Analytic Semiroots for Plane Branches and Singular Foliations

Abstract

The analytic moduli of equisingular plane branches has the semimodule of differential values as the most relevant system of discrete invariants. Focusing in the case of cusps, the minimal system of generators of this semimodule is reached by the differential values attached to the differential 1-forms of the so-called standard bases. We can complete a standard basis to an extended one by adding a last differential 1-form that has the considered cusp as invariant branch and the “correct” divisorial order. The elements of such extended standard bases have the “cuspidal” divisor as a “totally dicritical divisor” and hence they define packages of plane branches that are equisingular to the initial one. These are the analytic semiroots. In this paper we prove that the extended standard bases are well structured from this geometrical and foliated viewpoint, in the sense that the semimodules of differential values of the branches in the dicritical packages are described just by a truncation of the list of generators of the initial semimodule at the corresponding differential value. In particular they have all the same semimodule of differential values.

1 Introduction

The analytic classification of plane branches starts with Zariski (2006), who pointed the importance of the differential values in this problem. The semimodule of differential values was extensively described by Delorme (1978), although the complete analytic classification is due to Hefez and Hernandes (2011).

Geometrically, the “most interesting” differential values are viewed as the contact $\nu _{{\mathcal {C}}}(\omega )$ of a given branch ${\mathcal {C}}$ with the foliations defined by differential 1-forms $\omega$ without common factors in the coefficients. From the moduli view-point, the semimodule of differential values $\Lambda$ is interpreted as the “discrete structure” supporting the continuous part of the moduli. More precisely, the semimodule $\Lambda$ has a well defined basis $\{\lambda _j\}_{j=-1}^s$; so, it is reasonable to fix our attention in the differential forms that produce precisely the elements of the basis as differential values: these are the elements of the standard bases (for more details, see Hefez and Hernandes 2001, 2007).

In this paper we focus in the case of cusps, that is, branches with a single Puiseux pair (n, m). Our objective is to describe the cusps close to a cusp ${\mathcal {C}}$, in terms of a given standard basis ${\mathcal {H}}$ and the dicritical foliated behaviour of the elements of ${\mathcal {H}}$ in the final divisor E of the reduction of singularities of ${\mathcal {C}}$. Let us precise this.

We consider a cusp ${\mathcal {C}}$ with Puiseux pair (n, m). In view of Zariski Equisingularity Theory, we know that the semigroup $\Gamma =n{\mathbb {Z}}_{\ge 0}+m{\mathbb {Z}}_{\ge 0}$ of ${\mathcal {C}}$ is an equivalent data of the equisingularity class of ${\mathcal {C}}$. The differential values define a semimodule $\Lambda ^{\mathcal {C}}$ over $\Gamma$, that will have a strictly increasing basis

$$\begin{aligned} \lambda _{-1}=n,\lambda _0=m,\lambda _1,\ldots ,\lambda _s, \end{aligned}$$

to be the minimal one such that $\Lambda ^{\mathcal {C}}=\cup _{j=-1}^s(\lambda _j+\Gamma )$. By definition, an extended standard basis is a list of 1-forms

$$\begin{aligned} \omega _{-1},\omega _0,\omega _1,\ldots ,\omega _{s+1}, \end{aligned}$$

such that $\nu _{{\mathcal {C}}}(\omega _i)=\lambda _i$ for $i=-1,0,1,\ldots ,s$ and ${\mathcal {C}}$ is an invariant branch of $\omega _{s+1}$, that is $\nu _{{\mathcal {C}}}(\omega _{s+1})=\infty$, with some restrictions on the weighted order of $\omega _{s+1}$.

Associated to the final divisor E given by ${\mathcal {C}}$, we have a divisorial order $\nu _E(\omega )$ defined for functions and 1-forms. In adapted coordinates it is the weighted monomial order that assigns the weight $an+bm$ to the monomial $x^ay^b$. Both the differential values and the divisorial orders act “like” valuations and we have that $\nu _E(\omega )\le \nu _{{\mathcal {C}}}(\omega )$. For the case of a function we have that if $\nu _E(df)<nm$, then there is no resonance in the sense that $\nu _E(df)=\nu _{{\mathcal {C}}}(d f)$. Thus, the “new differential values” in $\Lambda ^{{\mathcal {C}}}$ will correspond to resonant 1-forms $\omega$ such that $\nu _E(\omega )<\nu _{{\mathcal {C}}}(\omega )$.

The structure of the semimodule $\Lambda ^{\mathcal {C}}$ is well known (see Delorme 1978; Alberich-Carramiñana et al. 2021; Almirón and Moyano-Fernández 2021); anyway, we provide complete proofs using another approach in the appendices of the paper. The key elements are the axes $u_i$, and the critical orders $t_i$, defined by

$$\begin{aligned} u_{i+1}=\min (\Lambda _{i-1}\cap (\lambda _i+\Gamma )),\quad t_{i+1}=t_i+u_{i+1}-\lambda _i, \end{aligned}$$

starting at $u_0=n$ and $t_{-1}=n,t_0=m$, where $\Lambda _{i-1}^{{\mathcal {C}}}=\cup _{j=-1}^{i-1}(\lambda _j+\Gamma )$. The axes are defined for $i=0,1,\ldots ,s+1$ and the critical orders for $i=-1,0,\ldots ,s+1$. We know that the semimodule is increasing in the sense that $\lambda _i>u_i$ for $i=1,2,\ldots ,s$ and the elements of any extended standard basis are characterized by the following properties

• (1) $\nu _E(\omega _i)=t_i$ and $\nu _{\mathcal {C}}(\omega _i)\notin \Lambda _{i-1}^{\mathcal {C}}$, for $i=-1,0,\ldots ,s$.

• (2) $\nu _E(\omega _{s+1})=t_{s+1}$ and $\nu _{\mathcal {C}}(\omega _{s+1})=\infty$.

Of course, the above properties assure that $\nu _{{\mathcal {C}}}(\omega _i)=\lambda _i$.

From the geometrical viewpoint, for each $i=1,2,\ldots ,s+1$, the elements $\omega _i$ of an extended standard basis are what we call basic and resonant. This property implies that the transform ${{\tilde{\omega }}}_i$ of the 1-form $\omega _i$ by the morphism $\pi$ of reduction of singularities of ${\mathcal {C}}$ has two remarkable properties:

• (a) The greatest common divisor of the coefficients of ${{\tilde{\omega }}}_i$ defines a normal crossings divisor at the points of E contained in the exceptional divisor of the morphism $\pi$.

• (b) The divisor E is dicritical (not invariant) for the foliation given by ${{\tilde{\omega }}}_i=0$. Moreover, this foliation is nonsingular and it has normal crossings with the exceptional divisor of $\pi$ at the points of E.

As a consequence of this, given an extended standard basis, we find a dicritical package $\{{\mathcal {C}}^i_P\}$ of cusps for each $i=1,2,\ldots ,s+1$ parameterized by the points $P\in E$ that are not corners of the exceptional divisor (that is, elements of ${\mathbb {C}}^*$). Each ${\mathcal {C}}^i_P$ corresponds to the invariant curve of ${{\tilde{\omega }}}_i=0$ through the point P. In particular, if $P_0$ is the infinitely near point of ${\mathcal {C}}$ at E, we have that ${\mathcal {C}}^{s+1}_{P_0}={\mathcal {C}}$. In a terminology inspired in Equisingularity Theory and Reduction of Singularities (see for instance Abhyankar and Moh 1973a, b; Wall 2004; Seidenberg 1968 for the case of foliations), we could say that $\{{\mathcal {C}}^i_{P_0}\}$ are the specific analytic semiroots and that $\{{\mathcal {C}}^i_{P}\}$ are the general analytic semiroots of ${\mathcal {C}}$ associated to the given extended standard basis.

The property of E to be dicritical for the 1-forms $\omega _i$ has been suggested to us by M. E. Hernandes. We have a work in progress with him in this direction (Corral et al. 2023).

The main objective of this paper is to describe the semimodule and extended standard bases of the analytic semiroots. The statement is the following one:

Theorem 1.1

Let $\Lambda ^{\mathcal {C}}=\cup _{j=-1}^{s}(\lambda _i+\Gamma )$ be the semimodule of differential values and consider an extended standard basis

$$\begin{aligned} \omega _{-1}=dx,\omega _0=dy,\omega _1,\ldots ,\omega _{s+1} \end{aligned}$$

of the cusp ${\mathcal {C}}$. Take an index $i\in \{1,2,\ldots ,s+1\}$ and an analytic semiroot ${\mathcal {C}}^{i}_P$ of ${\mathcal {C}}$ associated to the given extended standard basis. Then the semimodule of differential values of ${\mathcal {C}}^i_P$ is precisely $\Lambda _{i-1}^{\mathcal {C}}$ and

$$\begin{aligned} \omega _{-1}=dx,\omega _0=dy,\omega _1,\ldots ,\omega _{i} \end{aligned}$$

is a extended standard basis for ${\mathcal {C}}^i_P$.

The proof of this result uses as a main tool Delorme’s decomposition of the elements of a standard basis. In the appendices, we provide proofs, using a different approach to the one of Delorme, of the structure results for the semimodule of differential values and of Delorme’s decomposition.

Let us remark that it is possible to have curves of the dicritical package of the elements $\omega _j$, when $j\ge 2$, of an extended standard basis that are not analytically equivalent, although they have the same semimodule of differential values. This occurs for instance if we compute a standard basis for the curve

$$\begin{aligned} t\mapsto (t^7, t^{17}+t^{30}+t^{33}+t^{36}). \end{aligned}$$

as shown in Example 8.13. A natural question arises about “how many” analytic classes may be obtained in this way.

2 Cusps and Cuspidal Divisors

We are interested in the analytic moduli of branches with only one Puiseux pair, the analytic cusps. The last divisor of the minimal reduction of singularities of an analytic cusp is what we call a cuspidal divisor. As we shall see below, the study of the analytic moduli may be done through a fixed cuspidal divisor.

2.1 Cuspidal Sequences of Blowing-ups

Our ambient space is a two-dimensional germ of nonsingular complex analytic space $(M_0,P_0)$. We are going to consider a specific type of finite sequences of blowing-ups centered at points, that we call cuspidal sequences of blowing-ups and we introduce below.

First of all, let us establish some notations concerning a nonempty finite sequence of blowing-ups centered at points

$$\begin{aligned} {\mathcal {S}}=\{\pi _k:(M_k,K_k)\rightarrow (M_{k-1},K_{k-1});\; k=1,2,\ldots ,N\}, \end{aligned}$$

starting at $(M_0,P_0)=(M_0,K_0)$. For any $k=1,2,\ldots ,N$, the center of $\pi _k$ is denoted by $P_{k-1}$, note that $P_{k-1}\in K_{k-1}$. We denote the intermediary morphisms as $\sigma _k:(M_k,K_k)\rightarrow (M_0,P_0)$ and $\rho _k: (M_N,K_N)\rightarrow (M_k,K_k)$, where

$$\begin{aligned} \sigma _k=\pi _1\circ \pi _2\circ \cdots \circ \pi _k,\quad \rho _k=\pi _{k+1}\circ \pi _{k+2}\circ \cdots \circ \pi _N. \end{aligned}$$

We denote the exceptional divisor of $\pi _k$ as $E^k_k=\pi _{k}^{-1}(P_{k-1})$. By induction, for any $1\le j<k$ we denote by $E^k_j\subset M_k$ the strict transform of $E^{k-1}_j$ by $\pi _k$. In this way we have that

$$\begin{aligned} K_k =\sigma _{k}^{-1}(P_0)=E^k_1\cup E^k_2\cup \cdots \cup E^k_k. \end{aligned}$$

For any $P\in K_k$, we define $e(P)=\#\{j; P\in E^k_j\}$. Note that $e(P)\in \{1,2\}$. If $e(P)=1$, we say that P is a free point and if $e(P)=2$ we say that it is a corner point. Note that all the points in $E^1_1=K_1$ are free points. The last divisor $E^N_N$ will be denoted $E=E^N_N$. We will also denote $M=M_N$, $K=K_N$ and $\pi =\sigma _N:(M,K)\rightarrow (M_0,P_0)$.

Definition 2.1

Following usual Hironaka’s terminology, we say that the sequence ${\mathcal {S}}$ is a bamboo if $P_k\in E^k_k$ for any $k=1,2,\ldots ,N-1$. We say that ${\mathcal {S}}$ is a cuspidal sequence if it is a bamboo and $e(P_{k-1})\le e(P_{k})$, for any $2\le k\le N-1$. The last divisor E of a cuspidal sequence is called a cuspidal divisor.

Remark 2.2

In the frame of Algebraic Geometry, the cuspidal divisor E corresponds to a valuation $\nu _E$ of the field of rational functions and it determines completely the cuspidal sequence, once the starting ambient space is fixed. We will work with this valuation, but we present it in a direct way.

Given a cuspidal sequence ${\mathcal {S}}$ with $N\ge 2$, there is well defined index of freeness f with $1\le f \le N-1$ such that $P_1,P_2,\ldots , P_f$ are free points and $P_{f+1},P_{f+2},\ldots ,P_{N-1}$ are corner points. If $N=1$ we put $f=0$. A nonsingular branch $(Y,P_0)\subset (M_0,P_0)$ has maximal contact with ${\mathcal {S}}$ if and only if $P_k$ is an infinitely near point of $(Y,P_0)$ for each $k=1,2,\ldots ,P_f$.

Remark 2.3

For any cuspidal sequence ${\mathcal {S}}$ there is at least one nonsingular branch $(Y,P_0)$ having maximal contact with ${\mathcal {S}}$. Moreover, if $(Y,P_0)$ has maximal contact with ${\mathcal {S}}$ and $(Y',P_0)$ is another nonsingular branch, we have that $(Y',P_0)$ has maximal contact with ${\mathcal {S}}$ if and only if $i_{P_0}(Y,Y')\ge f+1$, where $i_{P_0}(Y,Y')$ stands for the intersection multiplicity.

We define intermediate cuspidal sequences of a cuspidal sequence ${\mathcal {S}}$ as follows. Given an index $0\le j\le N-1$, the intermediate $j\text {th}$-cuspidal sequence ${\mathcal {S}}^{(j)}$ of ${\mathcal {S}}$ is the sequence of length $N-j$, starting at $(M_j,P_j)$ such that the blowing ups

$$\begin{aligned} \pi _k^{(j)}: (M_{k+j},K_{k}^{(j)})\rightarrow (M_{k+j-1}, K^{(j)}_{k-1}),\quad k=1,2,\ldots ,N-j \end{aligned}$$

are obtained by restriction from $\pi _{k+j}$, where we put $K^{(j)}_0=\{P_j\}$ and $K_{k}^{(j)}\subset K_{k+j}$ is the image inverse of $P_j$ by $\pi _{j+1}\circ \pi _{j+2}\circ \cdots \circ \pi _{j+k}$.

Remark 2.4

Note that the (k, i)-divisor of ${\mathcal {S}}^{(j)}$ corresponds to the $(k+j,i+j)$ divisor of ${\mathcal {S}}$. In particular the last divisors of ${\mathcal {S}}^{(j)}$ and ${\mathcal {S}}$ are both equal to E.

The Puiseux pair (n, m) of ${\mathcal {S}}$ is defined by an inductive process that corresponds to Euclides’ algorithm as follows. If $N=1$, we put $(n,m)=(1,1)$. If $N>1$, we consider the intermediate cuspidal sequence ${\mathcal {S}}^{(1)}$ starting at $(M_1,P_1)$ that is supposed to have Puiseux pair $(n_1,m_1)$. Then

• (1) If $f\ge 2$, we have that $f_1=f-1$ and we put $(n,m)=(n_1,m_1+n_1)$.

• (2) If $f=1$, we put $(n,m)=(m_1,n_1+m_1)$.

We see that $1\le n\le m$ and n, m are without common factor. Note also that $f\ge 2$ if and only if $m\ge 2n$. Moreover, if $f=1$ and $N\ge 2$, we have that $2\le n<m<2n$.

Proposition 2.5

Consider $1\le n\le m$ without common factor and a nonsingular branch $(Y,P_0)\subset (M_0,P_0)$. There is a unique cuspidal sequence ${\mathcal {S}}$ starting at $(M_0,P_0)$ having maximal contact with $(Y,P_0)$ and such that (n, m) is the Puiseux pair of ${\mathcal {S}}$.

Proof

If $n=m=1$, the only possibility is that $N=1$ and then ${\mathcal {S}}$ consists in the blowing-up of $P_0$. Let us proceed by induction on $n+m$ and assume that $n+m>2$. We necessarily have that $N\ge 2$, the first blowing-up $\pi _1$ is centered in $P_0$ and $P_1$ is the infinitely near point of Y in $E^1_1$.

Assume first that $2n\le m$. We apply induction to $(Y_1,P_1)$ with respect to the pair $n',m'$ where $n'=n$, $m'=m-n$ and we obtain a cuspidal sequence ${\mathcal {S}}'$ over $(M_1,P_1)$ of length $N'$ with the required properties. We construct ${\mathcal {S}}$ of length $N=N'+1$ by taking $\pi _k$ centered at the point $P'_{k-2}$, for $k=2,3,\ldots ,N'+1$.

In the case that $n\le m<2n$ we consider the branch $(Y'_1,P_1)=(E^1_1,P_1)$, we apply induction to $(Y'_1,P_1)$ with respect to the pair $n',m'$ where $n'=m-n$, $m'=n$ and we obtain a cuspidal sequence ${\mathcal {S}}'$ over $(M_1,P_1)$ of length $N'$. We construct ${\mathcal {S}}$ os length $N=N'+1$ as before.

The uniqueness of ${\mathcal {S}}$ follows by an inductive invoking of the uniqueness after one blowing-up. $\square$

We denote by ${\mathcal {S}}^{n,m}_Y$ the sequence obtained in Proposition 2.5. Recall that $Y'$ has maximal contact with ${\mathcal {S}}^{n,m}_Y$ if and only if $i_{P_0}(Y,Y')\ge f+1,$ and hence in this case we have that ${\mathcal {S}}^{n,m}_Y={\mathcal {S}}^{n,m}_{Y'}$. Note also that given a cuspidal sequence ${\mathcal {S}}$ there is a nonsingular branch $(Y,P_0)$ and a Puiseux pair (n, m) in such a way that ${\mathcal {S}}={\mathcal {S}}^{n,m}_Y$.

2.2 Coordinates Adapted to a Cuspidal Sequence of Blowing-ups

Consider a cuspidal sequence ${\mathcal {S}}$ over $(M_0,P_0)$. A system (x, y) of local coordinates at $P_0$ is adapted to ${\mathcal {S}}$ if and only if $y=0$ has maximal contact with ${\mathcal {S}}$. In particular, we have that ${\mathcal {S}}={\mathcal {S}}^{n,m}_{y=0}$, where (n, m) is the Puiseux pair of ${\mathcal {S}}$.

The blowing-ups of ${\mathcal {S}}$ have a monomial expression in terms of adapted coordinates as we see below. Assume that ${\mathcal {S}}={\mathcal {S}}^{n,m}_{y=0}$, with $N\ge 2$. Let us describe a local coordinate system $(x_1,y_1)$ at $P_1$ and a pair $(n_1,m_1)$:

• If $f\ge 2$, we know that $2n\le m$ and we put

  $$\begin{aligned} n_1=n,\quad m_1=m-n, \quad x=x_1, \quad y=x_1y_1. \end{aligned}$$

• If $f=1$, we have that $2n>m>n\ge 2$ and we put

  $$\begin{aligned} n_1=m-n,\quad m_1=n, \quad y=x_1y_1, \quad x=y_1. \end{aligned}$$

The reader can verify that $(x_1,y_1)$ is a coordinate system adapted to ${\mathcal {S}}^{(1)}$ and that $(n_1,m_1)$ is its Puiseux pair. In this way, we have local coordinates $x_j,y_j$ at each $P_j$, for $0\le j\le N-1$.

Once we have an adapted coordinate system (x, y), we denote $(H_0,P_0)$ the normal crossings germ given by $xy=0$. Define $H_j=\sigma _j^{-1}(H_0)$, then the germ of $H_j$ at $P_j$ is given by $x_jy_j=0$, for any $0\le j\le N-1$. We can also consider $H=\pi ^{-1}(H_0)\subset M$; it is a normal crossings divisor of (M, K) containing K.

2.3 Cuspidal Analytic Module

Consider a cuspidal sequence ${\mathcal {S}}$ with Puiseux pair (n, m) with $2\le n$. Let E be the last divisor of ${\mathcal {S}}$. We say that an analytic branch $({\mathcal {C}},P_0)\subset (M_0,P_0)$ is an E-cusp, or a ${\mathcal {S}}$-cusp if the strict transform of $({\mathcal {C}},P_0)$ under the sequence of blowing-ups $\pi$ is nonsingular and cuts transversely E at a free point. Let us denote by ${\text {Cusps}}(E)= {\text {Cusps}}({\mathcal {S}})$ the family of E-cusps.

Each element of ${\text {Cusps}}({\mathcal {S}})$ is equisingular to the irreducible cusp $y^n-x^m=0$, where (n, m) is the Puiseux pair of ${\mathcal {S}}$. Moreover, we have the following result

Proposition 2.6

Consider a cuspidal sequence ${\mathcal {S}}$ with Puiseux pair (n, m) and last divisor E. Let $({{\mathcal {C}}},P_0)$ be a branch of $(M_0,P_0)$ equisingular to the irreducible cusp $y^n-x^m=0$. There is an E-cusp analytically equivalent to $({{\mathcal {C}}},P_0)$.

Proof

Choose a local coordinate system (x, y) adapted to ${\mathcal {S}}$.

If $n=1$, the branch $({{\mathcal {C}}},P_0)$ is nonsingular. Then, there is an automorphism $\phi :(M_0,P_0)\rightarrow (M_0,P_0)$ such that $\phi ({{\mathcal {C}}})=(y=0)$. We are done since in this case $y=0$ is an E-cusp.

Assume that $2\le n<m$. In view of the classical arguments of Hironaka (see for instance Aroca et al. 2018, p. 105), there is a nonsingular branch $(Z,P_0)$ having maximal contact with $({{\mathcal {C}}},P_0)$, that is with the property that

$$\begin{aligned} i_{P_0}(Z,{{\mathcal {C}}})= m. \end{aligned}$$

Take an automorphism $\phi :(M_0,P_0)\rightarrow (M_0,P_0)$ such that $\phi (Z)=(y=0)$. We have that $(\phi ({\mathcal {C}}),P_0)$ is an E-cusp. $\square$

According to the above result, the analytic moduli of the family of branches equisingular to the irreducible cusp $y^n-x^m=0$ is faithfully represented by the analytic moduli of the family ${\text {Cusps}}({\mathcal {S}})$.

Along the rest of this paper, we consider a fixed cuspidal sequence ${\mathcal {S}}$ where (n, m) is its Puiseux pair and E is the last divisor. Recall also that the composition of all the blowing-ups of ${\mathcal {S}}$ is denoted by

$$\begin{aligned} \pi :(M,K)\rightarrow (M_0,P_0). \end{aligned}$$

We also choose a local coordinate system (x, y) adapted to ${\mathcal {S}}$.

