_{P_{\mathcal{B},i}}(u,v)=v^{{\sum }_{t=i+1}^{m_{\mathcal{B}}}{a}_{\mathcal{B},t}{A}_{\mathcal{B},t}}{u}^{m(P[\mathcal{B}])+{\sum }_{t=1}^{i-1}{b}_{\mathcal{B},t}{A}_{\mathcal{B},t}}\prod _{j=1}^{r_{\mathcal{B},i}}(v^{a_{\mathcal{B},i}}+{\xi }_{\mathcal{B},i,j}{u}^{b_{\mathcal{B},i}}{)}^{A_{\mathcal{B},i,j}},$$

$$f_{P_{\mathcal{B},i,i+1}}(u,v)=v^{{\sum }_{t=i+1}^{m_{\mathcal{B}}}{a}_{\mathcal{B},t}{A}_{\mathcal{B},t}}{u}^{m(P[\mathcal{B}])+{\sum }_{t=1}^i{b}_{\mathcal{B},t}{A}_{\mathcal{B},t}},$$

$$X(\mathcal{B},i):=\{(u,v)\in {\mathbb{A}}_{\mathbb{C}}^2\mid f_{P_{\mathcal{B},i}}(u,v)=1\},$$

$$X(\mathcal{B},i,i+1):=\{(u,v)\in {\mathbb{A}}_{\mathbb{C}}^2\mid f_{P_{\mathcal{B},i,i+1}}(u,v)=1\}.$$

$${\mathcal{S}}_{f,O}=\sum _{\mathcal{B}}\left(\sum _{i=1}^{m_{\mathcal{B}}}[X(\mathcal{B},i)]-\sum _{i=1}^{m_{\mathcal{B}}-1}[X(\mathcal{B},i,i+1)]-(\mathbb{L}-1)\sum _{i=1}^{m_{\mathcal{B}}}\sum _{j=1}^{r_{\mathcal{B},i}}[{\mu }_{A_{\mathcal{B},i,j}}(\mathbb{C})]\right),$$

$$f(x,y)=\prod _{i=1}^m\prod _{j=1}^{r_i}\prod _{l=1}^{s_{ij}}{g}_{i,j,l}(x,y),$$

$$g_{i,j,l}(x,y)=(y^{a_i}+{\xi }_{i,j}{x}^{b_i}{)}^{A_{i,j,l}}+\text{(higher terms)}$$

$$f_{{\mathcal{B}}_{1,\mathbf{s}}}(x,y)=\prod _{i=1}^m\prod _{j=1}^{r_i}(y^{a_i}+{\xi }_{i,j}{x}^{b_i}{)}^{A_{i,j}},$$

$$A_{i,j}=\sum _{l=1}^{s_{ij}}{A}_{i,j,l}.$$

$$m(Q^{\mathrm{left}})=\sum _{t=1}^m{a}_t{A}_t,m(Q^{\mathrm{right}})=\sum _{t=1}^m{b}_t{A}_t$$

$$m(P_i)=a_i\sum _{1\le t\le i}{b}_t{A}_t+b_i\sum _{i+1\le t\le m}{a}_t{A}_t,1\le i\le m,$$

$${\Phi }^{\ast }f(u,v)=U(u,v)u^{m(P)}\prod _{i=1}^{m_{{\mathcal{B}}_{\mathbf{s}}}}\prod _{j=1}^{r_{{\mathcal{B}}_{\mathbf{s}},i}}\prod _{l=1}^{s_{{\mathcal{B}}_{\mathbf{s}},ij}}{g}_{{\mathcal{B}}_{\mathbf{s}},i,j,l}(u,v),$$

$$g_{{\mathcal{B}}_{\mathbf{s}},i,j,l}(u,v)=(v^{a_{{\mathcal{B}}_{\mathbf{s}},i}}+{\xi }_{{\mathcal{B}}_{\mathbf{s}},i,j}{u}^{b_{{\mathcal{B}}_{\mathbf{s}},i}}{)}^{A_{{\mathcal{B}}_{\mathbf{s}},i,j,l}}+\text{(higher terms)}$$

$$f_{{\mathcal{B}}_{\mathbf{s}}}(u,v)=u^{m(P)}\prod _{i=1}^{m_{{\mathcal{B}}_{\mathbf{s}}}}\prod _{j=1}^{r_{{\mathcal{B}}_{\mathbf{s}},i}}(v^{a_{{\mathcal{B}}_{\mathbf{s}},i}}+{\xi }_{{\mathcal{B}}_{\mathbf{s}},i,j}{u}^{b_{{\mathcal{B}}_{\mathbf{s}},i}}{)}^{A_{{\mathcal{B}}_{\mathbf{s}},i,j}},$$

$$A_{{\mathcal{B}}_{\mathbf{s}},i,j}=\sum _{l=1}^{s_{{\mathcal{B}}_{\mathbf{s}},ij}}{A}_{{\mathcal{B}}_{\mathbf{s}},i,j,l}.$$

$$m(P_{{\mathcal{B}}_{\mathbf{s}},i})=a_{{\mathcal{B}}_{\mathbf{s}},i}m(P)$$

$$m(Q_{{\mathcal{B}}_{\mathbf{s}}}^{\mathrm{right}})=m(P)+\sum _{t=1}^{m_{{\mathcal{B}}_{\mathbf{s}}}}{b}_{{\mathcal{B}}_{\mathbf{s}},t}{A}_{{\mathcal{B}}_{\mathbf{s}},t},$$

$$J_n=J_n(f):=\left\{(\phi ,\psi )\in {\left(t\mathbb{C}[t]/t^{n+1}\right)}^2\mid f(\phi ,\psi )=t^n\mathrm{mod}{t}^{n+1}\right\},$$

$$Z_{f,O}^{\mathrm{mot}}(T):=\sum _{n\ge 1}[J_n]{\mathbb{L}}^{-2n}{T}^n$$

$$\underset{T\to \infty }{\lim }:{\mathcal{M}}_{\mathbb{C}}^{\hat{\mu }}[[T]{]}_{\mathrm{sr}}\to {\mathcal{M}}_{\mathbb{C}}^{\hat{\mu }}$$

$${\pi }_S:{\tilde{E}}_S^{\circ }\to E_S^{\circ }$$

$$\{(u,v;z)\in E_S^{\circ }\times {\mathbb{A}}_{\mathbb{C}}^1\mid z^{m_S}=\tilde{U}(u,v{)}^{-1}\},$$

$${\mathcal{S}}_{f,O}=\sum _{i\in A}[{\tilde{E}}_i^{\circ }\cap {\pi }^{-1}(O)]-(\mathbb{L}-1)\sum _{i,j\in A}[{\tilde{E}}_{i,j}^{\circ }\cap {\pi }^{-1}(O)]$$

$${\mathcal{S}}_{f,O}=\sum _{E_i\text{ exceptional}}[{\tilde{E}}_i^{\circ }]-(\mathbb{L}-1)\sum _{i,j}[{\mu }_{m_{i,j}}(\mathbb{C})].$$

