The analytic classification of plane curves

Abstract

In this paper, we present a solution to the problem of the analytic classification of germs of plane curves with several irreducible components. Our algebraic approach follows precursive ideas of Oscar Zariski and as a subproduct allows us to recover some particular cases found in the literature.

1. Introduction

A remarkable step towards the understanding of the local structure of a germ of complex plane curve was the characterization of its topological type. Let $\mathcal {C}_1$ and $\mathcal {C}_2$ be germs of analytic reduced plane curves at the origin of $\mathbb {C}^2$. We say that $\mathcal {C}_1$ and $\mathcal {C}_2$ are topologically equivalent (as embedded germs) if there exist $U$ and $V$ neighborhoods at the origin of $\mathbb {C}^2$ and a homeomorphism $\Phi :U \to V$ such that $\Phi (\mathcal {C}_1 \cap U)= \mathcal {C}_2 \cap V$. In this case, Zariski in [Zar71, Zar06] says that $\mathcal {C}_1$ and $\mathcal {C}_2$ have the same topological type or they are equisingular. When $\Phi$ is an analytic isomorphism, the curves are called analytically equivalent.

The local topology of plane curves has been studied since the first decades of the last century, with important contributions from Brauner, Burau, Zariski, Milnor, and others (see [BK86]). For an irreducible plane curve (branch) the topological type of the curve is equivalent to the topology of the complement of the associated link, that is, the intersection of the curve with a small sphere centered at the origin. This intersection is an iterated torus knot. The topological type of the curve is completely described by the type of the knot which is characterized by pairs of integers called Puiseux pairs. There are many complete discrete invariants that determine and are determined by Puiseux pairs, for instance, the characteristic exponents, the value semigroup of the curve, the multiplicity sequence associated to the canonical resolution, and others. For plane curves with several branches, the local topology can be described by the value semigroup of the curve or, equivalently, by the value semigroup of each branch and the intersection multiplicities of pairs of branches (see [Wal72, Zar71]).

With regard to analytic equivalence, there are many hard problems. For instance, how we can decide if two curves are analytically equivalent? What is the associated moduli space?

Any introduction to the analytic classification subject of plane curves is hardly comparable to the review by Washburn presented in [Was88]. We just cite some contributions to the theme in chronological order.

In 1965, Ebey [Ebe65] presented normal forms for some classes of irreducible curves given by parameterizations. In a course given at the Centre de Mathématiques de l’École Polytechnique in 1973, Zariski [Zar06] considered an irreducible analytic plane curve $\mathcal {C}$ and denoted by $\mathbb {L}=\mathbb {L}(\mathcal {C})$ the equisingular class of $\mathcal {C}$, that is, the set of all branches equisingular to $\mathcal {C}$. The moduli space $\mathbb {M}$ is the quotient space of $\mathbb {L}$ by the analytic equivalence relation. Zariski studied the moduli space for some equisingular classes and gave a formula for the dimension of the generic component $\mathbb {M}_g$ of $\mathbb {M}$ for curves with semigroup $\langle n,m\rangle$ in which $1< n < m$ and $m \equiv 1 \ {\rm mod} \ n$. A few years later, Delorme (in 1978; see [Del78]), considering irreducible curves with semigroup $\langle n,m\rangle$, presented a combinatorial method to obtain the dimension of the generic component of the moduli space. In 1979, Granger presented normal forms for plane curves with nonsingular transversal branches (an ordinary multiple point) and gave a formula to compute the dimension of $\mathbb {M}_g$ (see [Gra79]).

Bruce and Gaffney in 1982 (see [BG82]), classified the simple irreducible plane curves, that is, when the moduli space is a zero-dimensional space. A more general approach for the moduli space was presented by Laudal and Pfister in 1988 (see [LP88]) where, for irreducible plane curves with semigroup $\langle n,m\rangle$, they fixed an analytic invariant (the Tjurina number) in order to describe normal forms given by elements in $\mathbb {C}[X,Y]$. In 1994, Greuel and Pfister (see [GP94]) developed a general method to construct coarse moduli spaces for singularities in the sense of Mumford Geometric Invariant Theory. Kang (in 2000; see [Kan00]) and Câmara and Scárdua (in 2018; see [CS18]), using different methods, classified analytic plane curves defined by weighted homogeneous polynomials. Kolgushkin and Sadykov (in 2001; see [KS01]) obtained normal forms for stably simple reducible curve singularities in complex spaces of any dimension.