3 Divisorial Order

Consider a holomorphic function h in (M, K) defined globally in $E\subset K$, the divisorial order $\nu _E(h)$ of h is obtained as follows. Take a point $P\in E$ and choose a reduced local equation $u=0$ of the germ (E, P), then

$$\begin{aligned} \nu _E(h)=\max \{a\in {\mathbb {Z}};\; u^{-a}h\in {\mathcal {O}}_{M,P}\}. \end{aligned}$$

This definition does not depend on the chosen point $P\in E$ nor on the local reduced equation of E. Take a point $P_j$, with $j\in \{0,1,\ldots ,N-1\}$ and a germ of holomorphic function $h\in {\mathcal {O}}_{M_j,P_j}$. Then $\rho _j^*h$ is a germ of function in (M, K) globally defined in E. We define the divisorial order $\nu _E(h)$ by $\nu _E(h)=\nu _E(\rho _j^*h)$.

Proposition 3.1

Consider a germ $h\in {\mathcal {O}}_{M_0,P_0}$ that we write as

$$\begin{aligned} h=\sum _{\alpha ,\beta }h_{\alpha ,\beta }x^\alpha y^\beta ,\quad h_{\alpha \beta }\in {\mathbb {C}}. \end{aligned}$$

Then $\nu _E(h)=\min \{n\alpha +m\beta ;\; h_{\alpha ,\beta }\ne 0\}$.

Proof

If $n=m=1$ we have a single blowing-up and we recover the usual multiplicity, that we visualize in E as $\nu _E(h)$. Let us work by induction on $n+m$ and assume that $n+m\ge 2$. We remark that $\nu _E(h)=\nu _E(\pi _1^*h)$. Consider the first intermediate sequence ${\mathcal {S}}^{(1)}$ of ${\mathcal {S}}$, with adapted coordinates $(x_1,y_1)$. Recalling how we obtain intermediate coordinate systems, we conclude that

$$\begin{aligned} \pi _1^*h= & {} \sum _{\alpha ,\beta }h_{\alpha ,\beta }x_1^{\alpha +\beta } y_1^\beta ; \text { if } f\ge 2; \text { here } n_1=n, m_1=m-n,\\ \pi _1^*h= & {} \sum _{\alpha ,\beta }h_{\alpha ,\beta }x_1^{\beta } y_1^{\alpha +\beta }; \text { if } f=1; \text { here } n_1=n-m, m_1=n. \end{aligned}$$

We end by applying induction hypothesis. $\square$

3.1 Divisorial Order of a Differential Form

Recall that we denote

$$\begin{aligned} H_0=(xy=0)\subset M_0, \quad H_j=\sigma _j^{-1}(H_0)\subset M_j \end{aligned}$$

and that $H_j$ is locally given at $P_j$ by $x_iy_j=0$ for $0\le j\le N-1$. We also consider $H_N=H=\pi ^{-1}(H_0)\subset M$. Each $H_j$ is a normal crossings divisor in $(M_j,K_j)$, containing $K_j$.

Take a point $Q\in K_j$, not necessarily equal to $P_j$, in particular we consider also the case $j=N$. Select local coordinates (u, v) such that $(u=0)\subset H_j\subset (uv=0)$, then we have that either $H_j=(u=0)$ or $H_j=(uv=0)$ locally at Q. The ${\mathcal {O}}_{M_j,Q}$-module $\Omega ^1_{M_j,Q}[\log H_j]$ of germs of $H_j$-logarithmic 1-forms is the rank two free ${\mathcal {O}}_{M_j,Q}$-module generated by

$$\begin{aligned} \begin{array}{ll} du/u, dv&{}\quad \text {if } H_j=(u=0),\\ du/u, dv/v&{}\quad \text {if }H_j=(uv=0). \end{array} \end{aligned}$$

Note that $\Omega ^1_{M_j,Q}\subset \Omega ^1_{M_j,Q}[\log H_j]$. Indeed, a differential 1-form $\omega =adu+bdv$ may be written as

$$\begin{aligned} \omega =ua\frac{du}{u}+bdv=ua\frac{du}{u}+vb\frac{dv}{v}. \end{aligned}$$

Now, let us consider a 1-form $\omega \in \Omega ^1_{M}[\log H]$ defined in the whole divisor E (we suppose that the reader recognizes the sheaf nature of $\Omega ^1_{M}[\log H]$). Select a point $Q\in E$ and a local reduced equation $u=0$ of E at Q. We define the divisorial order $\nu _E(\omega )$ by

$$\begin{aligned} \nu _E(\omega )=\max \{\ell \in {\mathbb {Z}};\; u^{-\ell }\omega \in \Omega ^1_{M,Q}[\log H]\}. \end{aligned}$$

The definition is independent of $Q\in E$ and of the reduced local equation of E.

Remark 3.2

Let $\omega \in \Omega ^1_{M}[\log E]$ be globally defined on E as before. Since E is one of the irreducible components of H, we have that

$$\begin{aligned} \Omega ^1_{M}[\log E]\subset \Omega ^1_{M}[\log H]. \end{aligned}$$

Let us choose a reduced local equation $u=0$ of E at a point $Q\in E$ as before. A direct computation shows that

$$\begin{aligned} \nu _E(\omega )= \max \{\ell \in {\mathbb {Z}};\; u^{-\ell }\omega \in \Omega ^1_{M,Q}[\log E]\}. \end{aligned}$$

(1)

This remark shows that the divisorial order, applied to 1-forms $\omega \in \Omega ^1_{M}[\log E]$ is independent of the choice of the adapted coordinate system that defines $H_0$. Anyway, this is only a remark for the case $n=1$, since when $n\ge 2$ the divisor H at the points of E is itself independent of the adapted coordinate system.

Definition 3.3

For any $\omega \in \Omega ^1_{M_j,P_j}$, the divisorial order $\nu _E(\omega )$ is defined by $\nu _E(\omega )=\nu _E(\rho _{j}^*\omega ).$

Proposition 3.4

Consider a differential 1-form $\omega =adx+bdy\in \Omega ^1_{M_0,P_0}$, that we can write as

$$\begin{aligned} \omega =xa({dx}/{x})+yb({dy}/{y})\in \Omega ^1_{M_0,P_0}[\log H_0]. \end{aligned}$$

Then, we have that $\nu _E(\omega )=\min \{\nu _E(xa),\nu _E(yb)\}$.

Proof

Write $\omega =f(dx/x)+g(dy/y)$. We proceed by induction on N. If $N=1$ we have that $E=(x'=0)$ where $x=x',y=x'y'$ and

$$\begin{aligned} \pi _1^*\omega =(f+g)(dx'/x')+g(dy'/y'). \end{aligned}$$

Then $\nu _E(\omega )= \min \{\nu _E(f+g),\nu _E(g)\}= \min \{\nu _E(f),\nu _E(g)\}$ and we are done. If $N\ge 2$, we have that

$$\begin{aligned} \nu _E(\omega )=\nu _E(\pi ^*\omega )= \nu _E(\rho _1^*(\pi _1^*\omega ))= \nu _E(\pi _1^*\omega ). \end{aligned}$$

By induction hypothesis, we have

$$\begin{aligned} \nu _E(\pi _1^*\omega )= \min \{\nu _E(f+g),\nu _E(g)\}= \min \{\nu _E(f),\nu _E(g)\} \end{aligned}$$

and we are done as before. $\square$

Corollary 3.5

If $f\in {\mathcal {O}}_{M_0,P_0}$ and $\omega =df$, then $\nu _E(\omega )=\nu _E(f)$.

Proof

It is enough to write $df=x({\partial f}/{\partial x})({dx}/{x})+ y({\partial f}/{\partial y})({dy}/{y}),$ recalling Euler’s identity $gP=xP_x+yP_y$ for degree g homogeneous polynomials. $\square$

3.2 Weighted Initial Parts

Consider a nonzero germ $h\in {\mathcal {O}}_{M_0,P_0}$, that we write as $h=\sum _{\alpha ,\beta }h_{\alpha \beta }x^\alpha y^\beta$. Suppose that $q\le \nu _E(h)$. We define the weighted initial part ${\text {In}}^q_{n,m; x,y}(h)$ by

$$\begin{aligned} {\text {In}}^q_{n,m; x,y}(h)=\sum _{n\alpha +m\beta =q}h_{\alpha \beta }x^\alpha y^\beta . \end{aligned}$$

Note that ${\text {In}}^q_{n,m; x,y}(h)=0$ if and only if $q<\nu _E(h)$. Anyway, we can write

$$\begin{aligned} h={\text {In}}^q_{n,m; x,y}(h)+{{\tilde{h}}},\quad \nu _E({{\tilde{h}}})>q. \end{aligned}$$

This definition extends to logarithmic differential 1-forms $\omega \in \Omega ^1_{M_0,P_0}[\log (xy=0)]$ as follows. Take $q\le \nu _E(\omega )$. Write $\omega =f(dx/x)+g(dy/y)$. We define

$$\begin{aligned} {\text {In}}^q_{n,m; x,y}(\omega )= {\text {In}}^q_{n,m; x,y}(f)(dx/x)+{\text {In}}^q_{n,m; x,y}(g)(dy/y). \end{aligned}$$

As before, we have $\omega = {\text {In}}^q_{n,m; x,y}(\omega )+{{\tilde{\omega }}}$, with $\nu _E({{\tilde{\omega }}})>q$.

Remark 3.6

The definition of initial part we have presented should be made in terms of graduated rings and modules to be free of coordinates. Anyway, this “coordinate-based” definition is enough for our purposes.

Proposition 3.7

Assume that $N>1$, take $\omega \in \Omega ^1_{M_0,P_0}[\log (xy=0)]$ and $q\in {\mathbb {Z}}_{\ge 0}$ with $q\le \nu _E(\omega )$. If $W={\text {In}}^q_{n,m; x,y}(\omega )$, then $\pi _1^*(W)={\text {In}}^q_{n_1,m_1; x_1,y_1}(\pi _1^*\omega )$.

Proof

Left to the reader. $\square$

4 Total Cuspidal Dicriticalness

This section is devoted to characterize the 1-forms $\omega \in \Omega ^1_{M_0,P_0}$ whose transform $\pi ^*\omega$ defines a foliation that is transversal to E and has normal crossings with K at each point of E. These 1-forms are the so-called pre-basic and resonant 1-forms. We detect these properties in terms of resonances of the initial part. The initial part is visible in the Newton polygon as the contribution of the 1-form to a single vertex (a, b), under the condition that the Newton polygon is contained in the particular region $R^{n,m}(a,b)$.

4.1 Reduced Divisorial Order and Basic Forms

Let us consider a nonnull differential 1-form $\omega \in \Omega ^1_{M_0,P_0}$. Let $V_\omega =x^a y^b$ be the monomial defined by the property that $\omega =V_\omega \eta$, where $\eta \in \Omega ^1_{M_0,P_0}[\log (xy=0)]$ is a logarithmic form that cannot be divided by any nonconstant monomial. We define the reduced divisorial order ${\text {rdo}}_E(\omega )$ to be ${\text {rdo}}_E(\omega )=\nu _E(\eta )$.

Definition 4.1

We say that $\omega \in \Omega ^1_{M_0,P_0}$ is a basic 1-form if and only if its reduced divisorial order satisfies that ${\text {rdo}}_E(\omega )<nm$.

Proposition 4.2

Assume that $N\ge 2$ and take $\omega \in \Omega ^1_{M_0,P_0}$. If $\omega$ is a basic 1-form, then $\pi _1^*\omega$ is also a basic 1-form.

Proof

Put $p={\text {rdo}}_E(\omega )=\nu _E(\eta )<nm$. Recall that $\nu _E(\eta )=\nu _E(\pi _1^*\eta )$. Since monomials are well behaved under $\pi _1$, it is enough to show that there are $c,d\ge 0$ such that $\pi _1^*\eta =x_1^{c}y_1^{d}\eta '$, with $\nu _E(\eta ')<n_1m_1$. Write

$$\begin{aligned} \eta =\sum _{\alpha ,\beta } x^{\alpha }y^{\beta }\eta _{\alpha \beta },\quad \eta _{\alpha \beta }= \mu _{\alpha \beta }\frac{dx}{x}+\zeta _{\alpha \beta }\frac{dy}{y}, \quad (\mu _{\alpha \beta },\zeta _{\alpha \beta })\in {\mathbb {C}}^2. \end{aligned}$$

Recall that $p=\min \{n\alpha +m\beta ;\; \eta _{\alpha \beta }\ne 0\}$. Put $r=\min \{\alpha +\beta ;\; \eta _{\alpha \beta }\ne 0\}$. We have two cases: $f=1$ and $f\ge 2$, where f is the index of freeness.

Assume first that $f\ge 2$ and hence $2n\le m$. In this situation, we have that $x=x_1$, $y=x_1y_1$, $n_1=n$, $m_1=m-n\ge n$ and

$$\begin{aligned} \pi _1^*(\eta )=x_1^r\eta ',\quad \eta '=\sum _{\alpha ,\beta }x_1^{\alpha +\beta -r}y_1^\beta \eta '_{\alpha \beta },\quad \eta '_{\alpha \beta }=\left( \mu _{\alpha \beta }+\zeta _{\alpha \beta }\right) \frac{dx_1}{x_1} +\zeta _{\alpha \beta }\frac{dy_1}{y_1}. \end{aligned}$$

Note that $\eta '_{\alpha \beta }\ne 0$ if and only if $\eta _{\alpha \beta }\ne 0$. Hence

$$\begin{aligned} \nu _E(\eta ')= & {} \min \{n_1(\alpha +\beta -r)+m_1\beta ;\eta _{\alpha \beta }\ne 0\}\\= & {} \min \{n (\alpha +\beta -r)+(m-n)\beta ;\eta _{\alpha \beta }\ne 0\}\\= & {} \min \{n\alpha +m\beta -nr;\eta _{\alpha \beta }\ne 0\}=p-nr. \end{aligned}$$

We have to verify that $p-nr<n_1m_1$, where $n_1m_1=n(m-n)=nm-n^2$. If $r\ge n$, we are done, since by hypothesis we have that $p<nm$. Assume that $r<n$. There are ${{\tilde{\alpha }}},{{\tilde{\beta }}}$ with $\eta _{{{\tilde{\alpha }}}{{\tilde{\beta }}}}\ne 0$ such that ${{\tilde{\alpha }}}+{{\tilde{\beta }}}=r$. Then

$$\begin{aligned} p-nr\le & {} n{{\tilde{\alpha }}}+m{{\tilde{\beta }}}-nr =n({{\tilde{\alpha }}}+{{\tilde{\beta }}})+(m-n){{\tilde{\beta }}}-nr\\= & {} (m-n){{\tilde{\beta }}}<(m-n)n, \end{aligned}$$

since ${{\tilde{\beta }}}\le r<n$.

Assume that $f=1$ and thus $n<m<2n$. We have $x=y_1$, $y=x_1y_1$, $n_1=m-n<n$, $m_1=n$ and

$$\begin{aligned} \pi _1^*(\eta )=y_1^r\eta '',\quad \eta ''=\sum _{\alpha ,\beta }x_1^{\beta } y_1^{\alpha +\beta -r}\eta ''_{\alpha \beta },\quad \eta ''_{\alpha \beta }=\zeta _{\alpha \beta }\frac{dx_1}{x_1} +\left( \mu _{\alpha \beta }+\zeta _{\alpha \beta }\right) \frac{dy_1}{y_1}. \end{aligned}$$

As before, we have that $\eta ''_{\alpha \beta }\ne 0$ if and only if $\eta _{\alpha \beta }\ne 0$. Hence

$$\begin{aligned} \nu _E(\eta '')= & {} \min \{n_1\beta +m_1(\alpha +\beta -r);\; \eta _{\alpha \beta }\ne 0\}\\= & {} \min \{(m-n) \beta +n(\alpha +\beta -r);\; \eta _{\alpha \beta }\ne 0\}\\= & {} \min \{m\beta +n\alpha -nr;\; \eta _{\alpha \beta }\ne 0\}=p-nr. \end{aligned}$$

We verify that $p-nr<n_1m_1$ exactly as before. $\square$

4.2 Resonant Basic Forms

Let $\omega \in \Omega ^1_{M_0,P_0}$ be a basic 1-form with $p={\text {rdo}}_E(\omega )$. This means that there is $\eta \in \Omega ^1_{M_0,P_0}[\log (xy=0)]$ and $a,b\ge 0$ such that $\omega =x^ay^b\eta$

$$\begin{aligned} \omega =x^ay^b\eta , \quad \eta \in \Omega ^1_{M_0,P_0}[\log (xy=0)], \end{aligned}$$

where $p=\nu _E(\eta )<nm$. The initial part of $\omega$ may be written

$$\begin{aligned} {\text {In}}^{p+na+mb}_{n,m;x,y}(\omega )=x^ay^b W, \quad W={\text {In}}^{p}_{n,m;x,y}(\eta ). \end{aligned}$$

Note that there is exactly one pair $(c,d)\in {\mathbb {Z}}^2_{\ge 0}$ such that $cn+dm=p$. Then we have that

$$\begin{aligned} W=x^cy^d\left\{ \mu \frac{dx}{x}+\zeta \frac{dy}{y}\right\} . \end{aligned}$$

We say that $\omega$ is resonant if and only if $n\mu +m\zeta =0$.

We have the following result that follows directly from the computations in the proof of Proposition 4.2.

Corollary 4.3

Assume that $N\ge 2$. A basic differential 1-form $\omega \in \Omega ^1_{M_0,P_0}$ is resonant if and only if $\pi _1^*\omega$ is resonant.

4.3 Pre-Basic Forms

Let us introduce a slightly more general class of 1-forms that we call pre-basic forms. Given a 1-form

$$\begin{aligned} \omega =\sum _{\alpha ,\beta }c_{\alpha \beta }x^\alpha y^\beta \omega _{\alpha \beta },\quad \omega _{\alpha \beta }= \left\{ \mu _{\alpha \beta }\frac{dx}{x}+\zeta _{\alpha \beta }\frac{dy}{y}\right\} , \end{aligned}$$

(2)

the cloud of points ${\text {Cl}}(\omega ;x,y)$ is ${\text {Cl}}(\omega ;x,y)= \{(\alpha ,\beta );\omega _{\alpha \beta }\ne 0\}$ and the Newton Polygon ${\mathcal {N}}(\omega ;x,y)$ is the positive convex hull of ${\text {Cl}}(\omega ;x,y)$ in ${\mathbb {R}}^2_{\ge 0}$.

Consider a pair (n, m) with $1\le n\le m$ such that n, m have no common factor. There are unique $b,d\in {\mathbb {Z}}_{\ge 0}$ such that $dn-bm=1$ with the property that $0\le b<n$ and $0<d\le m$. We call (b, d) the co-pair of (n, m).

Remark 4.4

Suppose that $1\le n\le m$ are without common factor and take b, d such that $dn-bm=1$. If $0\le b<n$, we have that $0<d\le m$ and then (b, d) is the co-pair of (n, m). In the same way, if $0<d\le m$, we have that $0\le b<n$ and then (b, d) is the co-pair of (n, m).

Definition 4.5

Given a pair $1\le n\le m$ without common factor, we define the region $R^{n,m}$ by $R^{n,m}= H^{n,m}_-\cap H^{n,m}_+$, where

$$\begin{aligned} H^{n,m}_-= & {} \{(\alpha ,\beta )\in {\mathbb {R}}^2;\; (n-b)\alpha +(m-d)\beta \ge 0\},\\ H^{n,m}_+= & {} \{(\alpha ,\beta )\in {\mathbb {R}}^2; \; b\alpha +d\beta \ge 0\}, \end{aligned}$$

and (b, d) is the co-pair of (n, m).

Remark 4.6

If $n=m=1$, the co-pair of (1, 1) is $(b,d)=(0,1)$. Then

$$\begin{aligned} H^{1,1}_-=\{(\alpha ,\beta );\; \alpha \ge 0\},\quad H^{1,1}_+=\{(\alpha ,\beta );\; \beta \ge 0\}. \end{aligned}$$

Thus, we have that $R^{1,1}$ is the quadrant $R^{1,1}={\mathbb {R}}^2_{\ge 0}$.

Remark 4.7

The slopes $-(n-b)/(m-d)$ and $-b/d$ satisfy that

$$\begin{aligned} -(n-b)/(m-d)<-n/m<-b/d. \end{aligned}$$

Indeed, we have $-n/m<-b/d \Leftrightarrow -dn<-mb=-dn+1$. On the other hand

$$\begin{aligned} -(n-b)/(m-d)<-n/m\Leftrightarrow & {} m(n-b)>n(m-d) \Leftrightarrow bm<dn=bm+1. \end{aligned}$$

We conclude that $R^{n,m}$ is a positively convex region of ${\mathbb {R}}^{2}$ such that (0, 0) is its only vertex and we have that

$$\begin{aligned} R^{n,m}\cap \{(\alpha ,\beta )\in {\mathbb {R}}^2;\; n\alpha +m\beta =0\}=\{(0,0)\}. \end{aligned}$$

Given a point $(a,b)\in {\mathbb {R}}_{\ge 0}^2$, we define $R^{n,m}(a,b)$ by $R^{n,m}(a,b)= R^{n,m}+(a,b)$.

Definition 4.8

We say that $\omega \in \Omega ^1_{M_0,P_0}$ is pre-basic if and only if there is a point $(a,b)\in {\text {Cl}}(\omega ;x,y)$ such that ${\text {Cl}}(\omega ;x,y)\subset R^{n,m}(a,b)$.

Remark 4.9

Note that $\omega$ is pre-basic if and only if $(a,b)\in {\mathcal {N}}(\omega ;x,y)$ and ${\mathcal {N}}(\omega ;x,y)\subset R^{n,m}(a,b)$.

If $\omega$ is pre-basic, we have that

$$\begin{aligned} {\text {Cl}}(\omega ;x,y) \cap \{(\alpha ,\beta )\in {\mathbb {R}}^2;\; n\alpha +m\beta =\nu _E(\omega )\}=\{(a,b)\}. \end{aligned}$$

Thus, the initial part W of $\omega$ has the form

$$\begin{aligned} W=x^ay^b\left\{ \mu _{ab}\frac{dx}{x}+\zeta _{ab}\frac{dy}{y}\right\} . \end{aligned}$$

(3)

As for basic forms, we say that $\omega$ is resonant if and only if $n\mu _{ab}+m\zeta _{ab}=0$.

Lemma 4.10

Assume that $1\le n<m$, where n, m are without common factor and let (b, d) be the co-pair of (n, m). Let us put $(n_1,m_1)=(n,m-n)$, if $m\ge 2n$ and $(n_1,m_1)=(m-n,n)$, if $m<2n$. Then, the co-pair $(b_1,d_1)$ of $(n_1,m_1)$ is given by $(b_1,d_1)= (b,d-b)$, if $m\ge 2$, and by $(b_1,d_1)=(m-n-d+b,n-b)$, if $m<2n$. Moreover, we have that $\Psi (R^{n,m})=R^{n_1,m_1}$, where $\Psi$ is the linear automorphism of ${\mathbb {R}}^2$ given by $\Psi (\alpha ,\beta )= (\alpha +\beta ,\beta )$, if $m\ge 2n$, and $\Psi (\alpha ,\beta )= (\beta ,\alpha +\beta )$, if $m<2n$.