$${\mathcal{S}}_{f,O}={\mathcal{S}}_{f_{{\mathcal{B}}_{1,\mathbf{s}}},O}+\sum _{\mathcal{B}\in \mathbf{B},\mathcal{B}\ne {\mathcal{B}}_1}$$

$$\gcd (m(E(P_{{\mathcal{B}}_{\mathbf{s}},i,})),A_{{\mathcal{B}}_{\mathbf{s}},i,j})=\gcd (m(E(P_{{\mathcal{B}}_{\mathbf{s}},i,})),m(E(Q_{{\mathcal{B}}_{\mathbf{s}}^{i,j},1}))).$$

$$m(E(Q_{{\mathcal{B}}_{\mathbf{s}}^{i,j},1}))=N_{\mathcal{B}}m(E(P_{{\mathcal{B}}_{\mathbf{s}},i,}))+A_{{\mathcal{B}}_{\mathbf{s}},i,j}.$$

$${\mathcal{S}}_{f_{{\mathcal{B}}_{1,\mathbf{s}}},O}=$$

$$-(\mathbb{L}-1)\sum _{i=1}^m\sum _{j=1}^{r_i}[{\mu }_{\gcd (m(E(P_i)),m(E(Q_{{\mathcal{B}}_{1,\mathbf{s}}^{i,j},1})))}(\mathbb{C})].$$

$${\mathcal{S}}_{f_{{\mathcal{B}}_{\mathbf{s}}},O}=$$

$$-(\mathbb{L}-1)\sum _{j=1}^{d_{\mathcal{B}}-1}[{\mu }_{\gcd (m(E(Q_{\mathcal{B},j})),m(E(Q_{\mathcal{B},j+1}))}(\mathbb{C})]$$

Full text

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number FWO.101.2015.02.

1. Introduction

For more than two decades geometric motivic integration [3, 4, 6] has been a powerful tool in algebraic geometry and many branches of mathematics, in particular it has several important applications to singularity theory. Indeed, the work [3] of Denef-Loeser gives a breakthrough point of view in studying singularity theory, with the philosophy that motivic Milnor fiber is a motivic incarnation of the classical Milnor fiber. One shows that many invariants in singularity theory such as Hodge-Euler characteristic, Hodge polynomials and Hodge-Steenbrink spectra can be read off from motivic zeta functions and motivic Milnor fibers by means of appropriate Hodge realizations (cf. [3, 4, 6], [9], [10]). Motivic zeta functions and motivic Milnor fibers recently also bring considerable advances in the study of monodromy conjecture. Therefore, the computation of motivic zeta functions and motivic Milnor fibers is widely taken care by geometers and singularity theorists. For instance, Denef-Loeser [4, 7] describe explicitly the motivic zeta function and the motivic Milnor fiber of a regular function using resolution of singularity, Guibert [9] and Steenbrink [21] compute the motivic Milnor fiber of a non-degenerate singularity in terms of its Newton boundary. For composition, several aspects of the computation problem have been achieved due to [5], [17], [10, 11, 12], [16], etc.

In connection with the Hodge-Steenbrink spectrum the best result was quickly obtained by Denef-Loeser [4], which states that the spectrum of the motivic Milnor fiber and that of the classical Milnor fiber of a singularity are the same. Combing this with the motivic formula of Guibert-Loeser-Merle [10] recovers a spectral formula conjectured in the 1980s by Steenbrink (proved earlier in [19]). Another interesting consequence of the result in [4] is the proof by Budur [2] on the equalities between the spectrum multiplicities and the inner jumping multiplicities with respect to the jumping numbers in $(0,1]\cap\mathbb{Q}$. In general, the motivic method in [4] for computing spectra seems not to be easier than classical methods in practice. However, that the motivic method is more effective is possible provided the singularity is non-degenerate with respect to its Newton polyhedron or the singularity is irreducible plane curve as known. The latter is realized in [9] and [8], in which the spectrum of an irreducible plane curve singularity is computed via its motivic Milnor fiber, and this allows to reduce the problem to computing the spectrum of a quasi-homogeneous singularity of the form $(y^{a}+x^{b})^{N}$, for some $a,b,N$ in $\mathbb{N}_{>0}$ and $(a,b)=1$.

In the present article we give a formula for the motivic Milnor fiber of an arbitrary complex plane curve singularity, which allows to express the spectrum of the singularity in terms of the spectra of quasi-homogeneous singularities of the form $\prod_{i=1}^{r}(\eta_{i}y^{a}+\xi_{i}x^{b})^{N_{i}}$, for some $a,b,N_{i}$ in $\mathbb{N}_{>0}$ and $\xi_{i},\eta_{i}$ in $\mathbb{C}$. Let $f=(f,O)$ be the germ of a holomorphic function at the origin of $\mathbb{C}^{2}$. The classical Milnor fiber $M_{f}$ of $f$ is a crucial object for studying the singularity of $f$, it is up to homotopy nothing else than a bouquet of circles (cf. [18]), the number of circles in the bouquet, i.e., the Milnor number of $f$, is a topological invariant. A generator of the fundamental group of a punctured disk, namely, the homotopic class of a small loop going once around the origin, which acts naturally on $H^{i}(M_{f},\mathbb{C})$, induces the monodromy $T_{f}$ of the singularity. To have a panorama picture about $f$ it requires at least profound knowledge from both $M_{f}$ and $T_{f}$ in the same importance. As explained in [4], the motivic Milnor fiber $\mathscr{S}_{f,O}$ of $f$, which lives in the Grothendieck ring $\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}$ (cf. Section 3), may carry information of both the Milnor fiber $M_{f}$ and the monodromy $T_{f}$.

Lê-Oka in [13] encode a resolution of singularity into a graph $\mathbf{G}$. Based on this, for $f$ plane curve singularity, our previous work [15] constructs a so-called extended simplified resolution graph $\mathbf{G}_{\mathbf{s}}$ of $f$, which is independent of the choice of resolution for $f$, and using $\mathbf{G}_{\mathbf{s}}$ the monodromy zeta function $Z_{f,O}^{\mathrm{mon}}(t)$ can be described combinatorially and inductively (cf. [15]). The present work is motivated by Denef-Loeser [4] and Thuong [15], where we obtain a formula for $\mathscr{S}_{f,O}$ in terms of $\mathbf{G}_{\mathbf{s}}$. As in [15] and Section 2, $\mathbf{G}_{\mathbf{s}}$ is arranged from simplified bamboos $\mathscr{B}$, in which each of the vertices $E(P_{\mathscr{B},i})$ of $\mathbf{G}_{\mathbf{s}}$ in $\mathscr{B}$ (ordered naturally by $i$) attaches with a multiplicity $m(P_{\mathscr{B},i})$ and a function of the form

[TABLE]

while each edge connecting $E(P_{\mathscr{B},i})$ and $E(P_{\mathscr{B},i+1})$ attaches with the function

[TABLE]

where $A_{\mathscr{B},i}=\sum_{j=1}^{r_{\mathscr{B},i}}A_{\mathscr{B},i,j}$, and $m(P[\mathscr{B}])$ is the multiplicity of $E(P[\mathscr{B}])$, the predecessor of $E(P_{\mathscr{B},1})$ in $\mathbf{G}_{\mathbf{s}}$. The multiplicities are computed inductively via $\mathbf{G}_{\mathbf{s}}$ in [15, Lemmas 3.1, 3.3]. Put $\mu_{n}(\mathbb{C})=\mathrm{Spec}(k[t]/(t^{n}-1))$, and

[TABLE]

and

[TABLE]

The following is the main result of the present article, which is proved in Section 4.