In 2011, Hefez and the first author (see [HH11]) presented a solution for the analytic classification of irreducible plane curves as considered by Zariski in [Zar06] (see also [HH21]). Stratifying each equisingularity class by the analytical invariant given by the set of values of Kähler differentials, they presented normal forms for each stratum that allowed them to describe the moduli space. Later, the authors in a joint work with Hefez (in 2015; see [HHR15]), generalized such method to obtain the analytic classification of plane curves with two irreducible components.

Genzmer and Paul (in 2016; see [GP16]), using tools of Foliation theory, described the moduli space for generic plane curves such that every branch admits semigroup $\langle n,m\rangle$ and they presented a method to obtain the normal form for the generic case. Ayuso and Ribón (in 2020; see [AR20]), using holomorphic flows recovered the normal forms for branches presented in [HH11]. In 2022, Genzmer [Gen22] obtained a formula for the dimension of $\mathbb {M}_g$ for any irreducible plane curve.

It is remarkable to observe that for about 50 years the questions proposed by Oscar Zariski in [Zar06], directly or indirectly, motivated these and so many other researchers that we could not possibly mention them all. The aim of this work is to present an answer to the intricate problem of the analytic classification of reduced plane curves in a fixed topological class. The combinatorial issues involved in this problem were probably our biggest challenge and makes this case not a simple generalization of the results about curves with two irreducible components presented in [HHR15].

We consider a topological class fixing the value semiring $\Gamma$ of a reduced plane curve $\mathcal {C}$. The value semiring is equivalent, as a set, to the classical value semigroup. The main difference is that the value semiring, equipped with the tropical operations, admits a finite minimal set of generators in contrast to the value semigroup (see [CH20a]), and such generators allow us to recover directly the topological data pointed out by Zariski, that is, the value semigroup of each branch and their mutual intersection multiplicities. With our approach we recover the normal forms for the irreducible and two branches cases, presented in [HH11] and [HHR15], but it is not a simple induction step. In fact, the passage $1 \to 2 \to r$ branches requires extra efforts and finer analysis in several situations. For example, understanding the semigroup for two branches as presented by Garcia (see [Gar82]) is not sufficient to describe the semigroup for $r\geqslant 3$. In fact, Mata (see [Mat87]) introduced new ingredients and non-trivial combinatorial aspects to perform this task. This important object and other invariants defined by a set of values of fractional ideals are described in § 2.

Our strategy is, under the action of a permutation group, to consider the branches of the curve in blocks according to their tangent lines, then we order the blocks and their elements by the multiplicities of the branches. In addition, we identify the appropriated group $\mathcal {G}$ of local diffeomorphisms that preserve such properties. This is done in § 3. In § 4 we obtain the Puiseux block form of a multigerm (Proposition 4.6) that is a convenient parameterization of each component of the multigerm.

Section 5 contains the main results of this work. We introduce the $\mathcal {G}$-invariant $\Lambda _{\mathcal {G}}$ that corresponds to values of elements in a fractional ideal $\mathcal {I}_{\mathcal {G}}$ of the local ring $\mathcal {O}$ of $\mathcal {C}$. The elements of $\mathcal {I}_{\mathcal {G}}$ are closely related to elements of the tangent space to the $\mathcal {G}$-orbit of the multigerm $\varphi$ associated to the curve $\mathcal {C}$ (Proposition 5.1). In Theorem 5.8 we present a $\mathcal {G}$-normal form of $\varphi$ by a reduction process using the set $\Lambda _{\mathcal {G}}$, more specifically by their fibers. Theorem 5.10 takes into account the homothety group action and by this result we can decide whether two curves are analytically equivalent or not as we discuss in the end of the section.

Finally, in § 6 we apply our techniques to recover some known results related with the analytic equivalence of plane curves: the irreducible case, bigerms, and the ordinary multiple point singularity. Concerning the case of plane curves in which each branch admits value semigroup $\langle n,m\rangle$ and mutual intersection multiplicities equal to $nm$ we apply our results to obtain a pre-normal form taking into account the value semiring $\Gamma$ (Proposition 6.7) and we discuss about the generic component of the corresponding moduli space (Proposition 6.8 and Corollary 6.10).