Proof

Let us show the first statement. If $m\ge 2n$, we have that

$$\begin{aligned} d_1n_1-b_1m_1= (d-b)n-b(m-n)=1. \end{aligned}$$

Moreover, since $0\le b_1=b<n_1=n$ we conclude that $(b_1,d_1)$ is the co-pair of $(n_1,m_1)$, in view of Remark 4.4. If $m<2n$, we have

$$\begin{aligned} d_1n_1-b_1m_1= (n-b)(m-n)-(m-n-d+b)n=1. \end{aligned}$$

We know that $0\le b<n$, hence $0<d_1=n-b\le m_1=n$; by Remark 4.4, we deduce that $(b_1,d_1)$ is the co-pair of $(n_1,m_1)$.

Let us show the second statement. Consider $(\alpha ,\beta )\in {\mathbb {R}}^2$ and put $(\alpha _1,\beta _1)=\Psi (\alpha ,\beta )$.

Case $m\ge 2n$. In order to prove that $\Psi (R^{n,m})=R^{n_1,m_1}$ it is enough to see that

$$\begin{aligned} (\alpha ,\beta )\in H^{n,m}_-\Leftrightarrow (\alpha _1,\beta _1)\in H^{n_1,m_1}_-\text { and } (\alpha ,\beta )\in H^{n,m}_+\Leftrightarrow (\alpha _1,\beta _1)\in H^{n_1,m_1}_+. \end{aligned}$$

We verify these properties as follows:

$$\begin{aligned} (\alpha _1,\beta _1)\in H^{n_1,m_1}_-&\Leftrightarrow (n_1-b_1)\alpha _1+(m_1-d_1)\beta _1\ge 0 \\&\Leftrightarrow (n-b)(\alpha +\beta )+(m-n-d+b)\beta \ge 0 \\&\Leftrightarrow (n-b)\alpha +(m-d)\beta \ge 0 \Leftrightarrow (\alpha ,\beta )\in H^{n,m}_-.\\ (\alpha _1,\beta _1)\in H^{n_1,m_1}_+&\Leftrightarrow b_1\alpha _1+d_1\beta _1\ge 0 \\&\Leftrightarrow b(\alpha +\beta )+(d-b)\beta \ge 0 \\&\Leftrightarrow b\alpha +d\beta \ge 0 \Leftrightarrow (\alpha ,\beta )\in H^{n,m}_+. \end{aligned}$$

Case $m<2n$. In this case, we have that

$$\begin{aligned} (\alpha ,\beta )\in H^{n,m}_+&\Leftrightarrow (\alpha _1,\beta _1)\in H^{n_1,m_1}_- \end{aligned}$$

(4)

$$\begin{aligned} (\alpha ,\beta )\in H^{n,m}_-&\Leftrightarrow (\alpha _1,\beta _1)\in H^{n_1,m_1}_+. \end{aligned}$$

(5)

and this also implies that $\Psi (R^{n,m})=R^{n_1,m_1}$. We verify the properties in Eqs. (4) and (5) as follows:

$$\begin{aligned} (\alpha _1,\beta _1)\in H^{n_1,m_1}_-&\Leftrightarrow (n_1-b_1)\alpha _1+(m_1-d_1)\beta _1\ge 0 \\&\Leftrightarrow (m-n-(m-n-d+b))\beta +(n-n+b)(\alpha +\beta )\ge 0 \\&\Leftrightarrow d\beta +b\alpha \ge 0 \Leftrightarrow (\alpha ,\beta )\in H^{n,m}_+.\\ (\alpha _1,\beta _1)\in H^{n_1,m_1}_+&\Leftrightarrow b_1\alpha _1+d_1\beta _1\ge 0 \\&\Leftrightarrow (m-n-d+b)\beta +(n-b)(\alpha +\beta )\ge 0 \\&\Leftrightarrow (m-d)\beta +(n-b)\alpha \ge 0 \Leftrightarrow (\alpha ,\beta )\in H^{n,m}_-. \end{aligned}$$

The proof is ended. $\square$

Proposition 4.11

Assume that $N\ge 2$. For any $\omega \in \Omega ^1_{M_0,P_0}$, we have

• (1) $\omega$ is pre-basic if and only if $\pi _1^*\omega$ is pre-basic.

• (2) $\omega$ is pre-basic and resonant if and only if $\pi _1^*\omega$ is pre-basic and resonant.

Proof

We consider two cases as in the statement of Lemma 4.10, the case $m\ge 2n$ and $m<2n$ and we define the linear automorphism $\Psi$ accordingly to these cases, as well as the Puiseux pair $(n_1,m_1)$. A monomial by monomial computation shows that

$$\begin{aligned} {\text {Cl}}(\pi _1^*\omega ;x_1,y_1)=\Psi ({\text {Cl}}(\omega ;x,y)). \end{aligned}$$

(6)

In view of Lemma 4.10, we have that

$$\begin{aligned} \Psi (R^{n,m}(a,b))=R^{n_1,m_1}(\Psi (a,b)). \end{aligned}$$

(7)

Statement (1) is now a direct consequence of Eqs. (6) and (7). Property (2) is left to the reader. $\square$

Proposition 4.12

Take a differential 1-form $\omega \in \Omega ^1_{M_0,P_0}$. We have

• (1) If $N=1$, then $\omega$ is pre-basic if and only if it is basic.

• (2) If $\omega$ is basic then it is pre-basic.

• (3) If $\omega$ is basic and resonant then it is pre-basic and resonant.

Proof

If $N=1$, we have $n=m=1$ and $R^{1,1}(a,b)={\mathbb {R}}^2_{\ge 0}+(a,b)$. Then being basic is the same property of being pre-basic: the Newton Polygon has a single vertex.

Assume now that $\omega$ is basic. In view of the stability result in Proposition  4.2, we have that ${{\tilde{\omega }}}$ is basic, where ${{\tilde{\omega }}}$ is the pull-back of $\omega$ in the last center $P_{N-1}$ of the cuspidal sequence. By the previous argument we have that ${{\tilde{\omega }}}$ is pre-basic. Now we apply Proposition  4.11 to conclude that $\omega$ is pre-basic.

The resonance for pre-basic 1-forms that are basic ones is the same property as for basic 1-forms. $\square$

4.4 Totally E-dicritical Forms

Consider a 1-form $\omega \in \Omega ^1_M$ defined around the divisor E. Recall that we have a normal crossings divisor H such that $H\supset E$, coming from our choice of adapted coordinates, although if $n\ge 2$ the divisor H around E is intrinsically defined and it coincides with K. We say that $\omega$ is totally E-dicritical with respect to H if for any point $P\in E$ there are local coordinates u, v such that $E=(u=0)$, $H\subset (uv=0)$ and $\omega$ has the form

$$\begin{aligned} \omega =u^av^bdv, \end{aligned}$$

where $b=0$ when $H=(u=0)$. Note that $\omega$ defines a non-singular foliation around E, this foliation has normal crossings with H and E is transversal to the leaves.

Proposition 4.13

For any $\omega \in \Omega ^1_{M_0,P_0}$, the following properties are equivalent:

• (1) $\pi ^*\omega$ is totally E-dicritical with respect to H.

• (2) The 1-form $\omega$ is pre-basic and resonant.

Proof

In view of the stability of the property “pre-basic and resonant” under the blowing-ups of ${\mathcal {S}}$ given in Proposition 4.11, it is enough to consider the case when $N=1$. In this case we have a single blowing-up, $H_0=(xy=0)$ and the property for $\pi ^*\omega$ of being totally E-dicritical with respect to H is equivalent to say that

$$\begin{aligned} \omega =h(x,y)x^a y^b\left[ \left\{ \frac{dx}{x}-\frac{dy}{y}\right\} + \sum _{\alpha +\beta \ge 1}x^\alpha y^\beta \left\{ \mu _{\alpha \beta }\frac{dx}{x}+\zeta _{\alpha \beta }\frac{dy}{y}\right\} \right] ,\quad a,b\ge 1, \end{aligned}$$

where $h(0,0)\ne 0$. That is, the 1-form $\omega$ is pre-basic and resonant. $\square$

Remark 4.14

If $n\ge 2$ the axes $x'y'=0$ around $P_{N-1}$ coincide with the germ of $K_{N-1}=\sigma _{N-1}^{-1}(P_0)$ at $P_{N-1}$. In this situation, the property of being basic and resonant does not depend on the chosen adapted coordinate system.

Definition 4.15

Given a resonant pre-basic 1-form $\omega$, we say that a branch $({\mathcal {C}},0)$ in $(M_0,P_0)$ is a $\omega$-cusp if and only if it is invariant by $\omega$ and the strict transform of $({\mathcal {C}},P_0)$ by $\pi$ cuts E at a free point.

Let us note that each free point of E defines a $\omega$-cusp and conversely, in view of the fact that $\pi ^*\omega$ is totally E-dicritical with respect to H.

One of the results in this paper is that any element of ${\text {Cusps}}({\mathcal {S}})$ is a $\omega$-cusp for certain resonant basic $\omega$ and hence can be included in the corresponding “dicritical package”.

5 Differential Values of a Cusp

Let us consider a branch $({\mathcal {C}},P_0)\subset (M_0,P_0)$ belonging to ${\text {Cusps}}(E)$. It has a Puiseux expansion of the form

$$\begin{aligned} (x,y)=\phi (t)=(t^n,\alpha t^m+t^{m+1}\xi (t)),\; \alpha \ne 0. \end{aligned}$$

defined by the fact that for any germ $h\in {\mathcal {O}}_{M_0,P_0}$ we have that $({\mathcal {C}},P_0)\subset (h=0)$ if and only if $h\circ \phi =0$. We recall that the intersection multiplicity of $({\mathcal {C}}, P_0)$ with a germ h is given by

$$\begin{aligned} {\text {i}}_{P_0}({\mathcal {C}}, h)={\text {order}}_t(h\circ \phi ). \end{aligned}$$

We also denote $\nu _{\mathcal {C}}(h)={\text {i}}_{P_0}({\mathcal {C}}, h)$. The semigroup $\Gamma$ of ${\mathcal {C}}$ is defined by

$$\begin{aligned} \Gamma \cup \{\infty \}=\{\nu _{\mathcal {C}}(h);\; h\in {\mathcal {O}}_{M_0,P_0}\}. \end{aligned}$$

As stated in Zariski’s Equisingularity Theory, this semigroup depends only on the equisingularity class (or topological class) of ${\mathcal {C}}$. In our case, we know that all the elements in ${\text {Cusps}}(E)$ are equisingular to the cusp $y^n-x^m=0$. Hence $\Gamma$ does not depend on the particular choice of ${\mathcal {C}}\in {\text {Cusps}}(E)$. More precisely, we know that $\Gamma$ is the subsemigroup of ${\mathbb {Z}}_{\ge 0}$ generated by n, m. That is

$$\begin{aligned} \Gamma =\{an+bm;\; a,b\in {\mathbb {Z}}_{\ge 0}\}. \end{aligned}$$

An important feature of $\Gamma$ is the existence of its conductor $c_\Gamma =(n-1)(m-1)$, which is the smallest element $c_\Gamma \in \Gamma$ such that any non-negative integer greater or equal to $c_\Gamma$ is contained in $\Gamma$. In a more algebraic way, the conductor ideal $(t^{c_\Gamma })$ is contained in the image of the morphism

$$\begin{aligned} \phi ^\#: {\mathbb {C}}\{x,y\}\rightarrow {\mathbb {C}}\{t\},\quad f\mapsto f\circ \phi . \end{aligned}$$

On the other hand, as it was pointed by Zariski, the differential values of ${\mathcal {C}}$ may strongly depend on the analytic class of ${\mathcal {C}}$. In fact, they are the main discrete invariants in the analytic classification of branches (see Hefez and Hernandes 2011).

Given a differential 1-form $\omega \in \Omega ^1_{M_0,P_0}$ with $\omega =gdx+hdy$. If we write $\phi (t)=(x(t),y(t))$, we have that $\phi ^*(\omega )=(g(\phi (t))x'(t)+h(\phi (t)) y'(t))dt$. We put $a(t)=t (g(\phi (t))x'(t)+h(\phi (t)) y'(t))$, hence

$$\begin{aligned} \phi ^*(\omega )=a(t)\frac{dt}{t} \end{aligned}$$

and we define the differential value $\nu _{{\mathcal {C}}}(\omega )$ by $\nu _{{\mathcal {C}}}(\omega )={\text {order}}_t(a(t))$.

We know that $({\mathcal {C}},P_0)$ is an invariant branch of $\omega$ if and only if $\phi ^*(\omega )=0$ and hence $\nu _{{\mathcal {C}}}(\omega )=\infty$. The semimodule $\Lambda ^{{\mathcal {C}}}$ of the differential values is defined by

$$\begin{aligned} \Lambda ^{\mathcal {C}} = \{\nu _{{\mathcal {C}}}(\omega );\quad \omega \in \Omega ^1_{M_0,P_0}, \nu _{{\mathcal {C}}}(\omega )\ne \infty \}\subset {\mathbb {Z}}_{\ge 0}. \end{aligned}$$

It is a $\Gamma$-semimodule in the sense that

$$\begin{aligned} p\in \Gamma , q\in \Lambda ^{\mathcal {C}}\Rightarrow p+q\in \Lambda ^{\mathcal {C}}. \end{aligned}$$

Remark 5.1

Note that $\nu _{{\mathcal {C}}}(\omega )\ge 1$ for any $\omega \in \Omega ^1_{M_0,P_0}$. Anyway, we have the important property that $\Gamma \subset \{0\}\cup \Lambda ^{{\mathcal {C}}}$. If $\Lambda ^{{\mathcal {C}}}\cup \{0\}=\Gamma$, we say that ${\mathcal {C}}$ is quasi-homogeneous and it is analytically equivalent to the cusp $y^n-x^m=0$. Otherwise, if $\lambda _1$ is the minimum of $\Lambda ^{\mathcal {C}}\setminus \Gamma$, we know that $\lambda _1-n$ is the Zariski invariant, the first nontrivial analytic invariant. This invariant was introduced by Zariski (1966).

Remark 5.2

Let us note that $\nu _E(\omega )\in \Gamma$, for any $\omega \in \Omega ^1_{M_0,P_0}$.

5.1 Divisorial Order and Differential Values

In view of the definition of the differential values, for any $\omega \in \Omega ^1_{M_0,P_0}$ we have that $\nu _E(\omega )\le \nu _{{\mathcal {C}}}(\omega )$. A useful consequence of this fact is the following one:

Lemma 5.3

A basic 1-form $\omega \in \Omega ^1_{M_0,P_0}$ is resonant if and only if $\nu _{{\mathcal {C}}}(\omega )>\nu _E(\omega )$.

Proof

Write $\omega =W+{{\tilde{\omega }}}$, where W is the initial form of $\omega$. Denote $d=\nu _E(\omega )<nm$. Recall that $\nu _E({{\tilde{\omega }}})>\nu _E(W)=d$, we conclude that $\nu _{\mathcal {C}}(\omega )>d$ if and only if $\nu _{{\mathcal {C}}}(W) >d$. Since $\omega$ is a basic 1-form, we can write

$$\begin{aligned} W=x^ay^b\left\{ \mu \frac{dx}{x}+\zeta \frac{dy}{y}\right\} ,\quad an+bm=d. \end{aligned}$$

We have

$$\begin{aligned} \phi ^*W=(t^n)^a(t^m+t^{m+1}\xi (t))^b\left( n\mu +m\zeta +t\psi (t)\right) \frac{dt}{t}. \end{aligned}$$

The fact that $\nu _{\mathcal {C}}(W)>d$ is equivalent to say that $n\mu +m\zeta =0$ and hence it is equivalent to say that $\omega$ is resonant. $\square$

Corollary 5.4

If $\nu _{\mathcal {C}}(\omega )\notin \Gamma$, then $\omega$ is a resonant basic 1-form.

Proof

Since $\nu _{{\mathcal {C}}}(\omega )\notin \Gamma$, this differential value is bounded by the conductor $c_\Gamma =(n-1)(m-1)$, hence we have that

$$\begin{aligned} \nu _E(\omega )\le \nu _{{\mathcal {C}}}(\omega )<(n-1)(m-1)<nm. \end{aligned}$$

Then $\omega$ is a basic 1-form. Moreover, since $\nu _E(\omega )\in \Gamma$ and $\nu _{{\mathcal {C}}}(\omega )\notin \Gamma$, we have that $\nu _E(\omega )<\nu _{{\mathcal {C}}}(\omega )$ and we conclude that $\omega$ is a resonant basic 1-form. $\square$

5.2 Reachability Between Resonant Basic Forms

Let $\omega ,\omega '$ be two 1-forms $\omega ,\omega '\in \Omega ^1_{M_0,P_0}$. We say that $\omega '$ is reachable from $\omega$ if and only if there are nonnegative integer numbers a, b and a constant $\mu \in {\mathbb {C}}$ such that

$$\begin{aligned} \nu _E(\omega '-\mu x^ay^b\omega )>\nu _E(\omega '). \end{aligned}$$

Note that the constant $\mu$ and the pair (a, b) are necessarily unique.

We are interested in the case when $\omega$ and $\omega '$ are basic and resonant. In this situation, the initial parts are respectively given by

$$\begin{aligned} W=\mu x^cy^d\left\{ m\frac{dx}{dy}-n\frac{dy}{y}\right\} ,\quad W'=\mu ' x^{c'}y^{d'}\left\{ m\frac{dx}{x}-n\frac{dy}{y}\right\} . \end{aligned}$$

Note that $a,a',b,b'\ge 1$ since $\omega ,\omega '$ are holomorphic 1-forms. We have that $\omega '$ is reachable from $\omega$ if and only if $c'\ge c$ and $d'\ge d$; in this case we have that

$$\begin{aligned} \nu _E\left( \omega '-\frac{\mu '}{\mu }x^{c'-c}y^{d'-d}\omega \right) >\nu _E(\omega ')=c'n+d'm. \end{aligned}$$

Note also that the minimum divisorial value of a basic and resonant 1-form is $n+m$ and its initial part is necessarily of the type

$$\begin{aligned} \mu xy\left\{ m\frac{dx}{x}-n\frac{dy}{y}\right\} =\mu (mydx-nxdy). \end{aligned}$$

If $\omega$ is basic and resonant with $\nu _E(\omega )=n+m$, then any basic and resonant 1-form is reachable from $\omega$.

6 Cuspidal Semimodules

In this section we develop certain features of semimodules over the semigroup $\Gamma$ generated by the Puiseux pair (n, m). We consider, unless it is specified, only the singular case $n\ge 2$; in this case the conductor is $c_\Gamma =(n-1)(m-1)$ and we have the interesting property that any $p\in \Gamma$ with $p<nm$ is written as $p=an+bm$ in a unique way, with $a,b\ge 0$.

We proceed in a self contained way in order to help the reader, several results are true for more general semigroups, but we focus on the cuspidal semigroup $\Gamma$ to shorten the arguments.

6.1 The Basis of a Semimodule

A nonempty subset $\Lambda \subset {\mathbb {Z}}_{\ge 0}$ is a $\Gamma$-semimodule if $\Lambda +\Gamma \subset \Lambda$. We say that $\Lambda$ is normalized if $0\in \Lambda$, this is equivalent to say that $\Gamma \subset \Lambda$. As for the case of semigroups, the conductor $c_\Lambda$ is defined by

$$\begin{aligned} c_\Lambda =\min \{p\in {\mathbb {Z}}_{\ge 0};\; \{q\in {\mathbb {Z}}; q\ge p \}\subset \Lambda \}. \end{aligned}$$

Note that if $\lambda _{-1}$ is the minimum of $\Lambda$, then we have that $c_\Lambda \le c_\Gamma +\lambda _{-1}$.

Definition 6.1

Let $\Lambda$ be a $\Gamma$-semimodule. A nonempty finite increasing sequence of nonnegative integer numbers $\frac{}{}{\mathcal {B}}=(\lambda _{-1},\lambda _0,\ldots ,\lambda _s)$ is a basis for $\Lambda$ if for any $0\le j\le s$ we have that $\lambda _j\notin \Gamma ({\mathcal {B}}_{j-1})$, where $\Gamma ({\mathcal {B}}_{j-1})=(\lambda _{-1}+\Gamma )\cup (\lambda _0+\Gamma )\cup \cdots \cup (\lambda _{j-1}+\Gamma )$.

If $\Lambda =\Gamma ({\mathcal {B}})$, we have a chain of semimodules

$$\begin{aligned} \lambda _{-1}+\Gamma =\Lambda _{-1}\subset \Lambda _{0}\subset \cdots \subset \Lambda _{s}=\Lambda , \end{aligned}$$

(8)

where $\Lambda _j=\Gamma ({\mathcal {B}}_j)$. We call decomposition sequence of $\Lambda$ to this chain of semimodules. Let us note that

$$\begin{aligned} \lambda _j=\min (\Lambda \setminus \Lambda _{j-1}),\quad 0\le j\le s. \end{aligned}$$

(9)

This definitions are justified by next Proposition 6.2

Proposition 6.2

Given a semimodule $\Lambda$, there is a unique basis ${\mathcal {B}}$ such that $\Lambda =\Gamma ({\mathcal {B}})$.

Proof

We start with $\lambda _{-1}=\min \Lambda$. Note that $\Gamma (\lambda _{-1})\subset \Lambda$. If $\Gamma (\lambda _{-1})=\Lambda$ we stop and we put $s=-1$. If $\Gamma (\lambda _{-1})\ne \Lambda$, we put $\lambda _0=\min (\Lambda \setminus \Gamma (\lambda _{-1}))$. Note that $\Gamma (\lambda _{-1},\lambda _0)\subset \Lambda$. We continue in this way and, since $\lambda _j \not \equiv \lambda _k \mod n$ for $j \ne k$, after finitely many steps we obtain that $\Lambda =\Gamma (\lambda _{-1},\lambda _0,\ldots ,\lambda _s)$. Let us show the uniqueness of ${\mathcal {B}}=(\lambda _{-1},\lambda _{0},\ldots ,\lambda _s)$. Assume that $\Lambda =\Gamma ({\mathcal {B}}')$, for another $\Gamma$-basis ${\mathcal {B}}'=(\lambda '_{-1},\lambda '_0,\ldots ,\lambda '_{s'})$. Note that $\lambda _{-1}=\min \Lambda =\lambda '_{-1}$. Assume that $\lambda _j=\lambda '_j$ for any $0\le j\le k-1$. In view of Eq. (9) we have that $\lambda _k=\lambda '_k=\min \left( \Lambda {\setminus } \Gamma ({\mathcal {B}}_{k-1})\right) = \min \left( \Lambda {\setminus } \Gamma ({\mathcal {B}}'_{k-1})\right)$. This ends the proof. $\square$

We say that ${\mathcal {B}}=(\lambda _{-1},\lambda _0,\ldots ,\lambda _s)$ is the basis of $\Lambda =\Gamma ({\mathcal {B}})$ and that s is the length of $\Lambda$.

Consider a semimodule $\Lambda =\Gamma ({\mathcal {B}})$, an element $\lambda \in {\mathbb {Z}}_{\ge 0}$ is said to be $\Lambda$-independent if and only if $\lambda \notin \Lambda$ and $\lambda >\lambda _s$, where $\lambda _s$ is the last element in the basis ${\mathcal {B}}$. In this case we obtain a basis ${\mathcal {B}}(\lambda )$, just by adding $\lambda$ to ${\mathcal {B}}$ as being the last element. The new semimodule is denoted $\Lambda (\lambda )$, thus we have $\Lambda (\lambda )=\Lambda \cup (\lambda +\Gamma )=\Gamma ({\mathcal {B}}(\lambda ))$.