Theorem 4.4.

The motivic Milnor fiber $\mathscr{S}_{f,O}$ is expressed as follows in the ring $\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}$:

[TABLE]

where the first sum $\sum_{\mathscr{B}}$ runs over the bamboos $\mathscr{B}$ of $\mathbf{G}_{\mathbf{s}}$.

We aimed to treat a bigger problem, that proves the Hertling conjecture for an arbitrary complex plane curve singularity $f$. The conjecture for plane curve states that if $\mu$ is the Milnor number of $f$ and $\alpha_{1}\leq\dots\leq\alpha_{\mu}$ are spectral numbers of $f$ counted with the spectrum multiplicities then $\frac{1}{\mu}\sum_{i=1}^{\mu}(\alpha_{i}-1)^{2}\leq\frac{1}{12}(\alpha_{\mu}-\alpha_{1})$. Saito [20] proves the conjecture for $f$ irreducible. To go further we need to compute the Hodge-Steenbrink spectrum $\mathrm{Sp}(f,O)$, or, modulo Theorem 4.4 and [4], to compute $\mathrm{Sp}(X(\mathscr{B},i))$ and $\mathrm{Sp}(X(\mathscr{B},i,i+1))$ (see Remark 4.6). However, it is well known that computing $\mathrm{Sp}(X(\mathscr{B},i))$ and $\mathrm{Sp}(X(\mathscr{B},i,i+1))$, as well as the spectrum of any quasi-homogeneous singularity, is still an open problem in general.

2. Preliminaries on complex plane curve singularity, after [15]

2.1. Extended resolution graphs

Let $f$ be the germ of a holomorphic function at the origin $O$ of $\mathbb{C}^{2}$, and let $C=f^{-1}(0)$. Remark that in the below arguments we need not to assume that $(f,O)$ is an isolated singularity. In [13], Lê-Oka define the extended resolution graph of a resolution of singularity, and their idea is the following. Let $\pi$ be a resolution of singularity of $f$ with $E_{i}$, $i\in A$, the irreducible components of $\pi^{-1}(C)_{\mathrm{red}}$. By [13], the extended resolution graph $\mathbf{G}=\mathbf{G}(f,\pi)$ of $\pi$ is a graph whose vertices are $E_{i}$, $i\in A$, such that two vertices $E_{i}$ and $E_{j}$ are connected by a single edge if and only if the intersection $E_{i}\cap E_{j}$ is nonempty. There is a derived graph of $\mathbf{G}$ that will be useful in the later part of this paper, it is obtained from $\mathbf{G}$ by removing all the vertices of degree $2$, which is called the simplified extended resolution graph and denoted by $\mathbf{G}_{\mathbf{s}}$.

In what follows we fix $\pi$ as a resolution of singularity of $f$ that is obtained by a set of toric modifications whose centers are determined canonically as in [15]. Note that the graphs $\mathbf{G}$ and $\mathbf{G}_{\mathbf{s}}$ admit an explicit description, as in [15], and we shall use this description in the rest of the present paper. Let $\mathbf{G}_{\mathrm{p}}$ be the graph whose vertices correspond bijectively to the total space of each toric modification and the base space and whose edges correspond to each toric modification. Consider the origin $O$ of the base space as the root of $\mathbf{G}$ (and of $\mathbf{G}_{\mathbf{s}}$). We identify each face $P_{i}$ ($1\leq i\leq m$) of the Newton polyhedron $\Gamma$ of $f$ with a primitive weight vector and assume that $\det(P_{i},P_{i+1})\geq 1$ for any $1\leq i\leq m-1$. Consider a regular simplicial cone subdivision $\{Q_{1},\dots,Q_{d}\}$ which contains $P_{1},\dots,P_{m}$. By definition, $Q_{0}=(1,0)^{t}$, $Q_{d+1}=(0,1)^{t}$, and the $Q_{j}$’s are primitive weight vectors and $\det(Q_{j},Q_{j+1})=1$ for all $0\leq j\leq d$. Let $Q^{\mathrm{left}}:=Q_{1}$ and $Q^{\mathrm{right}}:=Q_{d}$. The first toric modification $\pi_{O}$ centered at the origin $O$ of $\mathbb{C}^{2}$ provides first vertices $E(Q_{1}),\dots,E(Q_{d})$ of $\mathbf{G}$ and also provides first vertices $E(Q^{\mathrm{left}}),E(P_{1}),\dots,E(P_{m}),E(Q^{\mathrm{right}})$ of $\mathbf{G}_{\mathbf{s}}$. These give rise to a unique bamboo $\mathscr{B}_{1}$ (resp. $\mathscr{B}_{1,\mathbf{s}}$) in the first floor of $\mathbf{G}$ (resp. $\mathbf{G}_{\mathbf{s}}$). As explained in [15], we can take $Q^{\mathrm{left}}\not=P_{1}$ and $Q^{\mathrm{right}}\not=P_{m}$, and in view of (2.4) we can consider $\mathscr{B}_{1,\mathbf{s}}$ as independent of the choice of $\pi$.