2. Analytic equivalence and invariants

We denote by $\mathbb {C}\{X,Y\}$ the power series ring with complex coefficients in the variables $X$ and $Y$, which are absolutely convergent in a neighborhood of the origin in $\mathbb {C}^2$ and by $\mathcal {M}=\langle X,Y \rangle$ its maximal ideal. Let $\mathcal {C}$ be the germ of an analytic reduced plane curve in $(\mathbb {C}^2,0)$ defined by a reduced element $f\in \mathcal {M}\setminus \{0\}$, or in other words, $\mathcal {C}=f^{-1}(0)\cap U$, where $U$ is an open neighborhood at the origin $0$ in $\mathbb {C}^2$.

If $f=f_1 \cdots f_r$ is the decomposition of $f$ into irreducible factors, then each $f_i$ defines a branch, that is, an irreducible analytic plane curve, denoted by $\mathcal {C}_i$ for $i=1, \ldots, r$.

Given a branch $\mathcal {C}_i$ of $\mathcal {C}$ with multiplicity $n_i$, that is, $f_i\in \mathcal {M}^{n_i}\setminus \mathcal {M}^{n_i+1}$, we can parameterize it by $(a_it_i^{n_i}+\cdots, b_it_i^{n_i}+\cdots )\in \mathbb {C}\{t_i\}\times \mathbb {C}\{t_i\}$ with $a_i\neq 0$ or $b_i\neq 0$. In what follows, we consider primitive parameterizations, that is, those that cannot be obtained from others composed with higher powers of $t_i$. We often identify such parameterization with the smooth map-germ $\varphi _i:(\mathbb {C},0)\to (\mathbb {C}^2,0)$ defined by $t_i\mapsto (x_i,y_i):=(a_it_i^{n_i}+\cdots, b_it_i^{n_i}+\cdots )$.

We call

\[ [\varphi_1, \dots ,\varphi_r]=\left[\begin{array}{ccccc} x_1 & \cdots & x_i & \cdots & x_r \\ y_1 & \cdots & y_i & \cdots & y_r \end{array}\right] \]

a multigerm for $\mathcal {C}$ and we denote by $\mathcal {P}$ the set of multigerms of plane curves with $r$ branches.

The analytic equivalence of reduced plane curves with $r$ branches is translated, as we noted in [HHR15], into $\mathcal {S}\times {\mathcal {A}}$-equivalence on $\mathcal {P}$, where $\mathcal {S}$ is the symmetric group on $r$ elements and ${\mathcal {A}}=\{(\rho _1, \ldots, \rho _r , \sigma ); \ \rho _i \in {\rm Diff}(\mathbb {C},0) \ \mbox {and} \ \sigma \in {\rm Diff}(\mathbb {C}^2,0), \ 1 \leqslant i \leqslant r\}$ is the group of right–left equivalence, where ${\rm Diff}(\mathbb {C}^l,0)$ denotes the diffeomorphism group of $(\mathbb {C}^l,0)$.

Given $(\rho _1, \ldots, \rho _r, \sigma ) \in \mathcal {A}$ and $\varphi =[\varphi _1, \ldots,\varphi _r] \in \mathcal {P}$ the action of $\mathcal {A}$ on $\mathcal {P}$ is defined as

\begin{align*} (\rho_1, \ldots, \rho_r, \sigma)\cdot \varphi&:= [\sigma \circ \varphi_1 \circ \rho_1^{-1}, \ldots, \sigma \circ \varphi_r \circ \rho_r^{-1}]\\ &=\left[\begin{array}{ccc} \sigma_1 (x_1 \circ \rho_1^{-1}, y_1 \circ \rho_1^{-1}) & \cdots & \sigma_1 (x_r \circ \rho_r^{-1}, y_r \circ \rho_r^{-1}) \\ \sigma_2 (x_1 \circ \rho_1^{-1}, y_1 \circ \rho_1^{-1}) & \cdots & \sigma_2 (x_r \circ \rho_r^{-1}, y_r \circ \rho_r^{-1}) \end{array}\right], \end{align*}

where $\sigma =(\sigma _1,\sigma _2)$.