Given a semimodule $\Lambda =\Gamma (\lambda _{-1},\lambda _0,\ldots ,\lambda _s)$, we define the axes $u_i=u_i(\Lambda )$ by

$$\begin{aligned} u_0=\lambda _{-1};\quad u_i=\min \left( \Lambda _{i-2}\cap (\lambda _{i-1}+\Gamma )\right) , \; 1\le i\le s+1. \end{aligned}$$

(10)

Note that $u_i(\Lambda _j)=u_i(\Lambda )$, for $0\le i\le j+1\le s+1$.

Definition 6.3

A semimodule $\Lambda =\Gamma (\lambda _{-1},\lambda _0,\ldots ,\lambda _s)$ is increasing if and only if $\lambda _i>u_i$ for any $i=0,1,\ldots ,s$.

Remark 6.4

If $\Lambda$ is an increasing semimodule, each element $\Lambda _i$ of the decomposition sequence is also an increasing semimodule. Moreover, if $\lambda '$ is a $\Lambda$-independent value with $\lambda '>u_{s+1}$, then $\Lambda (\lambda ')$ is also an increasing $\Gamma$-semimodule.

Given a semimodule $\Lambda =\Gamma (\lambda _{-1},\lambda _0,\ldots ,\lambda _s)$, the semimodule ${\widetilde{\Lambda }}=\Lambda -\lambda _{-1}$ is called the normalization of $\Lambda$. Next features allow to deduce properties of $\Lambda$ from properties of its normalization:

• (1) The basis of ${\widetilde{\Lambda }}$ is $(0,\lambda _0-\lambda _{-1},\ldots ,\lambda _s-\lambda _{-1})$.

• (2) ${\widetilde{\Lambda }}_i=\Lambda _i-\lambda _{-1}$, for $i=-1,0,\ldots ,s$.

• (3) $u_i({\widetilde{\Lambda }})=u_i(\Lambda )-\lambda _{-1},\quad i=0,1,\ldots ,s+1$.

• (4) $c_{{\widetilde{\Lambda }}}=c_\Lambda -\lambda _{-1}$.

• (5) $\Lambda$ is increasing if and only if ${\widetilde{\Lambda }}$ is increasing.

6.2 Axes and Conductor

We precise the expressions of the axes and we bound them by the conductors.

Lemma 6.5

Consider a semimodule $\Lambda$ of length s and two indices $0\le k<i\le s$. Then we have that $u_{i+1}<c_{\Lambda _k}+n$.

Proof

We can assume that $i=s$, $k=s-1$. Note that $\lambda _{s}<c_{\Lambda _{s-1}}$, since $\lambda _s\notin \Lambda _{s-1}$. Then, there is a unique $\alpha \in {\mathbb {Z}}_{> 0}$ such that $0\le \lambda _{s}-c_{\Lambda _{s-1}}+\alpha n<n$. We have that $\lambda _{s}+\alpha n\in \lambda _{s}+\Gamma$ and $\lambda _{s}+\alpha n\ge c_{\Lambda _{s-1}}$. We obtain

$$\begin{aligned} \lambda _{s}+\alpha n\in \Lambda _{s-1}\cap (\lambda _{s}+\Gamma ). \end{aligned}$$

We deduce that $u_{s+1}\le \lambda _{s}+\alpha n<c_{\Lambda _{s-1}}+n$. $\square$

Corollary 6.6

Consider a semimodule $\Lambda$ of the form $\Lambda =\Gamma (n,m,\lambda _1,\ldots ,\lambda _s)$. Then $u_{i+1}<nm$, for any $0\le i\le s$.

Proof

By Lemma 6.5, we have that $u_{i+1}\le c_{\Lambda _0}+n$, but in this situation, we have that $\Lambda _0\cup \{0\}=\Gamma$ and thus

$$\begin{aligned} c_{\Lambda _0}=c_\Gamma =(n-1)(m-1). \end{aligned}$$

Hence $u_{i+1}\le c_{\Lambda _0}+n=(n-1)(m-1)+n<nm$. $\square$

Lemma 6.7

Consider $\Lambda =\Gamma (\lambda _{-1},\lambda _0,\ldots ,\lambda _s)$. There is a unique index k with $-1\le k\le s-1$ such that $u_{s+1}\in \lambda _k+\Gamma$ and there are unique expressions

$$\begin{aligned} u_{s+1}= & {} \lambda _{s}+na+mb,\quad a,b\in {\mathbb {Z}}_{\ge 0} \end{aligned}$$

(11)

$$\begin{aligned} u_{s+1}= & {} \lambda _k+n c+m d,\quad c,d\in {\mathbb {Z}}_{\ge 0}. \end{aligned}$$

(12)

In these expressions, we have $ac=bd=ab=cd=0$ and $(a,b)\ne (0,0)\ne (c,d)$.

Proof

The existence of the expressions (11) and (12) is given by the definition of $u_{s+1}$ as $u_{s+1}=\min (\Lambda _{s-1}\cap (\lambda _s+\Gamma ))$. By the minimality of $u_{s+1}$ and the fact that $u_{s+1}\ne \lambda _{s}$ and $u_{s+1}\ne \lambda _{k}$, we deduce the properties $ac=bd=0$ and $(a,b)\ne (0,0)\ne (c,d)$. Moreover, if $ab\ne 0$ we should have that $c=d=0$ which is not possible; in the same way we see that $cd=0$.

Let us show the uniqueness of the index k. Assume that there are two indices $-1\le k< k'\le s-1$ with $u_{s+1}=\lambda _k+c n+d m= \lambda _{k'}+c'n+d'm$. Take the case when $a\ne 0$, then we have that $c=c'=0$ and we can write

$$\begin{aligned} \lambda _{k'}=\lambda _k+m(d-d')\in \lambda _k+\Gamma , \end{aligned}$$

this is a contradiction. Same argument if $b\ne 0$.

If we normalize $\Lambda$, we have that $u_{s+1}-\lambda _{s}={{\tilde{u}}}_{s+1}-{{\tilde{\lambda }}}_{s}$. By Lemma 6.5 we have

$$\begin{aligned} {{\tilde{u}}}_{s+1}-{{\tilde{\lambda }}}_{s}\le {{\tilde{u}}}_{s+1}< c_{{\widetilde{\Lambda }}_{s-1}}+n\le c_\Gamma +n=(n-1)(m-1)+n<nm. \end{aligned}$$

Hence, we have that $u_{s+1}-\lambda _{s}$ (and with the same argument $u_{s+1}-\lambda _k$) are strictly smaller than nm. Thus, the expression of these elements of $\Gamma$ as a linear combination of n, m with non-negative coefficients is unique. $\square$

6.3 The Limits

Consider a semimodule $\Lambda =\Gamma (\lambda _{-1},\lambda _0,\ldots ,\lambda _s)$ with $s\ge 0$. The first and second limits $\ell _1$ and $\ell _2$ of $\Lambda$ are defined by

$$\begin{aligned} \ell _1= & {} \min \{p;\;np+\lambda _s\in \Lambda _{s-1}\}. \end{aligned}$$

(13)

$$\begin{aligned} \ell _2= & {} \min \{q;\; mq+\lambda _s\in \Lambda _{s-1}\}. \end{aligned}$$

(14)

Remark 6.8

We have that $\ell _1\ell _2\ne 0$ and

$$\begin{aligned} u_{s+1}=\min \{\ell _1n+\lambda _s,\; \ell _2m+\lambda _s\}. \end{aligned}$$

(15)

Indeed, by Lemma 6.7, we have either $u_{s+1}=an+\lambda _s$ or $u_{s+1}=bm+\lambda _{s}$; if $u_{s+1}=an+\lambda _s$, by minimality we have that $a=\ell _1$, in the same way, if $u_{s+1}=bm+\lambda _s$ we have that $b=\ell _2$. Moreover, there is a unique index k with $-1\le k<s$ such that

• (1) If $u_{s+1}=\ell _1n+\lambda _s$, then $u_{s+1}=\lambda _k+bm$.

• (2) If $u_{s+1}=\ell _2 m+\lambda _s$, then $u_{s+1}=\lambda _k+an$.

Lemma 6.9

If $an+bm+\lambda _s\in \Lambda _{s-1}$, then either $a\ge \ell _1$ or $b\ge \ell _2$.

Proof

Let us write $an+bm+\lambda _s=cn+dm+\lambda _{j}$ for a certain $j\le s-1$. If $ac\ne 0$, we find

$$\begin{aligned} (a-1)n+bm+\lambda _s=(c-1)n+dm+\lambda _j\in \Lambda _{s-1}. \end{aligned}$$

Repeating the argument and working in a similar way with the coefficients b, d, we find an element

$$\begin{aligned} {{\tilde{a}}} n+{{\tilde{b}}} m+\lambda _{s}= {{\tilde{c}}} n+{{\tilde{d}}} m+\lambda _j \end{aligned}$$

such that $a\ge {{\tilde{a}}}$ and $b\ge {{\tilde{b}}}$, with the property that ${{\tilde{a}}} {{\tilde{c}}}=0$ and ${{\tilde{b}}}{{\tilde{d}}}=0$. Moreover, we have that $({{\tilde{c}}},{{\tilde{d}}})\ne (0,0)$, since otherwise $\lambda _s\le \lambda _{s-1}$. Suppose that ${{\tilde{c}}}\ne 0$, then ${{\tilde{a}}}=0$ and

$$\begin{aligned} {{\tilde{b}}} m+\lambda _{s}= {{\tilde{c}}} n+{{\tilde{d}}} m+\lambda _j\in \Lambda _{s-1}. \end{aligned}$$

By the minimality property of $\ell _2$, we have that ${{\tilde{b}}}\ge \ell _2$ and then $b\ge \ell _2$. In a similar way, we show that if ${{\tilde{d}}}\ne 0$ we have that $a\ge \ell _1$. $\square$

Let us note that the limits of the normalization ${{\tilde{\Lambda }}}$ are the same ones as for $\Lambda$.

Example 6.10

Consider the semigroup $\Gamma =\langle 5,11\rangle$ and the $\Gamma$-semimodule $\Lambda =\Gamma (5,11,17,23,29)$. Let us compute the axes and the limits for this semimodule. Note that $s=3$. We have $u_0=\lambda _{-1}=n=5$, and $\lambda _0=m=11$. In order to compute the limits of $\Lambda _{0}$ we have to find the minimal non-negative integers

• $\ell _1^1$ such that $11+5 \ell _1^1=5 +11b$, hence $\ell _1^1=b=1$;

• $\ell _2^1$ such that $11+11 \ell _2^1=5+5a$ and we obtain $\ell _2^1=4$ and $a=10$.

Hence, $\lambda _0+n \ell _1^1=16$ and $\lambda _0+m \ell _2^1=55$ and $u_1=\min \{16,55\}=16$. Now, let us compute the limits of $\Lambda _{1}$ where $\lambda _1=17$. We search $\ell _1^2$ and $\ell _2^2$ minimal such that

• $17+5 \ell _1^2=11+11b$, hence $\ell _1^2=b=1$ and $\lambda _1+n\ell _1^2=22$;

• $17+11\ell _2^2=5 + 5a$ and we have that $\ell _2^2=3$ and $a=9$, then $\lambda _1+m\ell _2^2=50$.

We get that $u_2=22$. In a similar way, taking into account that $\lambda _2=23$ and $\lambda _3=29$, we get that $u_3=28$ and $u_4=34$. Since $u_i < \lambda _i$, for $i=0,1,2,3$, we obtain that the semimodule $\Lambda$ is increasing.

7 Standard Bases

From now on, we fix a cusp ${\mathcal {C}}$ in ${\text {Cusps}}(E)$ and we consider the semimodule $\Lambda$ of differential values of ${\mathcal {C}}$:

$$\begin{aligned} \Lambda =\Lambda ^{\mathcal {C}}=\{\nu _{\mathcal {C}}(\omega );\; \omega \in \Omega ^1_{M_0,P_0}\}\setminus \{\infty \}. \end{aligned}$$

(16)

We recall that $\Gamma \setminus \{0\}\subset \Lambda$.

Lemma 7.1

If $(\lambda _{-1},\lambda _0,\lambda _1,\ldots ,\lambda _s)$ is the basis of $\Lambda ^{\mathcal {C}}$, then $\lambda _{-1}=n$ and $\lambda _0=m$.

Proof

Let $(x,y)=(t^n, t^m\xi (t))$ be a Puiseux parametrization of ${\mathcal {C}}$, where $\xi (0)\ne 0$. Recall that $\nu _{\mathcal {C}}(adx+bdy)$ is the order in t of the expression

$$\begin{aligned} nt^{n}a(t^n, t^m\xi (t))+ t^{m}\xi (t)b(t^n, t^m\xi (t))\{m+t\xi '(t)/\xi (t)\}. \end{aligned}$$

(17)

We see that this order is $\ge n$ and that $\nu _{{\mathcal {C}}}(dx)=n$. Hence $n=\lambda _{-1}$. Moreover, the terms in Eq. (17) of degree $<m$ come only from the first part $nt^{n}a(t^n, t^m\xi (t))$ of the sum, so, they are values in $\Gamma$. Since $m=\nu _{\mathcal {C}}(dy)$, we conclude that $\lambda _0=m$. $\square$

Definition 7.2

Write $\Lambda ^{\mathcal {C}}=\Gamma (n,m,\lambda _1,\ldots ,\lambda _s)$. A standard basis for ${\mathcal {C}}$ is a list of 1-forms ${\mathcal {G}}=(\omega _{-1},\omega _0,\omega _1,\ldots ,\omega _s)$ such that $\nu _{{\mathcal {C}}}(\omega _i)=\lambda _i$, for $i=-1,0,1,\ldots ,s$.

Remark 7.3

There is at least one standard basis, by definition of the semimodule of differential values. The standard bases are not in general unique. For instance, we have that

$$\begin{aligned} \nu _{{\mathcal {C}}}(h dx)=n,\quad \nu _{{\mathcal {C}}}(h dy)=m,\quad h(0)\ne 0. \end{aligned}$$

On the other hand, for $i=1,2,\ldots ,s$, we have that $\nu _{\mathcal {C}}(\omega _i)=\lambda _i\notin \Gamma$, then, in view of Corollary  5.4, the 1-form $\omega _i$ is basic resonant.

Remark 7.4

Let ${\mathcal {G}}=(\omega _{-1},\omega _0,\omega _1,\ldots ,\omega _s)$ be a standard basis for ${\mathcal {C}}$. Then $\omega _{-1},\omega _0$ have the form

$$\begin{aligned} \omega _{-1}=h dx+gdy, \;h(0)\ne 0;\quad \omega _0= f dx+\psi dy,\; \psi (0)\ne 0,\; \nu _{\mathcal {C}}(fdx)>m. \end{aligned}$$

Thus, we can write any differential 1-form $\omega$ in a unique way as $\omega =a\omega _{-1}+b\omega _{0}$. Anyway, we are mainly interested in the study of the 1-forms $\omega _i$, for $1\le i\le s$. From the standard basis ${\mathcal {G}}$ we can obtain a new one “adapted to the coordinates” given by

$$\begin{aligned} {\mathcal {G}}=(dx,dy,\omega _1,\ldots ,\omega _s). \end{aligned}$$

Just for simplifying the presentation of the computations, we will consider only this kind of standard bases.

7.1 The Zariski Invariant

In this subsection, we deal with properties of divisorial orders and differential values around the element $\lambda _1$, where

$$\begin{aligned} \Lambda ^{\mathcal {C}}=\Gamma (n,m,\lambda _1,\ldots ,\lambda _s). \end{aligned}$$

This is the first step for a general result. Anyway, let us recall that $\lambda _1-n$ is the classical Zariski invariant. Let us cite the work of Gómez-Martínez (2021) that essentially contains several of the results in this section.

Proposition 7.5

We have the following properties:

• (1) If $s=0$, then $\infty = \sup \{\nu _{{\mathcal {C}}}(\omega );\; \omega \in \Omega ^1_{M_0,P_0},\;\nu _E(\omega )=n+m\}$.

• (2) If $s\ge 1$, then $\lambda _1= \sup \{\nu _{{\mathcal {C}}}(\omega );\; \omega \in \Omega ^1_{M_0,P_0},\;\nu _E(\omega )=n+m\}$.

Proof

Assume that $s=0$ and hence $\Lambda ^{\mathcal {C}}=\Gamma \setminus \{0\}$. Let us consider the 1-form $\eta =mydx-nxdy$. We have that $\nu _{{\mathcal {C}}}(\eta )>n+m=\nu _E(\eta )$. Moreover, since $s=0$ we have that $\nu _{{\mathcal {C}}}(\eta )\in \Gamma$; then there is a monomial function f such that

$$\begin{aligned} \nu _E(df)=\nu _{\mathcal {C}}(df)= \nu _{{\mathcal {C}}}(\eta )>n+m. \end{aligned}$$

In particular, there is a constant $\mu \ne 0$ such that $\nu _{\mathcal {C}}(\eta -\mu df)>\nu _{{\mathcal {C}}}(\eta )$. Write $\eta ^1=\eta -\mu df$; we have that $\nu _E(\eta ^1)=\nu _E(\eta )=n+m$ and $\nu _{\mathcal {C}}(\eta ^1)>\nu _{{\mathcal {C}}}(\eta )$. We repeat the argument with $\eta ^1$ and in this way we obtain 1-forms $\eta ^k$ with $\nu _E(\eta ^k)=n+m$ and $\nu _{{\mathcal {C}}}(\eta ^k)\ge n+m+1+k$. This proves the first statement.

Assume now that $s\ge 1$. Let us first show that

$$\begin{aligned} \lambda _1\le \sup \{\nu _{{\mathcal {C}}}(\omega );\; \omega \in \Omega ^1_{M_0,P_0},\;\nu _E(\omega )=n+m\}. \end{aligned}$$

If $\nu _{{\mathcal {C}}}(\eta )\notin \Gamma$, we have that $\nu _{{\mathcal {C}}}(\eta )\ge \lambda _1$ since $\lambda _1$ is the minimum of the differential values not in $\Gamma$, then we are done. Assume that $\nu _{{\mathcal {C}}}(\eta )\in \Gamma$ and hence

$$\begin{aligned} \nu _{{\mathcal {C}}}(\eta )=an+bm>n+m. \end{aligned}$$

Taking the function $f=x^ay^b$, up to multiply df by a constant $c_1$ we obtain that

$$\begin{aligned} \nu _{{\mathcal {C}}}(\eta _1)>\nu _{{\mathcal {C}}}(\eta )=an+bm, \quad \eta _1=\eta -c_1df. \end{aligned}$$

Note that $\nu _E(\eta _1)=n+m$, since $\nu _E(df)=an+bm>n+m.$ We restart with $\eta _1$ instead of $\eta$, noting that $\nu _{{\mathcal {C}}}(\eta )<\nu _{{\mathcal {C}}}(\eta _1)$. Repeating finitely many times this procedure, we obtain a new 1-form ${{\tilde{\eta }}}=\eta -d{{\tilde{f}}}$ such that $\nu _E({{\tilde{\eta }}})=n+m$ and either $\nu _{{\mathcal {C}}}({{\tilde{\eta }}})\ge c_\Gamma =(n-1)(m-1)$ or $\nu _{{\mathcal {C}}}({{\tilde{\eta }}})\notin \Gamma$, in both cases we have that $\nu _{{\mathcal {C}}}({{\tilde{\eta }}})\ge \lambda _1$ and we are done.

It remains to show that $\lambda _1\ge \sup \{\nu _{{\mathcal {C}}}(\omega );\; \omega \in \Omega ^1_{M_0,P_0},\;\nu _E(\omega )=n+m\}$. Let us consider $\omega _1$ such that $\nu _{{\mathcal {C}}}(\omega _1)=\lambda _1$ and let us show that it is not possible to have ${{\tilde{\omega }}}$ such that $\nu _E({{\tilde{\omega }}})=n+m$ and $\nu _{{\mathcal {C}}}({{\tilde{\omega }}})>\nu _{{\mathcal {C}}}(\omega _1)$. In this situation, both $\omega _1$ and ${{\tilde{\omega }}}$ are basic resonant. We know that $\omega _1$ is reachable from ${{\tilde{\omega }}}$ and thus there is a constant $\mu$ and $a,b\ge 0$ such that

$$\begin{aligned} \nu _E(\omega _1^1)>\nu _E(\omega _1), \quad \omega ^1_1= \omega _1-\mu x^ay^b{{\tilde{\omega }}}. \end{aligned}$$

We have that $\nu _{{\mathcal {C}}}(\omega _1^1)=\nu _{{\mathcal {C}}}(\omega _1)=\lambda _1$. We restart with the pair $\omega _1^1,{{\tilde{\omega }}}$; in this way, we obtain an infinite sequence of 1-forms $\omega _1,\omega _1^1,\omega _1^2,\ldots$ with strictly increasing divisorial orders. Up to a finite number of steps, we find an index k such that $\nu _E(\omega _1^k)>\lambda _1=\nu _{{\mathcal {C}}}(\omega _1^k)$. This contradicts with the fact $\nu _{{\mathcal {C}}}(\omega _1^k)\ge \nu _E(\omega _1^k)$. $\square$

Corollary 7.6

Any 1-form $w\in \Omega ^1_{M_0,P_0}$ such that $\nu _E(\omega )=n+m$ and $\nu _{{\mathcal {C}}}(\omega )\notin \Gamma$ satisfies that $\nu _{{\mathcal {C}}}(\omega )=\lambda _1$.

Proof

In view of the previous result, we have that $\nu _{{\mathcal {C}}}(\omega )\le \lambda _1$. Since $\nu _{{\mathcal {C}}}(\omega )\notin \Gamma$, we also have that $\nu _{{\mathcal {C}}}(\omega )\ge \lambda _1$. $\square$

Corollary 7.7

Any 1-form $w\in \Omega ^1_{M_0,P_0}$ such that $\nu _{\mathcal {C}}(\omega )=\lambda _1$ satisfies that $\nu _E(\omega )=n+m$.

Proof

Take $\omega _1$ such that $\nu _{\mathcal {C}}(\omega _1)=\lambda _1$ and $\nu _E(\omega _1)=n+m$. Assume that

$$\begin{aligned} \nu _E(\omega )>n+m \end{aligned}$$

in order to obtain a contradiction. Since $\lambda _1\notin \Gamma$, both $\omega$ and $\omega _1$ are basic resonant and $\omega$ is reachable from $\omega _1$. Then there is a function f with $\nu _{{\mathcal {C}}}(f)>0$ such that

$$\begin{aligned} \nu _E(\omega -f\omega _1)>\nu _E(\omega ). \end{aligned}$$

Put $\omega ^1=\omega -f\omega _1$, since $\nu _{{\mathcal {C}}}(f\omega _1)>\lambda _1$, we have that $\nu _{{\mathcal {C}}}(\omega ^1)=\lambda _1$. We restart with the pair $\omega ^1,\omega$. After finitely many repetitions we find $\omega ^k$ with $\nu _{\mathcal {C}}(\omega ^k)=\lambda _1$ and $\nu _E(\omega ^k)>\lambda _1$, contradiction. $\square$

The following two lemmas are necessary steps in order to prove an inductive version of Proposition 7.5 valid for all indices $i=1,2,\ldots ,s$:

Lemma 7.8

Assume that $s\ge 1$ and take $\omega _1$ such that $\nu _{{\mathcal {C}}}(\omega _1)=\lambda _1$. Consider an integer number $k=na+mb+\lambda _1\in \lambda _1+\Gamma$. The following statements are equivalent:

• (1) $k\notin \Gamma$.