Let $\pi_{\theta}:X_{i}\rightarrow X_{j}$ be an arbitrary toric modification different to $\pi_{O}$ which appears in $\mathbf{G}_{\mathrm{p}}$, where the center $\theta$ is one of the intersection points of $E(Q)$ and the strict transform of $C$ in $X_{j}$, with $Q$ a primitive weight vector of the previous toric modification $X_{j}\rightarrow X_{k}$. Assume by induction that the partial resolution graphs are already constructed and that $Q$ is the corresponding vertex of the simplified graph $\mathbf{G}_{\mathbf{s}}$. In [1], A’Campo-Oka give a canonical way to determine a local coordinates $(u,v)$ at $\theta$ so that $u=0$ defines $E(Q)$, and in the coordinates one obtains the Newton polyhedron of the pullback of $f$. We denote the faces of this Newton polyhedron by $P_{1}^{\theta},\dots,P_{m_{\theta}}^{\theta}$, which may be ordered due to the condition $\det(P_{i}^{\theta},P_{i+1}^{\theta})\geq 1$ for all $1\leq i\leq m_{\theta}$. Fitting them into a regular simplicial cone subdivision $Q_{1}^{\theta},\dots,Q_{d_{\theta}}^{\theta}$, we get a toric modification at $\theta$. Then $E(Q_{1}^{\theta}),\dots,E(Q_{d_{\theta}}^{\theta})$ are the vertices in a horizontal bamboo of $\mathbf{G}$, say, $\mathscr{B}$, and $E(P_{1}^{\theta}),\dots,E(P_{m_{\theta}}^{\theta}),E(Q^{\theta,\mathrm{right}})$ are the vertices in a horizontal bamboo $\mathscr{B}_{\mathbf{s}}$ of $\mathbf{G}_{\mathbf{s}}$, where $Q^{\theta,\mathrm{right}}$ stands for $Q_{d_{\theta}}^{\theta}$. For convenience, we shall write from now on $Q_{\mathscr{B},1},\dots,Q_{\mathscr{B},d_{\mathscr{B}}}$ for $Q_{1}^{\theta},\dots,Q_{d_{\theta}}^{\theta}$ and write $P_{\mathscr{B}_{\mathbf{s}},1},\dots,P_{\mathscr{B}_{\mathbf{s}},m_{\mathscr{B}_{\mathbf{s}}}},Q_{\mathscr{B}_{\mathbf{s}}}^{\mathrm{right}}$ for $P_{1}^{\theta},\dots,P_{m_{\theta}}^{\theta},Q^{\theta,\mathrm{right}}$. We can assume $Q_{\mathscr{B}_{\mathbf{s}}}^{\mathrm{right}}\not=P_{\mathscr{B}_{\mathbf{s}},m_{\mathscr{B}_{\mathbf{s}}}}$, since if $P_{\mathscr{B}_{\mathbf{s}},m_{\mathscr{B}_{\mathbf{s}}}}$ has the form $(1,b)$ we may choose $Q_{\mathscr{B}_{\mathbf{s}}}^{\mathrm{right}}=(1,b+1)^{t}$. By (2.10) we may consider $\mathscr{B}_{\mathbf{s}}$ not to depend on $\pi_{\theta}$. Now, we connect the vertex $E(Q_{\mathscr{B},1})$ (resp. $E(P_{\mathscr{B}_{\mathbf{s}},1})$) in the present floor, say, $n$th, with the vertex $E(Q)$ in the $(n-1)$th floor by a non-horizontal single edge for $\mathbf{G}$ (resp. for $\mathbf{G}_{\mathbf{s}}$). For convenience, the connection edge is taken into account of the predecessor bamboo. Inductively, this process describes the extended resolution graph $\mathbf{G}$ and the simplified extended resolution graph $\mathbf{G}_{\mathbf{s}}$ (cf. [15]).

2.2. Multiplicities

Let us write $f(x,y)$ as a finite product of irreducible germs, namely,

[TABLE]

where

[TABLE]

are irreducible in $\mathbb{C}\{x\}[y]$ and $\xi_{i,j}$ are nonzero and distinct. Then the $\Gamma$-principal part of $f$, that corresponds to the first floor bamboo $\mathscr{B}_{1}$, is the following

[TABLE]

where

[TABLE]

In fact, the $\Gamma$-principal part of $f$ only depends on $\mathscr{B}_{1,\mathbf{s}}$ and not on $\mathscr{B}_{1}$, hence the reason of notation $f_{\mathscr{B}_{1,\mathbf{s}}}$. The primitive weight vectors $P_{i}=(a_{i},b_{i})^{t}$, $1\leq i\leq m$, are ordered by the condition that $\det(P_{i},P_{i+1})\geq 1$ for all $1\leq i<m$. The first toric modification $\pi_{O}$ of $f$ provides the unique bamboo at the first floor of $\mathbf{G}_{\mathbf{s}}$ whose vertices are $E(Q^{\mathrm{left}}),E(P_{1}),\dots,E(P_{m})$ and $E(Q^{\mathrm{right}})$, in which the vertex $E(P_{i})$ has degree $r_{i}+2$ for each $1\leq i\leq m$. Let $m(P)$ be the multiplicity of $\pi_{O}^{*}f$ on $E(P)$. By [15, Lemma 3.1], we have

[TABLE]

and

[TABLE]

where $A_{t}=\sum_{j=1}^{r_{t}}\sum_{l=1}^{s_{t,j}}A_{t,j,l}$.

For a bamboo $\mathscr{B}\not=\mathscr{B}_{1}$ in $\mathbf{G}$, let $E(P)$ be the predecessor of $E(P_{\mathscr{B}_{\mathbf{s}},1})$ in $\mathbf{G}_{\mathbf{s}}$. Assume that the multiplicity $m(P)$ is already given. Let $\Phi$ be the resolution tower formed from the toric modifications earlier than the one corresponding to $\mathscr{B}_{\mathbf{s}}$ (with the center $\theta$ as above). Then, the pullback $\Phi^{*}f$ in the local coordinates $(u,v)$ at $\theta$ has the form

[TABLE]

where

[TABLE]

are irreducible in $\mathbb{C}\{u\}[v]$, in which $\xi_{\mathscr{B}_{\mathbf{s}},i,j}$ are nonzero and distinct, and $U(u,v)$ is a unit in the ring $\mathbb{C}\{u,v\}$. Thus the $\Gamma(\Phi^{*}f)$-principal part of $\Phi^{*}f(u,v)$ is the following

[TABLE]

where, by definition,

[TABLE]

Note that $f_{\mathscr{B}_{\mathbf{s}}}$ depends only on $\mathscr{B}_{\mathbf{s}}$ and not on $\mathscr{B}$. Furthermore, the primitive weight vectors $P_{\mathscr{B}_{\mathbf{s}},i}=(a_{\mathscr{B}_{\mathbf{s}},i},b_{\mathscr{B}_{\mathbf{s}},i})^{t}$, $1\leq i\leq m_{\mathscr{B}_{\mathbf{s}}}$, are ordered by the condition that $\det(P_{\mathscr{B}_{\mathbf{s}},i},P_{\mathscr{B}_{\mathbf{s}},i+1})\geq 1$ for all $1\leq i<m_{\mathscr{B}_{\mathbf{s}}}$. The corresponding toric modification gives rise to a bamboo of $\mathbf{G}_{\mathbf{s}}$ with vertices $E(Q_{\mathscr{B}_{\mathbf{s}}}^{\mathrm{left}}),E(P_{\mathscr{B}_{\mathbf{s}},1}),\dots,E(P_{\mathscr{B}_{\mathbf{s}},m})$ and $E(Q_{\mathscr{B}_{\mathbf{s}}}^{\mathrm{right}})$, and the degree of each $E(P_{\mathscr{B}_{\mathbf{s}},i})$ in $\mathbf{G}_{\mathbf{s}}$ is $r_{\mathscr{B}_{\mathbf{s}},i}+2$. By [15, Lemma 3.3], we have, for $1\leq i\leq m_{\mathscr{B}_{\mathbf{s}}}$,

[TABLE]

and

[TABLE]

where $A_{\mathscr{B}_{\mathbf{s}},t}=\sum_{j=1}^{r_{\mathscr{B}_{\mathbf{s}},t}}\sum_{l=1}^{s_{\mathscr{B}_{\mathbf{s}},tj}}A_{\mathscr{B}_{\mathbf{s}},t,j,l}$ and $P$ is the predecessor of the first vertex of $\mathbf{G}_{\mathbf{s}}$ in $\mathscr{B}_{\mathbf{s}}$.