We say that $\varphi,\psi \in \mathcal {P}$ are $\mathcal {A}$-equivalent, denoted by $\varphi \underset {\mathcal {A}}{\sim }\psi$, if and only if they are in the same $\mathcal {A}$-orbit. Now, the group $\mathcal {S}\times \mathcal {A}$ acts on $\mathcal {P}$ by

\[ \big( \pi, (\rho_1, \ldots, \rho_r, \sigma)\big)\cdot \varphi:= \pi ((\rho_1, \ldots, \rho_r, \sigma)\cdot \varphi):=[\sigma \circ \varphi_{\pi(1)} \circ \rho_{\pi(1)}^{-1}, \ldots, \sigma \circ \varphi_{\pi(r)} \circ \rho_{\pi(r)}^{-1}], \]

for all $\pi \in \mathcal {S}, \ (\rho _1, \ldots, \rho _r, \sigma ) \in \mathcal {A}$ and $\varphi \in \mathcal {P}$.

It is a well known fact that for a plane curve $\mathcal {C}$ defined by $f\in \mathbb {C}\{X,Y\}$, the isomorphism class of its local ring $\mathcal {O}:= {\mathbb {C}\{X,Y\}}/{\langle f\rangle }$ completely determines the analytic class of $\mathcal {C}$. The aim of this section is to present discrete $\mathcal {A}$-invariants related to some fractional ideals of $\mathcal {O}$.

A parameterization $\varphi _i =(x_i,y_i)\in \mathbb {C}\{t_i\}\times \mathbb {C}\{t_i\}$ of a branch $\mathcal {C}_i$ given by $f_i\in \mathbb {C}\{X,Y\}$ provides the exact sequence

\[ \{0\}\rightarrow \langle f_i\rangle\rightarrow \mathbb{C}\{X,Y\}\underset{\varphi_i^*}{\rightarrow} \mathbb{C}\{x_i,y_i\}\rightarrow \{0\}, \]

where $\varphi _i^*(h):=h(x_i,y_i)$. In what follows, we identify $\mathcal {O}_i:= {\mathbb {C}\{X,Y\}}/{\langle f_i\rangle }$ with the subalgebra $\mathbb {C}\{x_i,y_i\}\subseteq \mathbb {C}\{t_i\}$. In particular, the integral closure of $\mathcal {O}_i$ in its quotient field $\mathcal {Q}_i=\mathbb {C}((t_i))$ is $\overline {\mathcal {O}_i}=\mathbb {C}\{t_i\}$ and $\mathcal {O}_i=\overline {\mathcal {O}}_i$ if and only if $n_i=1$.

We consider the natural discrete normalized valuation

\[ \begin{array}{cccl} \nu_i: & \mathcal{Q}_i & \longrightarrow & \overline{\mathbb{Z}}:=\mathbb{Z}\cup\{\infty\} \\[6pt] & \dfrac{p}{q} & \longmapsto & \nu_i\bigg(\dfrac{p}{q}\bigg):=ord_{t_i}(p)-ord_{t_i}(q), \end{array} \]

where $p,q\in \mathcal {O}_i, q\neq 0$ and $\nu _i(0)=\infty$. The set

\[ \Gamma_i=\{\nu_i(p);\ p\in\mathcal{O}_i\}=\{\nu_i(h):=\nu_i(\varphi^*_i(h));\ h\in\mathbb{C}\{X,Y\} \} \]

is a submonoid of $\overline {\mathbb {N}}:=\mathbb {N}\cup \{\infty \}$ and $S_i=\Gamma _i\cap \mathbb {N}$ is the classical value semigroup of the branch $\mathcal {C}_i$, which is an $\mathcal {A}$-invariant and a complete topological invariant.