• (2) $\nu _{E}(\omega )\le \nu _{E}(x^ay^b\omega _1)$, for any $\omega \in \Omega ^1_{M_0,P_0}$ such that $\nu _{\mathcal {C}}(\omega )=k$.

Proof

Note that $k=\nu _{\mathcal {C}}(x^ay^b\omega _1)>(a+1)n+(b+1)m=\nu _E(x^ay^b\omega _1)$.

Assume that $k\in \Gamma$, then $k=na'+mb'>\nu _{E}(x^ay^b\omega _1)$. Taking $\omega =d(x^{a'}y^{b'})$, we have $\nu _{{\mathcal {C}}}(\omega )=\nu _E(\omega )=k>\nu _E(x^ay^b\omega _1)$.

Now assume that $k\notin \Gamma$. Let us reason by contradiction assuming that there is $\omega$ with $\nu _{{\mathcal {C}}}(\omega )=k$ with $\nu _{E}(\omega )> \nu _{E}(x^ay^b\omega _1)$. We have that $\omega$ is basic resonant, since $\nu _{\mathcal {C}}(\omega )\notin \Gamma$. Then $\omega$ is reachable from $\omega _1$. Then there is $a',b'\ge 0$ and a constant $\mu$ such that $\nu _E(x^{a'}y^{b'}\omega _1)=\nu _E(\omega )$ and

$$\begin{aligned} \nu _E(\omega -cx^{a'}y^{b'}\omega _1)>\nu _E(\omega )>\nu _E(x^{a}y^{b}\omega _1). \end{aligned}$$

Since $na'+mb'>na+mb$, we have that $\nu _{{\mathcal {C}}}(x^{a'}y^{b'}\omega _1)>k$ and hence $\nu _{{\mathcal {C}}}(\omega ^1)=k$, where $\omega ^1=\omega -cx^{a'}y^{b'}\omega _1$. Repeating the procedure with the pair $\omega ^1,\omega _1$, we obtain a sequence

$$\begin{aligned} \omega ,\omega ^1,\omega ^2,\ldots \end{aligned}$$

with strictly increasing divisorial order and such that $\nu _{\mathcal {C}}(\omega ^j)=k$ for any j. This is a contradiction. $\square$

Lemma 7.9

Take $\omega _1$ with $\nu _{{\mathcal {C}}}(\omega _1)=\lambda _1$. Let $\omega \in \Omega ^1_{M_0,P_0}$ be a 1-form such that $\nu _{\mathcal {C}}(\omega )=\lambda \notin \Gamma$. There are unique $a,b\ge 0$ such that $\nu _{E}(\omega )=\nu _{E}(x^ay^b\omega _1)$. Moreover, we have that $\lambda \ge na+mb+\lambda _1$.

Proof

Note that $\omega$ is basic resonant and thus the existence and uniqueness of a, b is assured. Moreover, if $\lambda <na+mb+\lambda _1$, we can find a constant $\mu$ such that

$$\begin{aligned} \nu _{E}(\omega -\mu x^ay^b\omega _1)> \nu _{E}(x^ay^b\omega _1) \end{aligned}$$

and $\nu _{\mathcal {C}}(\omega -\mu x^ay^b\omega _1)=\lambda$. Put $\omega ^1=\omega -\mu x^ay^b\omega _1$, we have that $\nu _{{\mathcal {C}}}(\omega ^1)=\lambda \notin \Gamma$. As before, we have that

$$\begin{aligned} \nu _E(\omega ^1)=\nu _E(x^{a_1}y^{b_1}\omega _1),\quad a_1n+b_1m>an+bm \end{aligned}$$

and thus $\lambda <a_1n+b_1m+\lambda _1$. We repeat the process with the pair $\omega ^1,\omega _1$, where $\omega ^1=\omega -\mu x^ay^b\omega _1$ in order to have a sequence $\omega ,\omega ^1,\omega ^2,\ldots$ with strictly increasing divisorial orders and such that $\nu _{{\mathcal {C}}}(\omega ^j)=\lambda$ for any j. This is a contradiction. $\square$

7.2 Critical Divisorial Orders

Recall that we are considering a cusp ${\mathcal {C}}$ in ${\text {Cusps}}(E)$, whose semimodule of differential values is

$$\begin{aligned} \Lambda ^{{\mathcal {C}}}=\Gamma (n,m,\lambda _1,\ldots ,\lambda _s). \end{aligned}$$

The critical divisorial orders $t_i$, for $i=-1,0,\ldots ,s+1$ are defined as follows:

• We put $t_{-1}=n$ and $t_0=m$.

• For $1\le i\le s+1$, we put $t_i=t_{i-1}+u_i-\lambda _{i-1}$.

Let us note that $t_1=m+(n+m)-m=n+m$.

Lemma 7.10

Consider the semimodule $\Lambda =\Gamma (n,m,\lambda _1,\ldots ,\lambda _s)$ and take an index $1\le i\le s$. If $\lambda _\ell >u_\ell$, for any $0\le \ell \le i$, we have that

$$\begin{aligned} \lambda _j-\lambda _k> t_j-t_k,\quad -1\le k< j\le i. \end{aligned}$$

(18)

Proof

We have that $\lambda _{j}-\lambda _{j-1}>t_{j}-t_{j-1}$ if and only if

$$\begin{aligned} t_j=t_{j-1}+u_j-\lambda _{j-1}>t_{j}+u_j-\lambda _j, \end{aligned}$$

which is true, since $u_j-\lambda _j<0$. Noting that

$$\begin{aligned} \lambda _j-\lambda _k=\sum _{\ell =k}^{j-1}(\lambda _{\ell +1}-\lambda _\ell )> \sum _{\ell =k}^{j-1}(t_{\ell +1}-t_\ell )=t_j-t_k, \end{aligned}$$

The proof is ended. $\square$

Lemma 7.11

If the semimodule $\Lambda =\Gamma (n,m,\lambda _1,\ldots ,\lambda _s)$ is increasing, we have that $t_i<nm$, for any $i=-1,0,1,\ldots ,s+1$.

Proof

If $i\in \{-1,0\}$ we have that $t_{-1}=n$, $t_0=m$ and we are done. Assume that $1\le i\le s$, we have that

$$\begin{aligned} t_i-n= t_i-t_{-1}=\sum _{\ell =0}^i(t_\ell -t_{\ell -1})\le \sum _{\ell =0}^i(\lambda _\ell -\lambda _{\ell -1}) =\lambda _{i}-\lambda _{-1}=\lambda _i-n. \end{aligned}$$

Then $t_i\le \lambda _i<c_\Gamma <nm$. Consider the case $i=s+1$. We have that

$$\begin{aligned} t_{s+1}=t_s+u_{s+1}-\lambda _s=u_{s+1}+(t_s-\lambda _s)\le u_{s+1}<c_\Gamma +n<nm. \end{aligned}$$

See Lemma 6.5. $\square$

Remark 7.12

As a consequence of Lemma 7.11 we have that any 1-form $\omega$ such that $\nu _E(\omega )=t_i$ is a basic 1-form; moreover, if $t_i=\nu _E(\omega )<\nu _{\mathcal {C}}(\omega )$, then it is basic and resonant.

The critical divisorial orders are the divisorial orders of the elements of a standard basis, in view of the following

Theorem 7.13

For each $1\le i\le s$ we have the following statements

• (1) $\lambda _i=\sup \{\nu _{\mathcal {C}}(\omega ):\nu _{E}(\omega )=t_i\}$.

• (2) If $\nu _{\mathcal {C}}(\omega )=\lambda _i$, then $\nu _{E}(\omega )=t_i$.

• (3) For each 1-form $\omega$ with $\nu _{\mathcal {C}}(\omega )\notin \Lambda _{i-1}$, there is a unique pair $a,b\ge 0$ such that $\nu _{E}(\omega )=\nu _{E}(x^ay^b\omega _i)$. Moreover, we have that $\nu _{\mathcal {C}}(\omega )\ge \lambda _i+na+mb$.

• (4) We have that $\lambda _i>u_i$.

• (5) Let $k=\lambda _i+na+mb$, then $k\notin \Lambda _{i-1}$ if and only if for all $\omega$ such that $\nu _{\mathcal {C}}(\omega )=k$ we have that $\nu _{E}(\omega )\le \nu _{E}(x^ay^b\omega _i)$.

In particular, the semimodules $\Lambda _i$ are increasing, for $i=1,2,\ldots ,s$.

A proof of this Theorem 7.13 is given in Appendix B.

Remark 7.14

Note that if ${\mathcal {B}}=(\omega _{-1}=dx,\omega _0=dy,\omega _1,\omega _2,\ldots ,\omega _s)$ is a standard basis, Theorem 7.13 says that $\nu _E(\omega _i)=t_i$ for any $i=-1,0,1,\ldots ,s$ and that $\omega _{j+1}$ is reachable from $\omega _j$, for any $1\le j\le s-1$. That is, the initial parts of the 1-forms $\omega _i$ are given by

$$\begin{aligned} W_i=\mu _ix^{a_i}y^{b_i}\left\{ m\frac{dx}{x}-n\frac{dy}{y}\right\} ,\quad \mu _i\ne 0, \end{aligned}$$

where $(a_1,b_1)=(1,1)$ and

$$\begin{aligned} 1=a_1\le a_2\le \cdots \le a_s,\quad 1=b_1\le b_2\le \cdots \le b_s. \end{aligned}$$

Moreover, we have that $na_i+mb_i=t_i$, for $i=1,2,\ldots ,s$.

Remark 7.15

Note that ${\mathcal {B}}=(\omega _{-1}=dx,\omega _0=dy,\omega _1,\omega _2,\ldots ,\omega _s)$ is a standard basis if and only if $\nu _E(\omega _i)=t_i$ and $\nu _{\mathcal {C}}(\omega _i)\notin \Lambda _{i-1}$, for any $i=1,2,\ldots ,s$. Moreover, Theorem 7.13 justifies an algorithm of construction of a standard basis as follows:

Assume we have obtained $\omega _j$, for $j=-1,0,1,\ldots ,s'$. We can produce the axis $u_{s'+1}$ and the critical divisorial order $t_{s'+1}=t_{s'}+u_{s'+1}-\lambda _{s'}$. There is an expression $t_{s'+1}=an+bm$. We consider the 1-form

$$\begin{aligned} \omega _{s'+1}^0=x^ay^b\left\{ m\frac{dx}{x}-n\frac{dy}{y}\right\} . \end{aligned}$$

If $\nu _{\mathcal {C}}(\omega _{s'+1}^0)\notin \Lambda _{s'}$, we know that $\lambda _{s'+1}=\nu _{\mathcal {C}}(\omega _{s'+1}^0)$ and $s\ge s'+1$. If $\nu _{\mathcal {C}}(\omega _{s'+1}^0)\in \Lambda _{s'}$, there is $j\le s'$ and $c,d\ge 0$ such that

$$\begin{aligned} \nu _{\mathcal {C}}(x^cy^d\omega _j)= \nu _{\mathcal {C}}(\omega _{s'+1}^0). \end{aligned}$$

We take a constant $\mu$ such that $\nu _{\mathcal {C}} (\omega _{s'+1}^0-\mu x^cy^d\omega _j)>\nu _{\mathcal {C}}(\omega _{s'+1}^0)$. Put $\omega _{s'+1}^1= \omega _{s'+1}^0-\mu x^cy^d\omega _j$. We have that $\nu _E(\omega _{s'+1}^1)=t_{s'+1}$. We repeat the procedure with $\omega _{s'+1}^1$. After finitely many steps we get that either $\nu _{\mathcal {C}}(\omega _{s'+1}^k)\notin \Lambda _{s'}$ or $\nu _{\mathcal {C}}(\omega _{s'+1}^k)\ge c_\Gamma$. In the first case, we put $\lambda _{s'+1}=\nu _{\mathcal {C}}(\omega _{s'+1}^k)$, in the second case we know that $s=s'$.

Example 7.16

Consider the semigroup $\Gamma =\langle 5,11\rangle$ and the $\Gamma$-semimodule $\Lambda =\Gamma (5,11,17,23,29)$ as in Example 6.10. The computation of the critical divisorial orders $t_i$, $i=-1,0,1,2,3$, gives

$$\begin{aligned} t_{-1}=5, \quad t_0=11, \quad t_1=16, \quad t_2=21, \quad t_3=26, \quad t_4=31. \end{aligned}$$

Since the semimodule $\Lambda$ is increasing, by Alberich-Carramiñana et al. (2021) there exists a curve whose semimodule is $\Lambda$. In particular, the curve ${\mathcal {C}}$ given by the Puiseux parametrization $\phi (t)=(t^5,t^{11}+t^{12}+t^{13})$ has semigroup $\Gamma$ and semimodule of differential values $\Lambda$. An extended standard basis is given by $\{\omega _{-1},\omega _0,\omega _1,\omega _2,\omega _3,\omega _4\}$ where

$$\begin{aligned}{} & {} \omega _{-1}=dx, \quad \omega _0=dy, \quad \omega _1=5x dy-11ydx, \\{} & {} \omega _2=11 x \omega _1 -5y dy, \quad \omega _3=x \omega _2+y\omega _1 \end{aligned}$$

and $\omega _4=x \omega _3 - 33 y \omega _2-1199x^6 dx -2035x^4 y dx -407 x^4 \omega _1 -1595 x^3 \omega _2 + \cdots$. The reader can check that $\nu _E(\omega _i)=t_i$ as stated in Theorem 7.13.

8 Extended Standard Basis and Analytic Semiroots

As in previous sections, we consider a cusp ${\mathcal {C}}$ in ${\text {Cusps}}(E)$, whose semimodule of differential values is

$$\begin{aligned} \Lambda =\Gamma (n,m,\lambda _1,\ldots ,\lambda _s). \end{aligned}$$

Let us recall that the axis $u_{s+1}=\min (\Lambda _{s-1}\cap (\lambda _s+\Gamma ))$ is well defined and we have also a well defined critical divisorial order

$$\begin{aligned} t_{s+1}=t_s+u_{s+1}-\lambda _s. \end{aligned}$$

Let us also remark that $t_\ell <nm$, for $0\le \ell \le s+1$, in view of Lemma 7.11.

Definition 8.1

We say that a differential 1-form $\omega$ is dicritically adjusted to ${\mathcal {C}}$ if and only if $\nu _E(\omega )=t_{s+1}$ and $\nu _{{\mathcal {C}}}(\omega )=\infty$. An extended standard basis for ${\mathcal {C}}$ is a list

$$\begin{aligned} \omega _{-1}=dx, \omega _0=dy, \omega _{1},\ldots ,\omega _{s};\, \omega _{s+1} \end{aligned}$$

where $\omega _{-1}, \omega _0, \omega _{1},\ldots ,\omega _{s}$ is a standard basis and $\omega _{s+1}$ is dicritically adjusted to ${\mathcal {C}}$.

Lemma 8.2

Assume that $\nu _E(\eta )=t_{s+1}$ and $\nu _{\mathcal {C}}(\eta )>u_{s+1}$. Then, there is a 1-form ${{\tilde{\eta }}}$ such that $\nu _E({{\tilde{\eta }}})=t_{s+1}$ and $\nu _{\mathcal {C}}({{\tilde{\eta }}})>\nu _{\mathcal {C}}(\eta )$.

Proof

Take a standard basis $dx,dy, \omega _{1},\ldots ,\omega _{s}$ for ${\mathcal {C}}$. There is an index k such that $\nu _{\mathcal {C}}(\eta )=an+bm+\lambda _k$. Consider the 1-form $x^ay^b\omega _k$. Note that $\nu _{\mathcal {C}}(x^ay^b\omega _k)=\nu _{{\mathcal {C}}}(\eta )$. If we show that $an+bm+t_k>t_{s+1}$, we are done, by taking ${{\tilde{\eta }}}=\eta -\mu x^ay^b\omega _k$ for a convenient constant $\mu$. We have

$$\begin{aligned} an+bm+\lambda _k> u_{s+1}\Rightarrow an+bm+t_k> u_{s+1}-\lambda _k+t_k. \end{aligned}$$

(19)

It remains to show that $u_{s+1}-\lambda _k+t_k\ge t_{s+1}$. We have

$$\begin{aligned} u_{s+1}-\lambda _k+t_k\ge t_{s+1}\Leftrightarrow u_{s+1}-\lambda _k+t_k\ge t_s+u_{s+1}-\lambda _s\Leftrightarrow \lambda _s-\lambda _k\ge t_s-t_k. \end{aligned}$$

We are done by Lemma 7.10. $\square$

Proposition 8.3

There is at least one 1-form $\omega$ dicritically adjusted to ${\mathcal {C}}$.

Proof

Take a standard basis $dx,dy, \omega _{1},\ldots ,\omega _{s}$ for ${\mathcal {C}}$. There is an index $k<s$ such that $u_{s+1}=an+bm+\lambda _s=cn+dm+\lambda _k$. Note that

$$\begin{aligned} an+bm+t_s<cn+dm+t_k, \end{aligned}$$

since $t_s-t_k<\lambda _s-\lambda _k$. In this way, we have

• (1) $t_{s+1}=\nu _E(x^ay^b\omega _s)< \nu _E(x^cy^d\omega _k)$.

• (2) $u_{s+1}=\nu _{\mathcal {C}}(x^ay^b\omega _s)= \nu _{\mathcal {C}}(x^cy^d\omega _k)$.

Taking $\eta =x^ay^b\omega _s-\mu x^cy^d\omega _k$, for a convenient constant $\mu$, we have that

$$\begin{aligned} \nu _E(\eta )=t_{s+1},\quad \nu _{\mathcal {C}}(\eta )>u_{s+1}. \end{aligned}$$

By a repeated application of Lemma 8.2, we find a 1-form ${{\tilde{\eta }}}$ such that

$$\begin{aligned} \nu _E({{\tilde{\eta }}})=t_{s+1}<\nu _{\mathcal {C}}({{\tilde{\eta }}}),\quad \nu _{\mathcal {C}}({{\tilde{\eta }}})>c_\Gamma +1. \end{aligned}$$

Now, we can integrate ${{\tilde{\eta }}}$ as follows. Let $\phi :t\mapsto \phi (t)=(\phi _1(t),\phi _2(t))$ be a reduced parametrization of the curve ${\mathcal {C}}$. Then $\phi$ induces a morphism

$$\begin{aligned} \phi ^{\#}:{\mathbb {C}}\{x,y\}\rightarrow {\mathbb {C}}\{t\},\quad h\mapsto h\circ \phi , \end{aligned}$$

whose kernel is generated by a local equation of ${\mathcal {C}}$ and moreover, the conductor ideal $t^{c_\Gamma }{\mathbb {C}}\{t\}$ is contained in the image of $\phi ^{\#}$. Let us write

$$\begin{aligned} \phi ^*{{\tilde{\eta }}}=\xi (t)dt,\quad \xi (t)\in t^{c_\Gamma }{\mathbb {C}}\{t\}. \end{aligned}$$

By integration, there is a series $\psi (t)$ such that $\psi '(t)=\xi (t)$, with $\psi (t)\in t^{c_\Gamma +1}{\mathbb {C}}\{t\}$. In view of the properties of the conductor ideal, there is a function $h\in {\mathbb {C}}\{x,y\}$ such that $h\circ \phi (t)=\psi (t)$. If we consider $\omega ={{\tilde{\eta }}}-dh$, we have that $\nu _E(\omega )=\nu _E({{\tilde{\eta }}})=t_{s+1}$ and $\nu _{\mathcal {C}}(\omega )=\infty$. $\square$

Proposition 8.4

If $\omega$ is dicritically adjusted to ${\mathcal {C}}$, we have that $\omega$ is basic and resonant (hence it is E-totally dicritical) and ${\mathcal {C}}$ is an $\omega$-cusp.

Proof

Recall that $t_{s+1}<nm$ and $\nu _{\mathcal {C}}(\omega )=\infty >t_{s+1}$. $\square$

8.1 Delorme’s Decompositions

Delorme’s decompositions are described in next Theorem 8.5. This result is the main tool we need to use in our statements on the analytic semiroots. We provide a proof of it, using a different approach to the one of Delorme, in Appendix C.

Theorem 8.5

(Delorme’s decomposition) Let ${\mathcal {C}}$ be a cusp in ${\text {Cusps}}(E)$, consider an extended standard basis $\omega _{-1},\omega _0,\omega _1,\ldots ,\omega _s;\omega _{s+1}$ of ${\mathcal {C}}$ and denote by

$$\begin{aligned} \Lambda =\Gamma (n,m,\lambda _1,\ldots , \lambda _s) \end{aligned}$$

the semimodule of differential values of ${\mathcal {C}}$. For any indices $0\le j\le i\le s$, there is a decomposition

$$\begin{aligned} \omega _{i+1}=\sum _{\ell =-1}^{j}f_\ell ^{ij}\omega _\ell \end{aligned}$$

such that, for any $-1\le \ell \le j$ we have $\nu _{{\mathcal {C}}}(f_\ell ^{ij}\omega _\ell )\ge v_{ij}$, where $v_{ij}=t_{i+1}-t_j+\lambda _j$ and there is exactly one index $-1\le k\le j-1$ such that $\nu _{{\mathcal {C}}}(f_k^{ij}\omega _k)=\nu _{{\mathcal {C}}}(f_j^{ij}\omega _j)= v_{ij}$.

Remark 8.6

Note that $v_{ii}= t_{i+1}-t_i+\lambda _i=u_{i+1}$. We also have that

$$\begin{aligned} v_{ij}=u_{i+1}-(\lambda _i-u_i)-\cdots -(\lambda _{j+1}-u_{j+1}). \end{aligned}$$

In particular we have that $v_{ij}\le u_{i+1}<c_\Gamma +n< nm$. See Lemma 6.5.

8.2 Analytic Semiroots

Let $\Lambda =\Gamma (n,m,\lambda _1,\ldots ,\lambda _s)$ be the semimodule of differential values of the E-cusp ${\mathcal {C}}$ and let us consider an extended standard basis

$$\begin{aligned} {\mathcal {E}}=(\omega _{-1},\omega _0,\omega _1,\ldots ,\omega _s;\omega _{s+1}) \end{aligned}$$

We recall that $\omega _1,\omega _2,\ldots ,\omega _{s+1}$ are basic and resonant. Fix a free point $P\in E$. For each $i=1,2,\ldots ,s+1$, there is an E-cusp ${\mathcal {C}}^i_P$ passing through P and invariant by $\omega _i$. Let us note that if P is the infinitely near point of ${\mathcal {C}}$ in E, we have that ${\mathcal {C}}^{s+1}_P={\mathcal {C}}$.

Definition 8.7

The cusps ${\mathcal {C}}^i_P$, for $i=1,2,\ldots ,s+1$ are called the analytic semiroots of ${\mathcal {C}}$ through P with respect to the extended standard basis ${\mathcal {E}}$.