3. The motivic Milnor fiber of a plane curve singularity

3.1. The Grothendieck ring of complex algebraic varieties with $\hat{\mu}$-action

Consider the group schemes $\mu_{n}(\mathbb{C})$ of $n$th roots of unity, the maps $\xi\mapsto\xi^{k}$, $n\geq 1$, $k\geq 1$, and let $\hat{\mu}$ be $\varprojlim\mu_{n}(\mathbb{C})$. Let $\mathrm{Var}_{\mathbb{C},\hat{\mu}}$ be the category of algebraic $\mathbb{C}$-varieties endowed with a $\hat{\mu}$-action. The Grothendieck group $K_{0}(\mathrm{Var}_{\mathbb{C},\hat{\mu}})$ is an abelian group generated by symbols $[X]$ for $X$ in $\mathrm{Var}_{\mathbb{C},\hat{\mu}}$ such that $[X]=[Y]$ whenever $X$ is $\hat{\mu}$-equivariant isomorphic to $Y$, $[X]=[Y]+[X\setminus Y]$ for $Y$ Zariski closed in $X$ with the $\hat{\mu}$-action induced from $X$ and $[X\times V]=[X\times\mathbb{A}_{\mathbb{C}}^{e}]$ if $V$ is an $e$-dimensional complex affine space with any linear $\hat{\mu}$-action and the action on $\mathbb{A}_{\mathbb{C}}^{e}$ is trivial. The abelian group $K_{0}(\mathrm{Var}_{\mathbb{C},\hat{\mu}})$ becomes a ring with unit with respect to cartesian product. Let $\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}$ denote $K_{0}(\mathrm{Var}_{\mathbb{C},\hat{\mu}})[\mathbb{L}^{-1}]$, where $\mathbb{L}$ is $[\mathbb{A}_{\mathbb{C}}^{1}]$.

3.2. The motivic Milnor fiber of $(C,O)$

We continue studying the plane curve singularity $(C,O)$ in Section 2. In the sequel, we shall express the motivic Milnor fiber $\mathscr{S}_{f,O}$ in terms of the extended simplified resolution graph $\mathbf{G}_{\mathbf{s}}$.

For $n\geq 1$, we define a $\mathbb{C}$-variety

[TABLE]

endowed with a $\hat{\mu}$-action via $\mu_{n}(\mathbb{C})$ defined by $\xi\cdot(\varphi(t),\psi(t))=(\varphi(\xi t),\psi(\xi t))$, $\xi$ in $\mu_{n}(\mathbb{C})$. Thus, it gives rise to an element $[J_{n}]$ in $\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}$. Consider the formal power series

[TABLE]

in $\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}[[T]]$, which is called the motivic zeta function of the singularity $f$. It was proved in [3] that $Z_{f,O}^{\mathrm{mot}}(T)$ is an element of $\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}[[T]]_{\mathrm{sr}}$, the $\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}$-submodule of $\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}[[T]]$ generated by 1 and by finite products of $\frac{\mathbb{L}^{p}T^{q}}{1-\mathbb{L}^{p}T^{q}}$ with $(p,q)$ in $\mathbb{Z}\times\mathbb{N}_{>0}$. Moreover, there exists a unique $\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}$-linear morphism

[TABLE]

such that $\lim_{T\rightarrow\infty}\frac{\mathbb{L}^{p}T^{q}}{1-\mathbb{L}^{p}T^{q}}=-1$. Then one defines the motivic Milnor fiber of the singularity $f$ to be $-\lim_{T\rightarrow\infty}Z_{f,O}^{\mathrm{mot}}(T)$ and denotes it by $\mathscr{S}_{f,O}$.

3.3. Denef-Loeser’s formula

Consider the resolution of singularity $\pi$ of $(C,O)$ in Subsection 2.1. Putting $E_{i}^{\circ}=E_{i}\setminus\bigcup_{j\not=i}E_{j}$ with $i$, $j$ in $A$, locally, on $E_{i}^{\circ}$, $f(\pi(u,v))=\widetilde{U}(u,v)u^{m_{i}}$, and on $E_{i,j}^{\circ}=E_{i}\cap E_{j}\not=\emptyset$, $f(\pi(u,v))=\widetilde{U}(u,v)u^{m_{i}}v^{m_{j}}$, where $\widetilde{U}(u,v)$ is a unit. The Denef-Loeser unramified Galois covering

[TABLE]

with Galois group $\mu_{m_{S}(\mathbb{C})}$ is defined locally by

[TABLE]

which is endowed with the $\mu_{m_{S}}(\mathbb{C})$-action induced by multiplying the $z$-coordinate with the $m_{S}$-roots of unity, where $S$ is $\{i\}$ or $\{i,j\}$, $m_{i,j}$ is the greatest common divisor of $m_{i}$ and $m_{j}$.

Theorem 3.1** (Denef-Loeser, [3, 6]).**

With the previous notation, the identity

[TABLE]

holds in $\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}$.

If $E_{i}$ is contained in $\pi^{-1}(O)$, we have $[\widetilde{E}_{i}^{\circ}\cap\pi^{-1}(O)]=[\widetilde{E}_{i}^{\circ}]$; otherwise, $[\widetilde{E}_{i}^{\circ}\cap\pi^{-1}(O)]=0$. If $E_{i,j}^{\circ}\not=\emptyset$, then $[\widetilde{E}_{i,j}^{\circ}\cap\pi^{-1}(O)]=[\mu_{m_{i,j}}(\mathbb{C})]$. One thus deduces from Theorem 3.1 that

[TABLE]

The following is a consequence of Theorem 3.1, via (3.3), together with using $\mathbf{G}_{\mathbf{s}}$.

Proposition 3.2**.**

In $\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}$, one has

[TABLE]

where $m(P[\mathscr{B}_{\mathbf{s}}])$ is the multiplicity of $E(P[\mathscr{B}_{\mathbf{s}}])$, the predecessor of $E(P_{\mathscr{B}_{\mathbf{s}},1})$ in $\mathbf{G}_{\mathbf{s}}$.

Remark 3.3**.**

The first bamboo is also the top bamboo if and only if $f(x,y)=x^{A}$ for some $A$ in $\mathbb{N}_{>0}$; in this case, $\mathscr{S}_{f,O}=[\mu_{A}(\mathbb{C})]$. If $\mathscr{B}\not=\mathscr{B}_{1}$ is a top bamboo, the term corresponding to $\mathscr{B}$ in the sum (3.4) vanishes in $\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}$.

Proof of Proposition 3.2.