Let us denote ${\mathcal {E}}_i=(\omega _{-1},\omega _0,\omega _1,\ldots ,\omega _{i-1};\omega _{i})$ for any $1\le i\le s+1$. The main objective of this paper is to show the following Theorem

Theorem 8.8

For any $1\le i\le s+1$ and any free point $P\in E$, we have

• (a) $\Lambda _{i-1}$ is the semimodule of differential values of ${\mathcal {C}}^i_P$.

• (b) ${\mathcal {E}}_i$ is an extended standard basis for ${\mathcal {C}}^i_P$.

Let us consider an index $1\le i\le s+1$ and an analytic semiroot ${\mathcal {C}}^i_P$ in order to prove Theorem 8.8.

Lemma 8.9

We have that $\nu _{{{\mathcal {C}}}^i_P}(\omega _j)=\lambda _j$, for any $j=-1,0,1,\ldots ,i-1$.

Proof

For any basic non resonant 1-form $\omega$, we have that

$$\begin{aligned} \nu _{\mathcal {C}}(\omega )=\nu _E(\omega )=\nu _{{{{\mathcal {C}}}^i_P}}(\omega ). \end{aligned}$$

This is particularly true for the case of 1-forms of the type $\omega =df$, where we know that $\nu _E(\omega )=\nu _E(f)$. Hence, for any germ of function $f\in {\mathbb {C}}\{x,y\}$ we have that

$$\begin{aligned} \min \{\nu _{{\mathcal {C}}}(df),nm\}=\min \{\nu _{{{{\mathcal {C}}}^i_P}}(df),nm\}. \end{aligned}$$

(20)

The statement of the Lemma is true for $\ell =-1,0$, since

$$\begin{aligned} \nu _{{\mathcal {C}}}(dx)= \nu _{{{{\mathcal {C}}}^i_P}}(dx)=n,\quad \nu _{{\mathcal {C}}}(dy)= \nu _{{{{\mathcal {C}}}^i_P}}(dy)=m. \end{aligned}$$

Let us assume that it is true for any $\ell =-1,0,1,\ldots ,j$, with $0\le j\le i-2$. We apply Theorem 8.5 to obtain a decomposition

$$\begin{aligned} \omega _{i}=\sum _{\ell =-1}^{j+1}f_\ell \omega _\ell \end{aligned}$$

such that $\nu _{{\mathcal {C}}}(f_\ell \omega _\ell )\ge v_{i-1,j+1}$, where $v_{i-1,j+1}<nm$ (see Remark 8.6) and there is a single $k\le j$ such that $\nu _{{\mathcal {C}}}(f_{j+1}\omega _{j+1})=\nu _{{\mathcal {C}}}(f_k\omega _k)=v_{i-1,j+1}.$ We deduce that

$$\begin{aligned} \nu _{{\mathcal {C}}}\left( \sum _{\ell =-1}^{j}f_\ell \omega _\ell \right) =\nu _{{\mathcal {C}}}(f_k\omega _k)=v_{i-1,j+1}. \end{aligned}$$

On the other hand, by induction hypothesis and noting that $v_{i-1,j+1}<nm$, we have that

$$\begin{aligned} \min \{\nu _{{{\mathcal {C}}}}(f_\ell \omega _\ell ),nm\}= \min \{\nu _{{{{\mathcal {C}}}^i_P}}(f_\ell \omega _\ell ),nm\},\quad \ell =-1,0,1,\ldots ,j. \end{aligned}$$

In particular, we have that

$$\begin{aligned} \nu _{{{{\mathcal {C}}}^i_P}}(f_k\omega _k)=v_{i-1,j+1},\quad \nu _{{{{\mathcal {C}}}^i_P}}\left( \sum _{\ell =-1}^{j}f_\ell \omega _\ell \right) =v_{i-1,j+1}. \end{aligned}$$

Since $\nu _{{\mathcal {C}}^i_P}(\omega _{i})=\infty$, we have that $\nu _{{{{\mathcal {C}}}^i_P}}(f_{j+1}\omega _{j+1})=v_{i-1,j+1}$. Hence we have

$$\begin{aligned} v_{i-1,j+1}=\nu _{{{\mathcal {C}}}}(f_{j+1}\omega _{j+1}) =\nu _{{{{\mathcal {C}}}^i_P}}(f_{j+1}\omega _{j+1}). \end{aligned}$$

Noting that $\nu _{{\mathcal {C}}}(f_{j+1})=\nu _{{{{\mathcal {C}}}^i_P}}(f_{j+1})$, we conclude that $\nu _{{\mathcal {C}}}(\omega _{j+1})=\nu _{{{{\mathcal {C}}}^i_P}}(\omega _{j+1})$. The proof is ended. $\square$

Corollary 8.10

$\Lambda ^{{{{\mathcal {C}}}^i_P}}\supset \Lambda _{i-1}$.

Proof

It is enough to remark that $\lambda _j\in \Lambda ^{{{{\mathcal {C}}}^i_P}}$ for any $j=-1,0,1,\ldots ,i-1$. $\square$

Proposition 8.11

$\Lambda ^{{{\mathcal {C}}}^i_P}=\Lambda _{i-1}$.

Proof

We already know that $\Lambda ^{{{{\mathcal {C}}}^i_P}}\supset \Lambda _{i-1}$. Assume that $\Lambda ^{{{{\mathcal {C}}}^i_P}}\ne \Lambda _{i-1}$ and take the number

$$\begin{aligned} \lambda =\min \left( \Lambda ^{{{{\mathcal {C}}}^i_P}}\setminus \Lambda _{i-1}\right) . \end{aligned}$$

Note that $\lambda >m$ and hence there is a maximum index $0\le j\le i-1$ such that $\lambda _j<\lambda$. We have that

$$\begin{aligned} \nu _E(\omega _j)=t_j=t_j({\mathcal {C}}^i_P),\quad t_j=t_j({\mathcal {C}}). \end{aligned}$$

where $t_j(*)$ denotes the critical divisorial order $t_j$ with respect to the curve $*$. Assume first that $j\le i-2$.

Let ${{\tilde{\omega }}}$ be a 1-form in a standard basis for ${{{\mathcal {C}}}^i_P}$ that corresponds to the differential value $\lambda$. Note that $\lambda$ is the differential value in the basis of $\Lambda ^{{\mathcal {C}}^i_P}$ that immediately follows $\lambda _j$ and the previous ones correspond to the values in the basis of $\Lambda$. We have that

$$\begin{aligned} \nu _E(\omega _{j+1})=t_{j+1}=t_j+u_{j+1}-\lambda _j=t_{j+1} ({\mathcal {C}}^i_P)=\nu _E({{\tilde{\omega }}}). \end{aligned}$$

In view of the property (1) in Theorem 7.13, we have that

$$\begin{aligned} \lambda =\max \{\nu _{{{{\mathcal {C}}}^i_P}}(\omega );\; \nu _E(\omega )=t_{j+1}({\mathcal {C}}^i_P)=t_{j+1}\} \end{aligned}$$

In view of Lemma 8.9, we know that $\nu _{{{{\mathcal {C}}}^i_P}}(\omega _{j+1})=\lambda _{j+1}$ and $\nu _E(\omega _{j+1})=t_{j+1}$. This implies that $\lambda _{j+1}<\lambda$, contradiction.

Let us consider now the case when $j=i-1$. We shall prove that it is not possible to have $s({\mathcal {C}}^i_P)>i-1$ where $s(*)$ refers to “concept s” with respect to the curve $*$ (that is, $s+2$ is the number of elements of the basis of the semimodule of the curve). If $s({\mathcal {C}}^i_P)>i-1$, we have that

$$\begin{aligned} \lambda =\max \{\nu _{{{{\mathcal {C}}}^i_P}}(\omega );\; \nu _E(\omega )= t_{i}({\mathcal {C}}^i_P)=t_{i}\}. \end{aligned}$$

But we know that $\nu _{{{{\mathcal {C}}}^i_P}}(\omega _{i})=\infty$ and $\nu _E(\omega _{i})=t_{i}$, this is the desired contradiction. $\square$

Example 8.12

Let us compute the analytic semiroots of the curve ${\mathcal {C}}$ given in Example 7.16. The Puiseux parametrizations for the E-cusps ${\mathcal {C}}_a^i$ of $\omega _i$, $i=1,2,3$, are given by

$$\begin{aligned} \phi _1^a(t)&=(t^5,at^{11}) \\ \phi _2^a(t)&= (t^5, a t^{11}+ a^2 t^{12} + \tfrac{23}{22}a^3 t^{13}+ \tfrac{136}{121} a^4 t^{14} + \cdots ) \\ \phi _3^a(t)&= \left( t^5, \sum _{i \ge 11} a^{i-10} t^{i}\right) \end{aligned}$$

with $a \in {\mathbb {C}}^*$. Hence, the analytic semiroots of ${\mathcal {C}}$ are the curves ${\mathcal {C}}^1_1$, ${\mathcal {C}}^2_1$ and ${\mathcal {C}}^3_1$ given by the above parametrizations $\phi _i^1(t)$, $i=1,2,3$, with $a=1$.

Note that in this example, for any $i=1,2,3$, all the E-cusps of the family $\{{\mathcal {C}}^i_a\}_a$ are analytically equivalent. To see this it is enough to consider the new parameter $t=a^{-1}u$ and the change of variables $x_1=a^5x$, $y_1=a^{10} y$.

Example 8.13

We would like to remark that, in general, the E-cusps of an element $\omega _i$, $i \ge 2$, of a standard basis are not analytically equivalent as the following example shows. Consider the curve ${\mathcal {C}}$ given by the Puiseux parametrization $\phi (t)=(t^7,t^{17}+t^{30}+t^{33}+t^{36})$ with $\Gamma =\langle 7,17\rangle$ and $\Lambda =\Gamma (7,17, 37,57)$. A standard basis is given by $\omega _{-1}=dx, \quad \omega _0=dy, \quad \omega _1=7xdy-17 y dx$ and

$$\begin{aligned} \omega _2=3757x^2ydx-1547x^3dy-4624y^2dx+1904xydy+1183 y^2dy. \end{aligned}$$

The E-cusps of $\omega _2$ are the curves given by the Puiseux parametrization

$$\begin{aligned} \varphi _a(t)=(t^7, at^{17} + a^3 t^{30} + a^4t^{33}+ \cdots ) \end{aligned}$$

with $a \in {\mathbb {C}}^*$. If we consider a new parameter $t=a^{-2/13}u$ and the we make the change of variables $x_1=a^{14/13}x, y_2=a^{21/13} y$, we obtain that the family of E-cusps of $\omega _2$ are the curves ${\mathcal {C}}^2_a$ given by the parametrizations

$$\begin{aligned} \phi _a(t)=(t^7,t^{17}+t^{30}+a^{7/13} t^{33} + \cdots ) \end{aligned}$$

From the results above, we have that $\Lambda ^{{\mathcal {C}}_a^2}=\Lambda _1=\Gamma (7,17,37)$ for all $a \in {\mathbb {C}}^*$. Since $33 \not \in \Lambda _1 - 7$, by Theorem 2.1 in Hefez and Hernandes (2011), two curves ${\mathcal {C}}^2_{a_1}$ and ${\mathcal {C}}^2_{a_2}$ are not, in general, analytically equivalent for $a_1,a_2 \in {\mathbb {C}}^*$.

Example 8.14

Let us consider the 1-form $\omega$ of example 4.7 in Gómez-Martínez (2021) given by

$$\begin{aligned} \omega{} & {} =(7y^5+2x^9y-2x^9y^2-9x^2y^4) dx + (4y^3x^3-x^{10} + 2x^{10} y -3xy^4-x^{8}y^2)dy. \end{aligned}$$

This 1-form is pre-basic and resonant for the pair (4, 9) since $\nu _E(\omega )=48$, the co-pair of (4, 9) is (3, 7) and $\text {Cl}(\omega ;x,y) \subset R^{4,9}(3,4)$ where

$$\begin{aligned} R^{4,9}(3,4)=\{(\alpha ,\beta )\in {\mathbb {R}}^2 \, \ \alpha + 2\beta \ge 11\} \cap \{(\alpha ,\beta )\in {\mathbb {R}}^2 \, \ 3 \alpha + 7 \beta \ge 37\}. \end{aligned}$$

Moreover, the weighted initial part of $\omega$ is given by $\text {In}_{4,9;x,y}^{48}=x^2y^3 (-9ydx+4xdy)$. Consequently $\omega$ is totally E-dicritical for the last divisor E associated to the cuspidal sequence ${\mathcal {S}}^{4,9}_{y=0}$. Note that $\omega$ is not a basic 1-form since $\nu _E(\omega ) > nm=36$.

The invariant curves of $\omega$ which are transversal to the dicritical component E are the curves ${\mathcal {C}}_a$, $a \in {\mathbb {C}}^*$, given by

$$\begin{aligned} y^4-ax^9+(a-1)x^7y+x^7y^2=0. \end{aligned}$$

Note that these curves do not have the same semimodule of differential values since the curves ${\mathcal {C}}_a$, with $a \ne 1$, have Zariski invariant equal to 10 whereas the curve ${\mathcal {C}}_1$ has Zariski invariant equal to 19.

Appendices

Appendix A: Bounds for the Conductor

In this Appendix we present some bounds for the conductor of the semimodules $\Lambda _i$ in the decomposition series of

$$\begin{aligned} \Lambda =\Gamma (\lambda _{-1},\lambda _0,\lambda _1,\ldots ,\lambda _s) \end{aligned}$$

that will be enough to prove the results we need relative to the structure of the semimodule of differential values. We recall that the semigroup $\Gamma$ is generated by a Puiseux pair (n, m), with $2\le n$.

Given two integer numbers $r\le s$, we denote [r, s] the set of the integer numbers $\ell$ such that $r\le \ell \le s$. For any $q\ge 0$ we denote by $I_q$ the interval $I_q=[nq,n(q+1)-1]$, in particular, we have that $I_0=\{0,1,2,\ldots ,n-1\}$. For any $r,s\in I_0$, we define the circular interval $\langle r,s\rangle$ by

$$\begin{aligned} \langle r,s\rangle =[r,s],\; \text { if } r\le s;\quad \langle r,s\rangle =[r,n-1]\cup [0,s],\; \text { if } r> s. \end{aligned}$$

We denote by $\rho :{\mathbb {Z}} \rightarrow {\mathbb {Z}}/(n)$ the canonical map and we also use the notation $\rho (p)={\bar{p}}$. Since $\gcd (n,m)=1$, there is a bijection

$$\begin{aligned} \xi :{\mathbb {Z}}/(n)\rightarrow I_0=\{0,1,2,\ldots ,n-1\},\quad \xi ^{-1}(k)= \rho (km). \end{aligned}$$

For any $q\ge 0$ and any subset $S \subset {\mathbb {Z}}_{\ge 0}$, we define the q-level set $R_q(S)\subset I_0$ by $R_q(S)=\xi \left( \rho ( S \cap I_q)\right)$.

Remark A.1

We have that $R_q(\Lambda )\subset R_{q'} (\Lambda )$ for all $q'\ge q$.

Remark A.2

For any $\mu \in {\mathbb {Z}}_{\ge 0}$ and $q\ge 1$, we have

$$\begin{aligned} \#R_{q}(\mu +\Gamma )\le \#R_{q-1}(\mu +\Gamma )+1. \end{aligned}$$

Indeed, this is equivalent to show that $\#\rho ((\mu +\Gamma )\cap I_q)\le \#\rho ((\mu +\Gamma )\cap I_{q-1})+1$. Assume that ${{\bar{p}}}_1,{{\bar{p}}}_2\in \rho ((\mu +\Gamma )\cap I_q){\setminus } \rho ((\mu +\Gamma )\cap I_{q-1})$. We can take representatives $p_1,p_2\in (\mu +\Gamma )\cap I_q$ of ${{\bar{p}}}_1$ and ${{\bar{p}}}_2$ of the form $p_1=\mu +b_1m$, $p_2=\mu +b_2m$. If $p_1\ne p_2$, we have that $|p_1-p_2|\ge m>n$ and this is not possible.

Lemma A.3

Consider $\mu \in I_v$, denote $r=\xi ({{\bar{\mu }}})$ and let q be such that $q\ge v$. For any $p\in R_q(\mu +\Gamma )$ we have that $\langle r,p\rangle \subset R_q(\mu +\Gamma )$. In particular, the set $R_q(\mu +\Gamma )$ is a circular interval.

Proof

The second statement is straightforward, since the union of circular intervals with a common point is a circular interval. To prove the first statement, we proceed by induction on the number $\ell$ of elements in $\langle r,p\rangle$. If $\ell \le 2$, we are done, since $\langle r,p\rangle \subset \{r,p\}\subset R_q(\mu +\Lambda )$. Assume that $\ell >2$; in particular we have that $r\ne p$. Consider the point ${{\tilde{p}}}\in I_0$ given by ${{\tilde{p}}}=p-1$, if $p\ge 1$ and ${{\tilde{p}}}=n-1$, if $p=0$. We have that $\langle r,p\rangle =\langle r,{{\tilde{p}}}\rangle \cup \{p\}$ and the length of $\langle r,{{\tilde{p}}}\rangle$ is $\ell -1$. Then, it is enough to show that ${{\tilde{p}}}\in R_q(\mu +\Gamma )$. Take an element $\mu +an+bm\in I_q\cap (\mu +\Gamma )$ such that $\rho ({\mu +an+bm})=\rho ({pm})$. Noting that $r\ne p$, we have that $b\ge 1$. There is $q'\le q$ such that $\mu +an+(b-1)m\in I_{q'}$ and hence

$$\begin{aligned} \mu +(a+q-q')n+(b-1)m\in I_{q}\cap (\mu +\Gamma ). \end{aligned}$$

We have that $\rho ({\mu +(a+q-q')n+(b-1)m})=\rho ({{{\tilde{p}}} m})$ and thus ${{\tilde{p}}}\in R_q(\mu +\Gamma )$.

$\square$

Definition A.4

We define the tops $q_1$ and $q_2$ of $\Lambda$ by the property that

$$\begin{aligned} \lambda _s+n\ell _1\in I_{q_1}, \quad \lambda _s+m\ell _2\in I_{q_2} \end{aligned}$$

where $\ell _1$ and $\ell _2$ are the limits of $\Lambda$. The main top $Q_\Lambda$ is the maximum of $q_1,q_2$.

Proposition A.5

Consider a normalized semimodule $\Lambda =\Gamma (0,\lambda _0,\lambda _1,\ldots ,\lambda _s)$. Let v be such that $\lambda _s\in I_v$ and assume that $R_q(\Lambda _{s-1})$ is a circular interval for any $q\ge v$. Denote by $q_1,q_2$ the tops of $\Lambda$ and put $r=\xi ({{\bar{\lambda }}}_s)$. Then we have that

• (1) $[0,r-1]\subset R_q(\Lambda _{s-1})$, for all $q\ge q_1-1$.

• (2) $[r,n-1]\subset R_q(\Lambda )$, for all $q\ge q_2-1$.

In particular $c_\Lambda \le n(Q_\Lambda -1)$.

Proof

Note that Statements (1) and (2) imply that

$$\begin{aligned} I_0=[0,n-1]=[0,r]\cup [r,n-1]\subset R_q(\Lambda ),\quad q\ge Q_\Lambda -1 \end{aligned}$$

and thus, we have that $c_\Lambda \le n(Q_\Lambda -1)$.

Proof of Statement (1) By Remark A.1 it is enough to show that we have $[0,r-1]\subset R_{q_1-1}(\Lambda _{s-1})$. Since $\lambda _s+n\ell _1\in \Lambda _{s-1}$, there is an index $k\le s-1$ such that $\lambda _s+n\ell _1=\lambda _k+an+bm$. By the minimality of $\ell _1$, we have that $a=0$ and hence $\lambda _s+n\ell _1=\lambda _k+bm$. Denote $r_k=\xi ({{\bar{\lambda }}}_k)$. Note that $r_k\ne r$, since $r\notin R_v(\Lambda _{s-1})$.

Assume that the next statements are true:

• (a) If $r_k>r$, then $[0,r]\subset R_{q_1}(\lambda _k+\Gamma )$.

• (b) If $r_k<r$, then $[r_k,r]\subset R_{q_1}(\lambda _k+\Gamma )$ and $[0,r_k]\subset R_{q_1-1}(\Lambda _{s-1})$.

If $r_k>r$, by the minimality of $\ell _1$, we have that $r\notin R_{q_1-1}(\lambda _k+\Gamma )$; now, in view of Remark A.2 and noting that $[0,r]=[0,r-1]\cup \{r\}$, we obtain that

$$\begin{aligned} {[}0,r-1]\subset R_{q_1-1}(\lambda _k+\Gamma )\subset R_{q_1-1}(\Lambda _{s-1}). \end{aligned}$$

If $r_k<r$, we obtain as above that $[r_k,r-1]\subset R_{q_1-1}(\lambda _k+\Gamma )$, then

$$\begin{aligned} {[}0,r-1]=[0,r_k]\cup [r_k,r-1]\subset R_{q_1-1}(\Lambda _{s-1}). \end{aligned}$$

If remains to prove (a) and (b).

Proof of (a): We can apply Lemma A.3 to have that $\langle r_k,r\rangle \subset R_{q_1}(\lambda _k+\Gamma )$. We end by noting that $[0,r]\subset \langle r_k,r\rangle$.

Proof of (b): We apply Lemma A.3 to have that $\langle r_k,r\rangle =[r_k,r]\subset R_{q_1}(\lambda _k+\Gamma )$. On the other hand, we know that $R_{q_1-1}(\Lambda _{s-1})$ is a circular interval since $q_1-1\ge v$ and it contains 0 and $r_k$. Moreover $r\notin R_{q_1-1}(\Lambda _{s-1})$ and $r>r_k$, then the circular interval $R_{q_1-1}(\Lambda _{s-1})$ contains $[0,r_k]$.

Proof of Statement (2): It is enough to show that $[r,n-1]\subset R_{q_2-1}(\Lambda )$. By an argument as before, there is an index $k\le s-1$ such that $\lambda _s+m\ell _2=\lambda _k+na$. Take $r_k\ne r$ as above. By Lemma A.3, we have that $\langle r,r_k\rangle \subset R_{q_2}(\lambda _s+\Gamma )$. Let us see that $r_k\notin R_{q_2-1}(\lambda _s+\Gamma )$. For this, let us show that the property

$$\begin{aligned} r_k\in R_{q_2-1}(\lambda _s+\Gamma ) \end{aligned}$$

leads to a contradiction. This property should imply that $\lambda _k+n(a-1)\in \lambda _s+\Gamma$ and hence there are nonnegative integer numbers $\alpha ,\beta$ such that

$$\begin{aligned} \lambda _s+n\alpha +m\beta = \lambda _k+n(a-1). \end{aligned}$$

If $a-1\le \alpha$, we have that $\lambda _k=\lambda _s+(\alpha -a+1)n+\beta m$ and this contradicts the fact that $\lambda _k<\lambda _s$; hence $a-1>\alpha$ and we have

$$\begin{aligned} \lambda _s+m\beta = \lambda _k+n(a-1-\alpha ). \end{aligned}$$

Since $a-1-\alpha <a$, we have that $\beta <\ell _2$. This contradicts the minimality of $\ell _2$.