For a bamboo $\mathscr{B}$ in $\mathbf{G}$ we consider the principal part $f_{\mathscr{B}_{\mathbf{s}}}$ given in (2.3) and (2.8). Let $C_{\mathscr{B}_{\mathbf{s}},i,j}$ be the strict transform of the germ (not necessarily reduced) $(\{f_{\mathscr{B}_{\mathbf{s}}}=0\},O)$, in the toric modification corresponding to $\mathscr{B}$, which intersects with $E(P_{\mathscr{B}_{\mathbf{s}},i})$, $1\leq i\leq m_{\mathscr{B}_{\mathbf{s}}}$, $1\leq j\leq r_{\mathscr{B}_{\mathbf{s}},i}$. Let $\mathscr{B}^{i,j}$ be the successor of $\mathscr{B}$ in the set $\mathbf{B}$ of bamboos of $\mathbf{G}$ whose indices $(i,j)$ correspond to those of $C_{\mathscr{B}_{\mathbf{s}},i,j}$. We remark that, if $\mathscr{B}$ is not a top bamboo, then $m(C_{\mathscr{B}_{\mathbf{s}},i,j})=A_{\mathscr{B}_{\mathbf{s}},i,j}$ and

[TABLE]

The latter comes from the fact that there exists a natural number $N_{\mathscr{B}}$ such that

[TABLE]

For convenience, we define $[\mu_{\mathrm{gcd}(m(E(P_{\mathscr{B}_{\mathbf{s}},i})),m(E(Q_{\mathscr{B}_{\mathbf{s}}^{i,j},1})))}(\mathbb{C})]=0$ if $\mathscr{B}$ a top bamboo.

We now apply Theorem 3.1 to $f_{\mathscr{B}_{\mathbf{s}}}$, with (3.5) used. If $\mathscr{B}=\mathscr{B}_{1}$,

[TABLE]

Also, if $\mathscr{B}\not=\mathscr{B}_{1}$,

[TABLE]

Thanks to [3, 6], in any case, $\mathscr{S}_{f_{\mathscr{B}_{\mathbf{s}}},O}$ is independent of the choice of the resolution of singularity, i.e., of $\mathscr{B}$. Now, using (3.6), (3.7) and (3.5), the right hand side of (3.4) equals

[TABLE]

which is nothing but $\mathscr{S}_{f,O}$, again by Theorem 3.1. ∎

4. The main result

4.1. Direct computations

In this paragraph, a direct computation of $\mathscr{S}_{f_{\mathscr{B}_{\mathbf{s}}}}$ for a bamboo $\mathscr{B}$ in $\mathbf{G}$ will be given. We define $m(P[\mathscr{B}_{1,\mathbf{s}}])=0$. For any bamboo $\mathscr{B}$, the face functions $f_{P_{\mathscr{B}_{\mathbf{s}},i}}(u,v)$ of the Newton polyhedron $\Gamma(f_{\mathscr{B}_{\mathbf{s}}})$ are equal to

[TABLE]

times

[TABLE]

for $1\leq i\leq m_{\mathscr{B}_{\mathbf{s}}}$, with convention $\sum_{t=1}^{0}=0$ and $\sum_{t=m_{\mathscr{B}_{\mathbf{s}}}+1}^{m_{\mathscr{B}_{\mathbf{s}}}}=0$. Let $P_{\mathscr{B}_{\mathbf{s}},i,i+1}$ be the intersection of two faces $P_{\mathscr{B}_{\mathbf{s}},i}$ and $P_{\mathscr{B}_{\mathbf{s}},i+1}$ for $0\leq i\leq m_{\mathscr{B}}$ (here $P_{\mathscr{B}_{\mathbf{s}},0,1}$ denotes the left end point of the face $P_{\mathscr{B}_{\mathbf{s}},1}$ and $P_{\mathscr{B}_{\mathbf{s}},m,m+1}$ denotes the right end point of the face $P_{\mathscr{B}_{\mathbf{s}},m}$). Then, using the previous convention,

[TABLE]

In what follows, by $g^{-1}(1)$ we shall mean the variety $\{(u,v)\in\mathbb{G}_{m,\mathbb{C}}^{2}\mid g(u,v)=1\}$ for a polynomial $g$ in $\mathbb{C}[u,v]$.

Proposition 4.1**.**

For $\mathscr{B}=\mathscr{B}_{1}$,

[TABLE]

For $\mathscr{B}\not=\mathscr{B}_{1}$,

[TABLE]

Proof.

Define a function $\ell_{\mathscr{B}_{\mathbf{s}}}:\mathbb{R}_{\geq 0}^{2}\to\mathbb{R}_{\geq 0}$ by setting $\ell(p,q)=\min_{(a,b)\in\Gamma_{\mathscr{B}_{\mathbf{s}}}}(ap+bq)$, where $\Gamma_{\mathscr{B}_{\mathbf{s}}}$ denotes $\Gamma(f_{\mathscr{B}_{\mathbf{s}}})$, for each bamboo $\mathscr{B}$ in $\mathbf{G}$. For $\Omega=(p,q)$ in $\mathbb{R}_{\geq 0}^{2}$, let $\gamma_{\Omega}$ be the face consisting of points $(a,b)$ in $\Gamma_{\mathscr{B}_{\mathbf{s}}}$ with $ap+bq=\ell_{\mathscr{B}_{\mathbf{s}}}(\Omega)$. For a face $P$, let $\sigma(P)$ be the sets of points $\Omega$ of $\mathbb{R}_{\geq 0}^{2}$ with $\gamma_{\Omega}=P$; thus the closure $\overline{\sigma}(P)$ is the sets of points $\Omega$ in $\mathbb{R}_{\geq 0}^{2}$ with $\gamma_{\Omega}\supset P$. When $P$ runs over the faces of $\Gamma_{\mathscr{B}_{\mathbf{s}}}$, $\overline{\sigma}(P)$ form a fan in $\mathbb{R}_{\geq 0}^{2}$ partitioning it into rational polyhedral cones (see [14]).

Let $J_{\mathscr{B}_{\mathbf{s}},n}$ be the scheme defined similarly as $J_{n}$ of (3.1) with $f$ replaced by $f_{\mathscr{B}_{\mathbf{s}}}$, that is, $J_{\mathscr{B}_{\mathbf{s}},n}:=J_{n}(f_{\mathscr{B}_{\mathbf{s}}})$. For $(p,q)$ in $\mathbb{N}_{>0}^{2}$, let $J_{\mathscr{B}_{\mathbf{s}},n}(p,q)$ be the set of $(\varphi,\psi)$ in $J_{\mathscr{B}_{\mathbf{s}},n}$ such that there exists $(\varphi^{\prime},\psi^{\prime})$ in $\mathbb{C}[[t]]^{2}$ of order $(p,q)$ with $\varphi=\varphi^{\prime}\mathrm{mod}\ t^{n+1}$ and $\psi=\psi^{\prime}\mathrm{mod}\ t^{n+1}$. It gives rise in a natural way to an element $[J_{\mathscr{B}_{\mathbf{s}},n}(p,q)]$ of $\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}$. Since $J_{\mathscr{B}_{\mathbf{s}},n}$ is a disjoint union of such these sets and $n\geq\ell_{\mathscr{B}_{\mathbf{s}}}(\Omega)$ for any $\Omega$ in $\mathbb{N}_{>0}^{2}$, we have

[TABLE]

where

[TABLE]

Now, by writing (4.1) as

[TABLE]

and by using Lemmas 4.2 and 4.3, we complete the proof of the proposition. ∎

Lemma 4.2**.**

The following identities hold:

[TABLE]

In particular, for $\mathscr{B}_{\mathbf{s}}\not=\mathscr{B}_{1,\mathbf{s}}$,

[TABLE]

Proof.