Since $r_k\notin R_{q_2-1}(\lambda _s+\Gamma )$, we can apply Remark A.2 that tells us that

$$\begin{aligned} \langle r,r_k\rangle \setminus \{r_k\}\subset R_{q_2-1}(\lambda _s+\Gamma )\subset R_{q_2-1}(\Lambda ). \end{aligned}$$

Note also that $r_k\in R_{q_2-1}(\Lambda )$. Then we have that $\langle r,r_k\rangle \subset R_{q_2-1}(\Lambda )$.

If $r>r_k$, then $[r,n-1]\subset \langle r,r_k\rangle \subset R_{q_2-1}(\Lambda )$. Assume now that $r<r_k$. Recall that $r\notin R_v(\Lambda _{s-1})$; since $R_v(\Lambda _{s-1})$ is a circular interval containing $r_k$ and 0, but not containing r, we have that

$$\begin{aligned} {[}r_k,n-1]\subset R_v(\Lambda _{s-1})\subset R_{q_2-1}(\Lambda _{s-1})\subset R_{q_2-1}(\Lambda ). \end{aligned}$$

We conclude that $[r,n-1]=\langle r,r_k\rangle \cup [r_k,n-1]\subset R_{q_2-1}(\Lambda )$. $\square$

Proposition A.6

Let $\Lambda$ be a normalized increasing semimodule of length s and let v be such that $u_{s+1}\in I_v$. Then $R_q(\Lambda )$ is a circular interval for any $q\ge v$.

Proof

Let us proceed by induction on the length s of $\Lambda$. If $s=-1$, we have $\Lambda =\Lambda _{-1}=\Gamma$. By Lemma A.3 applied to $\mu =0$, we are done. Let us suppose that $s\ge 0$ and assume by induction that the result is true for $\Lambda _{s-1}$. We have that $\Lambda =\Lambda _{s-1}\cup (\lambda _s+\Gamma )$. This implies that

$$\begin{aligned} R_q(\Lambda )=R_q(\Lambda _{s-1})\cup R_q(\lambda _s+\Gamma ),\quad q\ge 0. \end{aligned}$$

Let $v'$ be such that $u_{s}\in I_{v'}$. By induction hypothesis, we know that $R_q(\Lambda _{s-1})$ is a circular interval for any $q\ge v'$. Moreover, by the increasing property, we have that

$$\begin{aligned} u_{s+1}>\lambda _s>u_s\ge \lambda _{s-1}. \end{aligned}$$

In particular, we have that $v\ge v'$ and $R_q(\Lambda _{s-1})$ is a circular interval for any $q\ge v$. On the other hand, take $v''$ such that $\lambda _s\in I_{v''}$. By Lemma A.3, we know that $R_q(\lambda _s+\Gamma )$ is a circular interval for any $q\ge v''$. Since $v\ge v''$, we have that $R_q(\lambda _s+\Gamma )$ is a circular interval for any $q\ge v$. Thus, both

$$\begin{aligned} R_q(\Lambda _{s-1})\quad \text{ and }\quad R_q(\lambda _s+\Gamma ) \end{aligned}$$

are circular intervals for $q\ge v$. We need to show that their union is also a circular interval. Since $v\ge v''$, we have that $r\in R_q(\lambda _s+\Gamma )$ for $r=\xi ({\overline{\lambda }}_s)$. Noting that $0\in R_q(\Lambda _{s-1})$ and $r\in R_q(\lambda _s+\Gamma )$, in order to prove that $R_q(\Lambda )$ is a circular interval, it is enough to show that one of the following properties holds

$$\begin{aligned} (a):\quad [0,r]\subset R_q(\Lambda ); \quad \quad (b):\quad [r,n-1]\subset R_q(\Lambda ). \end{aligned}$$

We can apply Proposition A.5 to $\Lambda =\Lambda _{s-1}(\lambda _s)$. Indeed, by induction hypothesis we know that $R_{p}(\Lambda _{s-1})$ is a circular interval for any $p\ge v'$; since $v'\le v''$, we have that $R_{p}(\Lambda _{s-1})$ is a circular interval for any $p\ge v''$ and thus we are in the hypothesis of Proposition A.5. Now, by Eq. (15) in Remark 6.8, we have either $u_{s+1 }=\lambda _s+nl_1$ or $u_{s+1}=\lambda _s+ml_2$.

• (a) If we have $u_{s+1 }=\lambda _s+nl_1$, noting that $q_1=v$, we apply Proposition A.5 and we obtain that $[0,r]\subset R_q(\Lambda _{s-1})\subset R_q(\Lambda )$.

• (b) If we have $u_{s+1 }=\lambda _s+ml_2$, noting that $q_2=v$, we apply Proposition A.5 and we obtain that $[r,n-1]\subset R_q(\Lambda )$.

$\square$

Corollary A.7

Let $\Lambda$ be an increasing semimodule such that its minimum element $\lambda _{-1}$ is a multiple of n. Let $Q_\Lambda$ be the main top of $\Lambda$. Then $c_\Lambda \le n(Q_\Lambda -1)$.

Proof

Assume first that $\Lambda$ is normalized. Let $v'$ be such that $u_{s}\in I_{v'}$ and $v''$ such that $\lambda _s\in I_{v''}$. We know that $v''\ge v'$. By Proposition  A.6, we know that $R_q(\Lambda _{s-1})$ is a circular interval for any $q\ge v'$ and hence for any $q\ge v''$. Then we are in the hypothesis of Proposition A.5 and we conclude.

Assume now that $\lambda _{-1}=kn$ and consider the normalization ${\widetilde{\Lambda }}=\Lambda -kn$. Let us note that the tops are related by the property ${{\tilde{q}}}_j=q_j-k$, for $j=1,2$ and hence $Q_{{\widetilde{\Lambda }}}=Q_{\Lambda }-k$. On the other hand $c_{{\widetilde{\Lambda }}}=c_{\Lambda }-nk$. We conclude that

$$\begin{aligned} c_{\Lambda }=c_{{\widetilde{\Lambda }}}+nk\le n(Q_{{\widetilde{\Lambda }}}-1)+nk= n(Q_{\Lambda }-1). \end{aligned}$$

$\square$

Appendix B: Structure of the Semimodule

In this Appendix we present a proof, using a different approach to the one of Delorme, of the main results on the structure of the semimodule of differential values for an E-cusp ${\mathcal {C}}$. As before, we denote $\Lambda =\Gamma (n,m,\lambda _1,\ldots ,\lambda _s)$, $n\ge 2$, the semimodule of differential values of ${\mathcal {C}}$ and we select a standard basis $\omega _{-1}=dx,\omega _0=dy,\omega _1,\ldots ,\omega _s$ of the cusp ${\mathcal {C}}$.

Proposition B.1

For each $1\le i\le s$ we have the following statements

• (1) $\lambda _i=\sup \{\nu _{\mathcal {C}}(\omega ):\nu _{E}(\omega )=t_i\}$.

• (2) If $\nu _{\mathcal {C}}(\omega )=\lambda _i$, then $\nu _{E}(\omega )=t_i$.

• (3) For each 1-form $\omega$ with $\nu _{\mathcal {C}}(\omega )\notin \Lambda _{i-1}$, there is a unique pair $a,b\ge 0$ such that $\nu _{E}(\omega )=\nu _{E}(x^ay^b\omega _i)$. Moreover, we have that $\nu _{\mathcal {C}}(\omega )\ge \lambda _i+na+mb$.

• (4) We have that $\lambda _i>u_i$.

• (5) Let $k=\lambda _i+na+mb$, then $k\notin \Lambda _{i-1}$ if and only if for all $\omega$ such that $\nu _{\mathcal {C}}(\omega )=k$ we have that $\nu _{E}(\omega )\le \nu _{E}(x^ay^b\omega _i)$.

In particular, the semimodules $\Lambda _i$ are increasing, for $i=1,2,\ldots ,s$.

Proof

Assume that $i=1$ and then $t_1=n+m=u_1$. We have

• Statement (1) is proven in Proposition 7.5.

• Statement (2) is proven in Corollary 7.7.

• Statement (3) is proven in Lemma 7.9.

• Statement (4) follows from the fact that $\lambda _1>n+m=\nu _E(\omega _1)=u_1$.

• Statement (5) is proven in Lemma 7.8.

Now, let us assume that $i\ge 2$ and take the induction hypothesis that the statements (1)-(5) are true for indices $\ell$ with $1\le \ell \le i-1$.

Denote by $\ell _1$ and $\ell _2$ the $\Lambda _{i-1}$-limits. By Eq. (15) in Remark 6.8, we have two possibilities: either $u_i=\lambda _{i-1}+n\ell _1$ or $u_i=\lambda _{i-1}+m\ell _{2}$. We assume that $u_i=\lambda _{i-1}+n\ell _1$, the computations in the case $u_i=\lambda _{i-1}+m\ell _2$ are similar ones.

The proof is founded in three claims as follows:

• Claim 1: There is a 1-form $\eta$ with $\nu _E(\eta )=t_i$ , whose initial part is proportional to the initial part of $x^{\ell _1}\omega _{i-1}$ and such that either $\nu _{\mathcal {C}}(\eta )\ge c_\Gamma$ or $\nu _{\mathcal {C}}(\eta )\notin \Lambda _{i-1}$ .

• Claim 2: Any 1-form $\omega$ with $\nu _{{\mathcal {C}}}(\omega )\notin \Lambda _{i-1}$ is reachable from $x^{\ell _1}\omega _{i-1}$.

• Claim 3: Let $\eta$ be a 1-form such that $\nu _E(\eta )=t_i$ whose initial part is proportional to the initial part of $x^{\ell _1}\omega _{i-1}$ and such that either $\nu _{\mathcal {C}}(\eta )\ge c_\Gamma$ or $\nu _{\mathcal {C}}(\eta )\notin \Lambda _{i-1}$ . Then $\nu _{\mathcal {C}} (\eta )=\lambda _i$.

We recall to the reader that the notion “initial part” refers to the concept of weighted initial part defined in Sect. 3.2.

Proof of Claim 1 Recall that $t_i=\nu _E(\omega _{i-1})+u_i-\lambda _{i-1}=\nu _E(\omega _{i-1})+n\ell _1$. Let us start with $\eta _1=x^{\ell _1}\omega _{i-1}$. We have that

$$\begin{aligned} \nu _E(\eta _1)=n\ell _1+\nu _E(\omega _{i-1})=t_i,\quad \nu _{{\mathcal {C}}}(\eta _1)=n\ell _1+\lambda _{i-1}=u_i\in \Lambda _{i-2}. \end{aligned}$$

By Statement (5) applied to $\nu _{\mathcal {C}}(\eta _1)\in \Lambda _{i-2}$, there is $\eta _1'$ with $\nu _{\mathcal {C}}(\eta '_1)=\nu _{{\mathcal {C}}}(\eta _1)$ and $\nu _E(\eta '_1)>\nu _E(\eta _1)$. Since $\nu _{{\mathcal {C}}}(\eta '_1)=\nu _{{\mathcal {C}}}(\eta _1)$, there is a non-null constant $\mu$ such that

$$\begin{aligned} \nu _{{\mathcal {C}}}({{\tilde{\eta }}})>\nu _{{\mathcal {C}}}(\eta _1),\quad \text { where } {{\tilde{\eta }}}=\eta _1-\mu \eta '_1. \end{aligned}$$

Since $\nu _E(\eta '_1)>\nu _E(\eta _1)$, we have that $\nu _E({{\tilde{\eta }}})=\nu _E(\eta _1)=t_i$ and the initial part of ${{\tilde{\eta }}}$ is the same one as the initial part of $\eta _1=x^{\ell _1}\omega _{i-1}$. If $\nu _{{\mathcal {C}}}({{\tilde{\eta }}})\ge c_\Gamma$ or $\nu _{{\mathcal {C}}}({{\tilde{\eta }}})\notin \Lambda _{i-1}$, we put $\eta ={{\tilde{\eta }}}$ and we are done. Assume that $\nu _{\mathcal {C}}({{\tilde{\eta }}})\in \Lambda _{i-1}$. Let us write

$$\begin{aligned} \nu _{{\mathcal {C}}}({{\tilde{\eta }}})=an+bm+\lambda _{\ell },\quad \ell \le i-1. \end{aligned}$$

Let us see that $\nu _E({{\tilde{\eta }}})<\nu _E(x^ay^b\omega _\ell )$; this is equivalent to verify that $t_i-t_\ell <na+mb$. Since $\nu _{\mathcal {C}}({{\tilde{\eta }}})>u_i$, in view of Lemma 7.10 we have

$$\begin{aligned} na+mb>u_i-\lambda _\ell =n\ell _1+\lambda _{i-1}-\lambda _\ell \ge n\ell _1+t_{i-1}-t_\ell =t_{i}-t_\ell . \end{aligned}$$

On the other hand, we have that $\nu _{{\mathcal {C}}}({{\tilde{\eta }}})=\nu _{{\mathcal {C}}}(x^ay^b\omega _{\ell })$. Thus, there is a constant $\mu _1$, such that $\nu _{\mathcal {C}}({{\tilde{\eta }}}_1)>\nu _{{\mathcal {C}}}({{\tilde{\eta }}})$, $\nu _E({{\tilde{\eta }}}_1)=\nu _E({{\tilde{\eta }}})$, where ${{\tilde{\eta }}}_1={{\tilde{\eta }}}-\mu _1 x^ay^b\omega _{\ell },$ and the initial part of ${{\tilde{\eta }}}_1$ is the same one as the initial part of $x^{\ell _1}\omega _{i-1}$.

If $\nu _{{\mathcal {C}}}({{\tilde{\eta }}}_1)\in \Lambda _{i-1}$, we repeat the procedure starting with ${{\tilde{\eta }}}_1$, to obtain ${{\tilde{\eta }}}_2$ such that $\nu _{{\mathcal {C}}}({{\tilde{\eta }}}_2)>\nu _{{\mathcal {C}}}({{\tilde{\eta }}}_1)$ and $\nu _E({{\tilde{\eta }}}_2)=t_i$. After finitely many repetitions, we get a 1-form $\eta$ such that $\nu _E(\eta )=t_i$, whose initial part is the same one as $x^{\ell _1}\omega _{i-1}$ and either $\nu _{{\mathcal {C}}}(\eta )\ge c_\Gamma$ or $\nu _{{\mathcal {C}}}(\eta )\notin \Lambda _{i-1}$. This proves Claim 1.

Proof of Claim 2 Take $\omega$ such that $\lambda =\nu _{{\mathcal {C}}}(\omega )\notin \Lambda _{i-1}$. Note that $\lambda \notin \Lambda _{i-2}$. By Statement (3), we have that $\omega$ is reachable from $\omega _{i-1}$. Thus, there are $a,b\ge 0$ and a constant $\mu$ such that

$$\begin{aligned} \nu _E(\omega -\mu x^ay^b\omega _{i-1})>\nu _E(\omega )=\nu _E(x^ay^b\omega _{i-1})=an+bm+t_{i-1}. \end{aligned}$$

and moreover, we have that $\lambda =\nu _{{\mathcal {C}}}(\omega )>an+bm+\lambda _{i-1}=k$ (note that $\lambda \ne k$, since $\lambda \notin \Lambda _{i-1}$).

Consider the 1-form $\omega '=\omega -\mu x^ay^b\omega _{i-1}$. We know that

$$\begin{aligned} \nu _{{\mathcal {C}}}(\omega ')=k,\quad \nu _E(\omega ')>\nu _E(x^ay^b\omega _{i-1}). \end{aligned}$$

By Statement (5), we conclude that $k\in \Lambda _{i-2}$. Hence $k\in \Lambda _{i-2}\cap (\lambda _{i-1}+\Gamma )$. Let us show that we necessarily have that $a\ge \ell _1$. Write

$$\begin{aligned} k=an+bm+\lambda _{i-1}={{\tilde{a}}} n+{{\tilde{b}}} m+\lambda _j,\quad j\le i-2. \end{aligned}$$

Since $\lambda _{i-1}>\lambda _j$, we have that $an+bm<{{\tilde{a}}} n+{{\tilde{b}}} m$. Thus, we have either $a<{{\tilde{a}}}$ or $b<{{\tilde{b}}}$. If $b<{{\tilde{b}}}$, we have that $an+\lambda _{i-1}={{\tilde{a}}} n+({{\tilde{b}}}-b)m +\lambda _{j}\in \Lambda _{i-2}\cap (\lambda _{i-1}+\Gamma )$. In view of the minimality of $\ell _1$ we should have that $\ell _1\le a$ and then $\omega$ is reachable from $x^{\ell _1}\omega _{i-1}$. Assume that $a<{{\tilde{a}}}$ and let us obtain a contradiction. We have

$$\begin{aligned} bm+\lambda _{i-1}=({{\tilde{a}}}-a) n+{{\tilde{b}}}m +\lambda _{j}\in \Lambda _{i-2}\cap (\lambda _{i-1}+\Gamma ). \end{aligned}$$

We deduce that $b\ge \ell _2$. By Statement (4), we know that $\Lambda _{i-1}$ is an increasing semimodule, starting at $\lambda _{-1}=n$. By Corollary  A.7, we know that $c_{\Lambda _{i-1}}\le n(Q_{\Lambda _{i-1}}-1)$, where $Q_{\Lambda _{i-1}}=\max \{q_1,q_2\}$ and $q_1,q_2$ are the tops of $\Lambda _{i-1}$. Suppose that $\lambda \in I_{d}$, we have

• (1) $\lambda >k=an+bm+\lambda _{i-1}\ge u_i=\ell _1n+\lambda _{i-1}$ and hence $d\ge q_1$.

• (2) $\lambda >k=an+bm+\lambda _{i-1}\ge \ell _2m+\lambda _{i-1}$ and hence $d\ge q_2$.

We conclude that $\lambda \in \Lambda _{i-1}$, contradiction. This ends the proof of Claim 2.

Proof of Claim 3 Note that $\nu _{\mathcal {C}}(\eta )\ge \lambda _i$. Assume that $\lambda =\nu _{{\mathcal {C}}}(\eta )>\lambda _i$. Recalling that $\nu _{{\mathcal {C}}}(\omega _i)=\lambda _i\notin \Lambda _{i-1}$ and that the initial part of $\eta$ is proportional to the initial part of $x^{\ell _1}\omega _{i-1}$, we can apply Claim 2 and we get that $\omega _i$ is reachable from $\eta$. Then there are $a,b\ge 0$ and a constant $\mu$ such that $\nu _E(\omega _i-\mu x^ay^b\eta )>\nu _E(\omega _i)$. Put $\omega _i^1=\omega _i-\mu x^ay^b\eta$. We have that $\nu _{{\mathcal {C}}}(\omega _i^1)=\lambda _i$, since $\nu _{{\mathcal {C}}}(\mu x^ay^b\eta )\ge \lambda >\lambda _i$. In this way we produce an infinite list of strictly increasing divisorial order 1-forms

$$\begin{aligned} \omega _i=\omega _i^0,\omega _i^1,\omega _i^2,\ldots \end{aligned}$$

such that $\nu _{{\mathcal {C}}}(\omega _i^j)=\lambda _i$, for any $i\ge 0$. For an index j we have that $\nu _{E}(\omega _i^j)\ge c_\Gamma$ and then $\lambda _i\ge \nu _{E}(\omega _i^j)\ge c_\Gamma$ and this is a contradiction. So we necessarily have that $\nu _{{\mathcal {C}}}(\eta )=\lambda _i$. This ends the proof of Claim 3.

Proof of Statements (1) and (2): In view of Claim 1 and Claim 3, there is a 1-form $\eta$ with $\nu _E(\eta )=t_i$ such that $\nu _{\mathcal {C}}(\eta )=\lambda _i$, whose initial part is proportional to the initial part of $x^{\ell _1}\omega _{i-1}$. In order to prove Statement (1), it remains to prove that if $\nu _E(\omega )=t_i$ then $\nu _{{\mathcal {C}}}(\omega )\le \lambda _i$. Assume that $\lambda =\nu _{\mathcal {C}}(\omega )>\lambda _i=\nu _{\mathcal {C}}(\eta )$. The 1-form $\omega$ is basic and resonant and it has the same divisorial order as $\eta$. Hence there is a constant $\mu \ne 0$ such that

$$\begin{aligned} \nu _E(\eta ^1)>t_i=\nu _E(\eta )=\nu _E(\omega ), \quad \eta ^1=\eta -\mu \omega . \end{aligned}$$

The 1-form $\eta ^1$ satisfies that $\nu _{\mathcal {C}}(\eta ^1)=\lambda _i\notin \Lambda _{i-1}$; by Claim 2, there are $a,b\ge 0$ and a constant $\mu '$ such that

$$\begin{aligned} \nu _E(\eta ^2)>\nu _E(\eta ^1),\quad \eta ^2=\eta ^1-\mu 'x^ay^b\eta . \end{aligned}$$

We have that $\nu _{{\mathcal {C}}}(\eta ^2)=\lambda _i$ and $\nu _E(\eta ^2)>\nu _E(\eta ^1)$. Repeating this procedure, we have a list of 1-forms $\eta ^1,\eta ^2,\ldots$ with strictly increasing divisorial order such that $\nu _{{\mathcal {C}}}(\eta ^j)=\lambda _i$ for any j. We find a contradiction just by considering one of such $\eta ^j$ with $\nu _E(\eta ^j)\ge c_\Gamma$. This ends the proof of Statement (1).

Let us prove Statement (2). Choose $\omega$ with $\nu _{{\mathcal {C}}}(\omega )=\lambda _i$. By Claim 2, we have that $\omega$ is reachable from $\eta$ and hence $\nu _E(\omega )\ge t_i$. Assume by contradiction that $\nu _E(\omega )>t_i$. There is a constant $\mu$ and $a,b\ge 0$ with $a+b\ge 1$ such that

$$\begin{aligned} \nu _E(\omega ^1)>\nu _E(\omega ),\quad \omega ^1=\omega -\mu x^ay^b\eta . \end{aligned}$$

Since $\nu _{{\mathcal {C}}}(\mu x^ay^b\eta )=an+bm+\lambda _i>\lambda _i$, we have that $\nu _{\mathcal {C}}(\omega ^1)=\lambda _i$. Repeating the argument, we get a sequence of 1-forms $\omega ^0=\omega ,\omega ^1,\ldots$ with strictly increasing divisorial order such that $\nu _{\mathcal {C}}(\omega ^j)=\lambda _i$ for any j. This is a contradiction.