Note that an element of $\sigma(P_{\mathscr{B}_{\mathbf{s}},i})$ has the form $\Omega=(p,q)=(\alpha a_{\mathscr{B}_{\mathbf{s}},i},\alpha b_{\mathscr{B}_{\mathbf{s}},i})$, $\alpha>0$, and for such an $\Omega$, $\ell_{\mathscr{B}_{\mathbf{s}}}(\Omega)=\alpha m(P_{\mathscr{B}_{\mathbf{s}},i})$. Thus $p<\ell_{\mathscr{B}_{\mathbf{s}}}(p,q)$ and $q<\ell_{\mathscr{B}_{\mathbf{s}}}(p,q)$ for any $(p,q)$ in $\mathbb{N}_{>0}^{2}$. For $(\varphi(t),\psi(t))$ in $J_{\mathscr{B}_{\mathbf{s}},\ell_{\mathscr{B}_{\mathbf{s}}}(\Omega)}(\Omega)$, we have

[TABLE]

with $(c_{p},d_{q})$ in $\mathbb{G}_{\mathbb{C}}^{2}$ the coefficients of $(t^{p},t^{q})$ in $(\varphi(t),\psi(t))$. It turns out that, in $\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}$,

[TABLE]

where $|\Omega|=p+q$. By [9, Lemma 2.1.5],

[TABLE]

the first identity is proved.

For $1\leq i<m_{\mathscr{B}_{s}}$, the cone $\sigma(P_{\mathscr{B}_{\mathbf{s}},i,i+1})$ consists of $\Omega=(\alpha a_{\mathscr{B}_{\mathbf{s}},i}+\beta a_{i+1},\alpha b_{\mathscr{B}_{\mathbf{s}},i}+\beta b_{\mathscr{B}_{\mathbf{s}},i+1})$ for $\alpha$ and $\beta$ in $\mathbb{N}_{>0}$, and

[TABLE]

Meanwhile, for $\mathscr{B}_{\mathbf{s}}\not=\mathscr{B}_{1,\mathbf{s}}$, the cone $\sigma(P_{\mathscr{B}_{\mathbf{s}},0,1})$ consists of $\Omega=(\alpha+\beta a_{\mathscr{B}_{\mathbf{s}},1},\beta b_{\mathscr{B}_{\mathbf{s}},1})$ for $\alpha$ and $\beta$ in $\mathbb{N}_{>0}$, and

[TABLE]

Then the results follow by using the same arguments as for the first identity.

Now, we consider the case where $\mathscr{B}_{\mathbf{s}}=\mathscr{B}_{1,\mathbf{s}}$ and $i=0$, and rewrite for simplicity that $\ell=\ell_{\mathscr{B}_{1,\mathbf{s}}}$, $J_{n}^{1}=J_{\mathscr{B}_{1,\mathbf{s}},n}$. Then, for any $\Omega=(p,q)=(\alpha+\beta a_{1},\beta b_{1})$ in $\sigma(P_{0,1})$ ($\alpha,\beta$ in $\mathbb{N}_{>0}$), we have $\ell(\Omega)=qm(Q^{\mathrm{left}})$. If $p>qm(Q^{\mathrm{left}})$, an element of $J_{\ell(\Omega)}(\Omega)$ is of the form

[TABLE]

with $f(0,\psi)$ led by the term $d_{q}^{m(Q^{\mathrm{left}})}t^{qm(Q^{\mathrm{left}})}$, hence

[TABLE]

If $p\leq qm(Q^{\mathrm{left}})$, $J_{\ell(\Omega)}^{1}(\Omega)$ consists of pairs $(\varphi,\psi)$ with $\mathrm{ord}_{t}\varphi=p$ and $\psi$ as previous, and the leading term of $f(\varphi,\psi)$ is $d_{q}^{m(Q^{\mathrm{left}})}t^{qm(Q^{\mathrm{left}})}$. It follows that

[TABLE]

Thus $\sum_{\Omega\in\sigma(P_{0,1})}\Phi_{1}(T)$ is the sum of the series

[TABLE]

and

[TABLE]

Taking $\lim_{T\to\infty}$, the former has image $-[\mu_{m(Q^{\mathrm{left}})}(\mathbb{C})]$ by a direct computation or due to the proof of [9, Lemma 3.3.3], the latter has image zero by [10, Lemma 2.10].

The case where $i=m_{\mathscr{B}_{s}}$ is proved similarly. ∎

Lemma 4.3**.**

The following identities hold:

[TABLE]

Proof.

We only need to write down a proof for the first identity because the second one is trivial. For each $\Omega$ in $\sigma(P_{\mathscr{B}_{\mathbf{s}},i})\cap\mathbb{Z}^{2}$ and each $n=\ell_{\mathscr{B}_{\mathbf{s}}}(\Omega)+k$, $k\geq 1$, let us consider the projection

[TABLE]

which sends $(\varphi(t),\psi(t))$ to their leading nonzero coefficients. Defining $J_{\mathscr{B}_{\mathbf{s}},n}^{(j)}(\Omega)$ to be

[TABLE]

we can decompose $J_{\mathscr{B}_{\mathbf{s}},n}(\Omega)$ into a disjoint union of subsets $J_{\mathscr{B}_{\mathbf{s}},n}^{(j)}(\Omega)$ for all $1\leq j\leq r_{\mathscr{B}_{\mathbf{s}},i}$. Recall that, since $\Omega\in\sigma(P_{\mathscr{B}_{\mathbf{s}},i})\cap\mathbb{Z}^{2}$, it has the form $(\alpha a_{\mathscr{B}_{\mathbf{s}},i},\alpha b_{\mathscr{B}_{\mathbf{s}},i})$ for some $\alpha\in\mathbb{N}_{>0}$, and $\ell_{\mathscr{B}_{\mathbf{s}}}(\Omega)=\alpha m(P_{\mathscr{B}_{\mathbf{s}},i})$. Thus, for such an $\Omega$ and $k\geq 1$, the class $[J_{\mathscr{B}_{\mathbf{s}},\ell_{\mathscr{B}_{\mathbf{s}}}(\Omega)+k}^{(j)}(\Omega)]$ equals

[TABLE]

By definition, we have

[TABLE]

if $A_{\mathscr{B}_{\mathbf{s}},i,j}$ does not divide $\alpha a_{\mathscr{B}_{\mathbf{s}},i}b_{\mathscr{B}_{\mathbf{s}},i}A_{\mathscr{B}_{\mathbf{s}},i,j}+k$. Otherwise, we put