Proof of Statement (3): By Claim 2, we have that $\omega _i$ is reachable from $x^{\ell _1}\omega _{i-1}$. By Statement (2) (already proved) we have that $\nu _E(\omega _i)=t_i$. Hence the initial part of $\omega _{i}$ is proportional to the initial part of $x^{\ell _1}\omega _{i-1}$. Consider a 1-form $\omega$ with $\nu _{{\mathcal {C}}}(\omega )\notin \Lambda _{i-1}$. By Claim 2 the 1-form $\omega$ is reachable from $x^{\ell _1}\omega _{i-1}$ and hence it is reachable from $\omega _i$. Then, there are $a,b\ge 0$ such that

$$\begin{aligned} \nu _E(x^ay^b\omega _i)=an+bm+t_i=\nu _E(\omega ). \end{aligned}$$

Since $\nu _{\mathcal {C}}(\omega )\notin \Lambda _{i-1}$, we have that $nm>\nu _{\mathcal {C}}(\omega )>\nu _E(\omega )>an+bm$, this implies the uniqueness of a, b. Let us show that $\nu _{{\mathcal {C}}}(\omega )\ge an+bm+\lambda _i$. Assume by contradiction that $\nu _{{\mathcal {C}}}(\omega )< an+bm+\lambda _i$. Consider $\omega ^1=\omega -\mu x^ay^b\omega _{i}$ such that $\nu _E(\omega ^1)>\nu _E(\omega )$. In view of the contradiction hypothesis, we have that $\nu _{{\mathcal {C}}}(\omega ^1)=\nu _{{\mathcal {C}}}( \omega )$. Moreover, if $\nu _E(\omega ^1)=\nu _E(x^{a_1}y^{b_1}\omega _i)$ we also have that $\nu _{{\mathcal {C}}}(\omega )< a_1n+b_1m+\lambda _i$. The situation repeats and we obtain an infinite sequence of 1-forms $\omega ^0=\omega ,\omega ^1,\omega ^2,\ldots$ with strictly increasing divisorial orders, such that $\nu _{{\mathcal {C}}}(\omega ^j)=\nu _{{\mathcal {C}}}(\omega )$ for any $j\ge 0$. This is a contradiction.

Proof of Statement (4): Noting that $\nu _E(x^{\ell _1}\omega _{i-1})=t_i$, by Statement (1) we have $\lambda _i\ge \nu _{\mathcal {C}}(x^{\ell _1}\omega _{i-1})=n\ell _1+\lambda _{i-1}=u_i$. On the other hand, since $\lambda _i\notin \Lambda _{i-1}$, we have that $\lambda _i\ne u_i$ and hence $\lambda _i>u_i$.

Proof of Statement (5): Consider $k=\lambda _i+na+mb$. Assume first that $k\notin \Lambda _{i-1}$. Let $\omega$ be such that $\nu _{{\mathcal {C}}}(\omega )=k$. We have to prove that

$$\begin{aligned} \nu _{E}(\omega )\le \nu _E(x^ay^b\omega _i)=an+bm+t_i. \end{aligned}$$

In view of Statement (3), we know that $\omega$ is reachable from $\omega _{i}$. Hence there are $a',b'\ge 0$ and a constant $\mu$ such that $\nu _E(\omega -\mu x^{a'}y^{b'}\omega _i)>\nu _E(\omega )$. Hence

$$\begin{aligned} \nu _E(\omega )=\nu _E(x^{a'}y^{b'}\omega _i)=a'n+b'm+t_i. \end{aligned}$$

Assume by contradiction that $\nu _{E}(\omega )>\nu _E(x^ay^b\omega _i)=an+bm+t_i$. This implies that $a'n+b'm>an+bm$ and thus

$$\begin{aligned} \nu _{{\mathcal {C}}}(x^{a'}y^{b'}\omega _i)=a'n+b'm+\lambda _i>k=an+bm+\lambda _i=\nu _{{\mathcal {C}}}(x^ay^b\omega _i)=\nu _{{\mathcal {C}}}(\omega ). \end{aligned}$$

Put $\omega ^1=\omega -\mu x^{a'}y^{b'}\omega _i$. We have that $\nu _{{\mathcal {C}}}(\omega ^1)=k$. Repeating the argument with $\omega ^1$, we obtain an infinite list of increasing divisorial orders 1-forms $\omega ^0=\omega ,\omega ^1,\omega ^2,\ldots$ such that $\nu _{\mathcal {C}}(\omega ^j)=k\notin \Lambda _{i-1}$. This is a contradiction.

Assume now that $k\in \Lambda _{i-1}$. There is an index $\ell \le i-1$ such that

$$\begin{aligned} k=an+bm+\lambda _{i}=a'n+b'm+\lambda _{\ell }. \end{aligned}$$

By Lemma 7.10, we have that $\lambda _i-\lambda _\ell >t_i-t_\ell$ and hence $an+bm+t_{i}<a'n+b'm+t_{\ell }.$ The 1-form $x^{a'}y^{b'}\omega _\ell$ satisfies that $k=\nu _{\mathcal {C}}(x^{a'}y^{b'}\omega _\ell )$ and

$$\begin{aligned} \nu _E(x^{a'}y^{b'}\omega _\ell )=a'n+b'm+t_{\ell }>an+bm+t_{i}=\nu _E(x^ay^b\omega _i). \end{aligned}$$

This ends the proof. $\square$

Appendix C: Delorme’s Decompositions

In this Appendix, we provide a proof, using another approach, of Delorme’s decompositions stated in Theorem 8.5. That is, we consider a cusp ${\mathcal {C}}\in {\text {Cusps}}(E)$, an extended standard basis $\omega _{-1},\omega _0,\omega _1,\ldots ,\omega _s;\omega _{s+1}$ of ${\mathcal {C}}$, where $\Lambda =\Gamma (n,m,\lambda _1,\ldots , \lambda _s)$ is the semimodule of differential values of ${\mathcal {C}}$. We have to prove that for any indices $0\le j\le i\le s$, there is a decomposition

$$\begin{aligned} \omega _{i+1}=\sum _{\ell =-1}^{j}f_\ell ^{ij}\omega _\ell \end{aligned}$$

such that, for any $-1\le \ell \le j$ we have $\nu _{{\mathcal {C}}}(f_\ell ^{ij}\omega _\ell )\ge v_{ij}$, where $v_{ij}=t_{i+1}-t_j+\lambda _j$ and there is exactly one index $-1\le k\le j-1$ such that $\nu _{{\mathcal {C}}}(f_k^{ij}\omega _k)=\nu _{{\mathcal {C}}}(f_j^{ij}\omega _j)= v_{ij}$.

Note that the case $i=0$ is straightforward. Indeed, we have $v_{00}=n+m$ and if we write $\omega _1=adx+bdy$, we necessarily have that $\nu _{{\mathcal {C}}}(adx)=\nu _{\mathcal {C}}(bdy)=n+m$ in view of the fact that the initial part of $\omega _1$ is proportional to $mydx-nxdy$.

Thus, we assume that $i\ge 1$.

Lemma C.1

Given a 1-form $\eta$ with $\nu _{\mathcal {C}}(\eta )>u_{i+1}$ and $\nu _E(\eta )>t_{i+1}$, we have

• (a) If $\nu _E(\eta )<nm$, there is a 1-form $\alpha$ such that $\nu _E(\eta -\alpha )>\nu _E(\eta )$ that can be decomposed as $\alpha =\sum _{\ell =-1}^{i}g_\ell \omega _\ell$, where $\nu _{\mathcal {C}}(g_\ell \omega _\ell )>u_{i+1}$ for $-1\le \ell \le i$.

• (b) If $\nu _E(\eta )\ge nm$, there is a decomposition $\eta =\sum _{\ell =-1}^{i}h_\ell \omega _\ell$ where each summand $h_\ell \omega _\ell$ satisfies that $\nu _{\mathcal {C}}(h_\ell \omega _\ell )>u_{i+1}$.

Proof

(b) Assume that $\nu _E(\eta )\ge nm$, we have $\eta =fdx+gdy=f\omega _{-1}+g\omega _0$, where $\nu _E(fdx)\ge nm$ and $\nu _E(gdy)\ge nm$. In view of Lemma 6.5, we have that $u_{i+1}<c_\Gamma +n<nm$. We are done by taking the decomposition $\eta =f\omega _{-1}+g\omega _0$.

(a) Assume that $\nu _E(\eta )<nm$. Note that $\eta$ is a basic 1-form. There are two possible cases: $\eta$ is resonant or not. Assume first that $\eta$ is not resonant. Then

$$\begin{aligned} \nu _{\mathcal {C}}(\eta )=\nu _E(\eta )=\nu _E(\alpha )>u_{i+1}, \end{aligned}$$

where $\alpha$ is the initial part of $\eta$. Note that $\nu _E(\eta -\alpha )>\nu _E(\eta )$. We can write $\alpha$ $=g_{-1}dx+g_0dy=g_{-1}\omega _{-1}+g_0\omega _0$, where

$$\begin{aligned} \nu _{\mathcal {C}}(g_\ell \omega _\ell )\ge \nu _E(g_\ell \omega _\ell )\ge \nu _E(\alpha )=\nu _E(\eta )=\nu _{\mathcal {C}}(\eta )>u_{i+1},\quad \ell =-1,0. \end{aligned}$$

This is the desired decomposition.

Assume that $\eta$ is resonant. Define $k=\max \{\ell \le i; \; \eta \text { is reachable from } \omega _\ell \}$. The fact that $\eta$ is resonant implies that $k\ge 1$ (recall that $i\ge 1$). Consider $a,b\ge 0$ and a constant $\varphi$ such that

$$\begin{aligned} \nu _E({{\tilde{\eta }}})>\nu _E(\eta ),\quad {{\tilde{\eta }}}=\eta -\varphi x^ay^b\omega _k. \end{aligned}$$

(21)

If we show that $\nu _{\mathcal {C}}(x^ay^b\omega _k)>u_{i+1}$, we are done. Let us do it. Assume first that $k=i$. We know that

$$\begin{aligned} \nu _E(\eta )=\nu _E(x^ay^b\omega _{i})=an+bm+t_{i}>t_{i+1} =t_{i}+u_{i+1}-\lambda _{i}. \end{aligned}$$

This implies that $an+bm>u_{i+1}-\lambda _{i}$ and then $\nu _{{\mathcal {C}}}(x^ay^b\omega _{i})=an+bm+\lambda _{i}>u_{i+1}$.

Assume now that $1\le k\le i-1$. Let us reason by contradiction assuming that $\nu _{{\mathcal {C}}}(x^ay^b\omega _k)\le u_{i+1}$. Denote by ${{\tilde{\eta }}}=\eta -\varphi x^ay^b\omega _k$. By Eq. (21), we know that

$$\begin{aligned} \nu _E({{\tilde{\eta }}})>\nu _E(x^ay^b\omega _k)=an+bm+t_k. \end{aligned}$$

(22)

Since $\nu _{\mathcal {C}}(x^ay^b\omega _k)\le u_{i+1}<\nu _{{\mathcal {C}}}(\eta )$, we have that

$$\begin{aligned} \nu _{\mathcal {C}}({{\tilde{\eta }}})=\nu _{\mathcal {C}}(x^ay^b\omega _k) =an+bm+\lambda _k. \end{aligned}$$

(23)

In view of Eqs. (23) and (22), we can apply Statement (5) in Proposition B.1 to conclude that

$$\begin{aligned} an+bm+\lambda _k\in \Lambda _{k-1}. \end{aligned}$$

(24)

Let $\ell _1$ and $\ell _2$ be the $\Lambda _k$-limits. Since $an+bm+\lambda _k\in \Lambda _{k-1}$, we have that $a\ge \ell _1$ or $b\ge \ell _2$, by Lemma  6.9. We have four cases to be considered:

$$\begin{aligned} u_{k+1}= & {} n\ell _1+\lambda _k \text { and } a\ge \ell _1;\quad u_{k+1}= n\ell _1+\lambda _k \text { and } b\ge \ell _2;\\ u_{k+1}= & {} m\ell _2+\lambda _k \text { and } a\ge \ell _1;\quad u_{k+1}= m\ell _2+\lambda _k \text { and } b\ge \ell _2. \end{aligned}$$

Assume that $u_{k+1}= n\ell _1+\lambda _k$ and $a\ge \ell _1$. This implies that $x^ay^b\omega _k$, and hence $\eta$, is reachable from $x^{\ell _1}\omega _k$ and hence from $\omega _{k+1}$. This contradicts the maximality of the index k.

Assume that $u_{k+1}= n\ell _1+\lambda _k$ and $b\ge \ell _2$ and $a<\ell _1$. We have that

$$\begin{aligned} \nu _{\mathcal {C}}(x^ay^b\omega _k)\ge \nu _{{\mathcal {C}}}(y^{\ell _2}\omega _k)=m\ell _2 +\lambda _k>u_{k+1}=n\ell _1+\lambda _k. \end{aligned}$$

Let $q_1$ and $q_2$ be the tops of $\Lambda _{k}$ and $Q_{\Lambda _k}$ the main top, we have that

$$\begin{aligned} \nu _{\mathcal {C}}(x^ay^b\omega _k)\ge nQ_{\Lambda _k}\ge c_{\Lambda _k}+n. \end{aligned}$$

Recall that $c_{\Lambda _k}\le n(Q_{\Lambda _k}-1)$ in view of Proposition A.5. On the other hand, we know by Lemma  6.5 that $u_{i+1}<c_{\Lambda _i}+n\le c_{\Lambda _k}+n$. We have the contradiction $u_{i+1}<c_{\Lambda _k}+n \le \nu _{\mathcal {C}}(x^ay^b\omega _k)\le u_{i+1}$.

The two remaining cases with $u_{k+1}= m\ell _2+\lambda _k$ may be considered in a similar way to the previous ones. $\square$

Proposition C.2

We can write $\omega _{i+1}=\sum _{\ell =-1}^{i}f_\ell \omega _\ell$ where $\nu _{\mathcal {C}}(f_\ell \omega _\ell )\ge u_{i+1}$ for $-1\le \ell \le i$ and such that $\nu _{{\mathcal {C}}}(f_{i}\omega _{i})=u_{i+1}$ and there is exactly one index $k\in \{-1,0,1,\ldots ,i-1\}$ satisfying that $\nu _{{\mathcal {C}}}(f_k\omega _k)= u_{i+1}$.

Proof

Let us consider first the case $i=0$. We know that $u_1=t_1=n+m$ and that $\omega _1$ is basic resonant, with $\nu _{E}(\omega _1)=n+m$. Then, there is a constant $\mu$ such that $\nu _E(\eta )>n+m$, where $\eta =\omega _1-\mu (mydx-nxdy)$. We can write $\eta =g_{-1}dx+g_0dy$, where $\nu _E(g_{-1})>m$ and $\nu _E(g_0)>n$. Let us put $f_{-1}= \mu my+g_{-1}$ and $f_0=-\mu nx+g_{0}$. We have that $\nu _{{\mathcal {C}}}(f_{-1})=m$ and $\nu _{{\mathcal {C}}}(f_0)=n$; hence $\omega _1=f_{-1}dx+f_0dy= f_{-1}\omega _{-1}+f_0\omega _0$ is the desired decomposition.

Assume now that $1\le i\le s$. Let $\ell _1,\ell _2$ be the limits of $\Lambda _i$. By Remark 6.8, there is exactly one index k with $-1\le k\le i-1$ such that

• (1) If $u_{i+1}=\ell _1n+\lambda _i$, then $u_{i+1}=\lambda _k+bm$ (note that $b\ge 1$).

• (2) If $u_{i+1}=\ell _2m+\lambda _i$, then $u_{i+1}=\lambda _k+an$ (note that $a\ge 1$).

Assume that $u_{i+1}=\ell _1n+\lambda _i$, the case $u_{i+1}=\ell _2m+\lambda _i$ is symmetric to this one. We have that

$$\begin{aligned} \nu _{{\mathcal {C}}}(x^{\ell _1}\omega _{i})=u_{i+1} =\nu _{{\mathcal {C}}}(y^{b}\omega _k). \end{aligned}$$

(25)

On the other hand, we have that

$$\begin{aligned} \nu _E(x^{\ell _1}\omega _{i})= & {} t_i+ \ell _1n= t_{i} +(u_{i+1}-\lambda _{i})=t_{i+1};\\ \nu _E(y^b\omega _{k})= & {} t_k+bm= t_k+u_{i+1}-\lambda _{k}. \end{aligned}$$

By Lemma 7.10, we have that

$$\begin{aligned} t_k+u_{i+1}-\lambda _{k}-t_{i+1}= (\lambda _{i}-\lambda _k)-(t_{i}-t_k)>0, \end{aligned}$$

and hence $t_{i+1}=\nu _E(x^{\ell _1}\omega _{i})<\nu _E(y^b\omega _{k})$. Since both $\omega _{i+1}$ and $x^{\ell _1}\omega _{i}$ are basic resonant with the same divisorial order $t_{i+1}$, there is a constant $\varphi$ such that

$$\begin{aligned} \nu _E(\omega _{i+1}-\varphi x^{\ell _1}\omega _{i})>\nu _E(\omega _{i+1})=t_{i+1}. \end{aligned}$$

By Eq. (25), there is a constant $\mu$ such that

$$\begin{aligned} \nu _{\mathcal {C}}(\omega _{i+1}^{0})>u_{i+1}, \text { where } \omega _{i+1}^{0}=\varphi x^{\ell _1}\omega _{i}-\mu y^b\omega _{k}. \end{aligned}$$

Put $\eta ^0=\omega _{i+1}-\omega _{i+1}^0=\omega _{i+1}-\varphi x^{\ell _1}\omega _{i}+\mu y^b\omega _{k}$. We have that $\nu _E(\eta ^0)>t_{i+1}$ and $\nu _{{\mathcal {C}}}(\eta ^0)>u_{i+1}$ in view of the following facts:

• (1) $\nu _E(\eta ^0)\ge \min \{\nu _E(\omega _{i+1}-\varphi x^{\ell _1}\omega _{i}),\nu _E(\mu y^b\omega _{k})\}>t_{i+1}$.

• (2) $\nu _{{\mathcal {C}}}(\eta ^0)\ge \min \{\nu _{\mathcal {C}}(\omega _{i+1}),\nu _{{\mathcal {C}}}(\omega _{i+1}^0)\}= \min \{\lambda _{i+1},\nu _{{\mathcal {C}}}(\omega _i^0)\}>u_{i+1}$. Recall that $\Lambda$ is an increasing semimodule; (here we put $\lambda _{s+1}=\infty$).

The proof is now a consequence of Lemma C.1 as follows. We start with $\eta ^0$ as before. If $\nu _E(\eta ^0)\ge nm$, we apply Lemma C.1 (b). We are done by taking the decomposition

$$\begin{aligned} \omega _{i+1}=\omega _{i+1}^0+ \sum _{\ell =-1}^{i}h_\ell \omega _\ell = \varphi x^{\ell _1}\omega _{i}-\mu y^b\omega _{k}+ \sum _{\ell =-1}^{i}h_\ell \omega _\ell . \end{aligned}$$

If $\nu _E(\eta ^0)<nm$, we apply Lemma C.1 (a) and we obtain $\eta ^1=\eta ^0-\sum _{\ell =-1}^{i}g_\ell \omega _\ell$ such that $\nu _E(\eta ^1)>\nu _E(\eta ^0)>t_{i+1}$ and $\nu _{\mathcal {C}}(\eta ^1)>u_{i+1}$. If $\nu _E(\eta ^1)< nm$, we re-apply Lemma C.1 (a) to $\eta ^1$. After finitely many steps, we obtain

$$\begin{aligned} {{\tilde{\eta }}}=\eta ^0- \sum _{\ell =-1}^{i}{{\tilde{g}}}_\ell \omega _\ell , \end{aligned}$$

such that $\nu _E({{\tilde{\eta }}})\ge nm$, $\nu _{{\mathcal {C}}}({{\tilde{\eta }}})>u_{i+1}$ and $\nu _{E}({{\tilde{\eta }}})>t_{i+1}$, where $\nu _{{\mathcal {C}}}({{\tilde{g}}}_\ell \omega _\ell )>u_{i+1}$ for any $-1\le \ell \le i$. We apply Lemma C.1 (b) to ${{\tilde{\eta }}}$ to obtain that ${{\tilde{\eta }}}=\sum _{\ell =-1}^{i}{{\tilde{h}}}_\ell \omega _\ell$ with $\nu _{{\mathcal {C}}}({{\tilde{h}}}_\ell \omega _\ell )>u_{i+1}$. The desired decomposition is given by

$$\begin{aligned} \omega _{i+1}= \omega _{i+1}^0 + \sum _{\ell =-1}^{i}({{\tilde{g}}}_\ell +{{\tilde{h}}}_\ell )\omega _\ell = \varphi x^{\ell _1}\omega _{i}-\mu y^b\omega _{k} + \sum _{\ell =-1}^{i}({{\tilde{g}}}_\ell +{{\tilde{h}}}_\ell )\omega _\ell . \end{aligned}$$

This ends the proof. $\square$

Let us end the proof of Theorem 8.5. We already know that it is true when $j=i$, in view of Proposition C.2. We assume that the result is true for $j+1\le i$ and let us show that it is true for $0\le j\le i$. In order to simplify notations, let us write

$$\begin{aligned} v_j= t_{i+1}-t_j+\lambda _j,\quad v_{j+1}= t_{i+1}-t_{j+1}+\lambda _{j+1}. \end{aligned}$$

Recall that $v_{j+1}=v_j+\lambda _{j+1}-u_{j+1}$. By induction hypothesis, we have that

$$\begin{aligned} \omega _{i+1}=\sum _{\ell =-1}^{j+1}h_\ell \omega _{\ell }, \end{aligned}$$

where $\nu _{\mathcal {C}}(h_\ell \omega _{\ell })\ge v_{j+1}$ for any $-1\le \ell \le j+1$ and $\nu _{\mathcal {C}}(h_{j+1}\omega _{j+1})= v_{j+1}$. We apply Proposition C.2 to write $\omega _{j+1}=\sum _{\ell =-1}^{j}g_\ell \omega _\ell$, where $\nu _{\mathcal {C}}(g_\ell \omega _\ell )\ge u_{j+1}$ for any $-1\le \ell \le j$ and there is exactly one index k such that $\nu _{\mathcal {C}}(g_{j}\omega _{j})=\nu _{\mathcal {C}}(g_{k}\omega _{k})= u_{j+1}$. Now, we have an expression

$$\begin{aligned} \omega _{i+1}=\sum _{\ell =-1}^{j}f_\ell \omega _\ell ,\quad f_\ell =h_\ell +h_{j+1}g_\ell . \end{aligned}$$

We have the following properties:

• (1) $\nu _{\mathcal {C}}(h_\ell \omega _\ell )> v_{j}$, for any $-1\le \ell \le j$. Indeed, we know that

  $$\begin{aligned} v_{j+1}= v_j+(\lambda _{j+1}-u_{j+1})>v_j, \end{aligned}$$

  recall that the semimodule is increasing and then $\lambda _{j+1}>u_{j+1}$.

• (2) $\nu _{\mathcal {C}}(h_{j+1}g_\ell \omega _\ell )\ge v_{j}$ and k, j are the unique indices such that

  $$\begin{aligned} \nu _{\mathcal {C}}(h_{j+1}g_j\omega _j)=\nu _{\mathcal {C}}(h_{j+1}g_k\omega _k)=v_j. \end{aligned}$$

  In order to prove this, it is enough to note that

  $$\begin{aligned} \nu _{\mathcal {C}}(h_{j+1}g_\ell \omega _\ell )=(v_{j+1}-\lambda _{j+1})+\nu _{\mathcal {C}}(g_\ell \omega _\ell )\ge (v_{j+1}-\lambda _{j+1})+u_{j+1}=v_j \end{aligned}$$

  and the equality holds exactly for the indices $\ell =j,k$.

The desired result comes from the above properties (1) and (2), noting that

$$\begin{aligned} \nu _{\mathcal {C}}(f_\ell \omega _\ell )\ge \min \{\nu _{\mathcal {C}}(h_\ell \omega _\ell ), \nu _{\mathcal {C}}(h_{j+1}g_\ell \omega _\ell )\} \end{aligned}$$

and the equality holds when the two values are different.