[TABLE]

Since $y^{a_{\mathscr{B}_{\mathbf{s}},i}}+\xi_{\mathscr{B}_{\mathbf{s}},i,j}x^{b_{\mathscr{B}_{\mathbf{s}},i}}$ is non-degenerate with respect to its Newton polyhedron, we may use the argument in the proof of [9, Lemma 2.1.1] and obtain

[TABLE]

It implies that, for $n=\ell_{\mathscr{B}_{\mathbf{s}}}(\Omega)+k=\ell_{\mathscr{B}_{\mathbf{s}}}(\Omega)+(\lambda-\alpha a_{\mathscr{B}_{\mathbf{s}},i}b_{\mathscr{B}_{\mathbf{s}},i})A_{\mathscr{B}_{\mathbf{s}},i,j}$ and $\lambda>\alpha a_{\mathscr{B}_{\mathbf{s}},i}b_{\mathscr{B}_{\mathbf{s}},i}$, we have

[TABLE]

Then

[TABLE]

Now it follows from [9, Lemma 2.1.5] that

[TABLE]

which proves the lemma. ∎

4.2. Description of $\mathscr{S}_{f,O}$ via $\mathbf{G}_{\mathbf{s}}$

Let $\mathbf{B}_{\mathbf{s}}$ denote the set of bamboos of $\mathbf{G}_{\mathbf{s}}$. Let us recall the face functions corresponding to each bamboo $\mathscr{B}$ in $\mathbf{B}_{\mathbf{s}}$ as in Subsection 4.1, up to factor in $\mathbb{G}_{m,\mathbb{C}}$:

[TABLE]

for $1\leq i\leq m_{\mathscr{B}}$, and

[TABLE]

for $0\leq i\leq m_{\mathscr{B}}$. Here $m(P[\mathscr{B}])$ is the multiplicity of $E(P[\mathscr{B}])$, the predecessor of $E(P_{\mathscr{B},1})$ in $\mathbf{G}_{\mathbf{s}}$, $m(P[\mathscr{B}])=0$ for $\mathscr{B}$ being the first bamboo of $\mathbf{G}_{\mathbf{s}}$, and, by convention, $\sum_{t=1}^{0}=0$, $\sum_{t=m_{\mathscr{B}}+1}^{m_{\mathscr{B}}}=0$. To each $\mathscr{B}\in\mathbf{B}_{\mathbf{s}}$ we associate varieties

[TABLE]

for $1\leq i\leq m_{\mathscr{B}}$, and

[TABLE]

for $1\leq i<m_{\mathscr{B}}$.

Theorem 4.4**.**

The motivic Milnor fiber $\mathscr{S}_{f,O}$ is expressed via $\mathbf{G}_{\mathbf{s}}$ as follows

[TABLE]

Proof.

For any bamboo $\mathscr{B}$ of $\mathbf{G}_{\mathbf{s}}$ and $1<i<m_{\mathscr{B}}$, the complex variety $X(\mathscr{B},i)$ is nothing else than $\{(u,v)\in\mathbb{G}_{m,\mathbb{C}}^{2}\mid f_{P_{\mathscr{B},i}}(u,v)=1\}$, which is exactly $f_{P_{\mathscr{B},i}}^{-1}(1)$ as denoted in Proposition 4.1. In these notation we have

[TABLE]

and

[TABLE]

The theorem now follows directly from Propositions 3.2 and 4.1. ∎

Examples 4.5**.**

a) Let $f(x,y)=(y^{a}+x^{b})^{A}$, where $a$, $b$, $A$ are in $\mathbb{N}_{>0}$ and $(a,b)=1$. Applying Theorem 4.4 we get

[TABLE]

b) Let $(f,O)$ be an irreducible singularity with weight vectors $(a_{i},b_{i})^{t}$, $1\leq i\leq g$. The graph $\mathbf{G}_{\mathbf{s}}$ is obtained in terms of a sequence of toric modifications with respect to these weight vectors.

P_{\mathscr{B}_{1},1}$$\mathscr{B}_{1}$$Q_{\mathscr{B}_{1}}^{\mathrm{right}}$$Q_{\mathscr{B}_{1}}^{\mathrm{left}}$$P_{\mathscr{B}_{2},1}$$Q_{\mathscr{B}_{2}}^{\mathrm{right}}$$\mathscr{B}_{2}$$P_{\mathscr{B}_{g},1}$$Q_{\mathscr{B}_{g}}^{\mathrm{right}}$$\mathscr{B}_{g}$$\mathscr{B}_{g+1}

Let $\mathscr{B}_{i},\dots,\mathscr{B}_{g+1}$ denote the bamboos of $\mathbf{G}_{\mathbf{s}}$, where $\mathscr{B}_{i}$ is the unique bamboo of the $i$th floor. Note that, in this case, $P_{\mathscr{B}_{i},1}=(a_{i},b_{i})^{t}$, $A_{\mathscr{B}_{i},1,1,1}=a_{i+1}\cdots a_{g}$, and the multiplicities of the resolution on the exceptional divisors $E(P_{\mathscr{B}_{i},1})$ are computed as follows:

[TABLE]

Then by Theorem 4.4 we have

[TABLE]

Remark 4.6**.**

It is a fact that Hodge-Steenbrink spectrum is a crucial invariant in singularity theory. By [21], $H^{j}(M_{f},\mathbb{C})$ carries a canonical mixed Hodge structure compatible with the semisimple part of the monodromy $T_{f}$. This gives rise to the Hodge-Steenbrink spectrum $\mathrm{Sp}(f,O)$ of the singularity $f$, which is a fractional Laurent polynomial $\sum_{\alpha\in\mathbb{Q}}n_{\alpha}(f)t^{\alpha}$, where $n_{\alpha}(f)=\sum_{j\in\mathbb{Z}}(-1)^{j}\dim_{\mathbb{C}}Gr_{F}^{\lfloor{2-\alpha}\rfloor}H^{1+j}(M_{f},\mathbb{C})_{e^{-2\pi i\alpha}}$, with $F^{\bullet}$ the Hodge filtration. Using a Hodge realization Denef-Loeser [3] construct a linear map $\mathrm{Sp}:\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}\to\mathbb{Z}[\mathbb{Q}]$, which is a ring homomorphism with respect to the convolution product $\ast$ in $\mathscr{M}_{\mathbb{C}}^{\hat{\mu}}$ (see [10]), and by that work we have $\mathrm{Sp}(f,O)=\mathrm{Sp}(\mathscr{S}_{f,O})$. Now, it follows from [9, Lemme 3.4.2], [19, (2.1.2)] and Theorem 4.4 that

[TABLE]

Therefore in order to compute $\mathrm{Sp}(f,O)$ it suffices to study the spectrum of a quasi-homogeneous plane curve singularity.

Acknowledgement**.**

The author would like to thank The Abdus Salam International Centre for Theoretical Physics (ICTP), The Vietnam Institute for Advanced Study in Mathematics (VIASM) and Department of Mathematics - KU Leuven for warm hospitality during his visits.