The conductor ideal of $\mathcal {O}_i$ in $\overline {\mathcal {O}_i}$ is $(\mathcal {O}_i:\overline {\mathcal {O}_i})=\{h \in \mathcal {O}_i;\ h \overline {\mathcal {O}_i}\subseteq \mathcal {O}_i\}=\langle t_i^{\mu _i}\rangle$, where $\mu _i$, called conductor of $\Gamma _i$, satisfies $\mu _i-1\not \in \Gamma _i$ and $\mu _i+\overline {\mathbb {N}}\subset \Gamma _i$. In this case, the conductor of $\Gamma _i$ coincides with the Milnor number of $\mathcal {C}_i$, that is, $\mu _i=\dim _{\mathbb {C}} ({\mathbb {C}\{X,Y\}}/{\langle (f_i)_X,(f_i)_Y\rangle })$ where $(f_i)_X$ and $(f_i)_Y$ indicate the derivatives of $f_i$ with respect to $X$ and $Y$, respectively.

The previous concepts can be extended for a reduced curve $\mathcal {C}$ with $r$ branches defined by $f=f_1\cdots f_r$.

Considering $I=\{1,\ldots,r\}$ and the monomorphism

\[ \begin{array}{ccc} \mathcal{O} & \rightarrow & \bigoplus_{i\in I}\mathcal{O}_i \\ p & \mapsto & (p_1,\ldots ,p_r),\end{array} \]

where $p_i$ denotes the class of $p\in \mathcal {O}$ in $\mathcal {O}_i$, it is possible to verify that the total ring of fractions of $\mathcal {O}$ is $\mathcal {Q}=\bigoplus _{i\in I}\mathcal {Q}_i$ and the integral closure of $\mathcal {O}$ in $\mathcal {Q}$ is $\overline {\mathcal {O}}=\bigoplus _{i\in I}\mathbb {C}\{t_i\}$. As before, if $\varphi =[\varphi _1,\ldots, \varphi _r]\in \mathcal {P}$ is a multigerm associated to $\mathcal {C}$, then we identify $\mathcal {O}$ with the subalgebra $\{\varphi ^*(h):=(\varphi ^*_1(h),\ldots,\varphi ^*_r(h)),\ h\in \mathbb {C}\{X,Y\}\}\subseteq \bigoplus _{i\in I}\mathcal {O}_i$ and we set

\[ \Gamma = \big\{\nu(h):=\big(\nu_1(h),\dots,\nu_r(h)\big); \ h \in \mathbb{C}\{X,Y\} \big\}\subset \Gamma_1\times\cdots \times\Gamma_r. \]

The conductor ideal of $\mathcal {O}$ in $\overline {\mathcal {O}}$ is

\[ (\mathcal{O}:\overline{\mathcal{O}})=\bigoplus_{i\in I}\bigg((\mathcal{O}_i:\overline{\mathcal{O}_i})\cdot \prod_{\substack{{j\in I}\\ {j\neq i}}}\varphi_i^*(f_j) \bigg)=\bigoplus_{i\in I}\langle t_i^{\kappa_i}\rangle, \]

where $\kappa _i=\mu _i+ \sum _{{j\in I\atop j \neq i} }\nu _i(f_j)$. The element $\kappa =(\kappa _1, \ldots, \kappa _r)$, called the conductor of $\Gamma$, satisfies $\kappa + \overline {\mathbb {N}}^r \subset \Gamma$ and $\kappa -e_i\not \in \Gamma$ for any element $e_i$ in the canonical basis of $\mathbb {Q}^r$ (see [Gor52]).

Zariski in [Zar71] showed that (up to permutation of the branches) the topological class of $\mathcal {C}$ is totally characterized by $\Gamma _i\cap \mathbb {N}$ and the intersection multiplicity $\nu _i(f_j)\ (=\nu _j(f_i))$ of $\mathcal {C}_i$ and $\mathcal {C}_j$ for $i,j\in I$ with $i\neq j$. On the other hand, Waldi (see [Wal72]) proved that $\Gamma \cap \mathbb {N}^r$ is also a complete topological invariant and Mata, in [Mat87], showed that $\Gamma$ determines and it is determined by $\Gamma _i$ and $\nu _i(f_j)$ for all $i,j\in I,\ i\neq j$, connecting the results of Waldi and Zariski.

In contrast to $\Gamma _i$, the (additive) semigroup $\Gamma$ is not finitely generated, but equipped with the tropical operations, the set $(\Gamma,\inf,+)$ is a finitely generated semiring¹ (see [CH20a]), where $\inf \{(\alpha _1,\ldots,\alpha _r),(\beta _1,\ldots,\beta _r)\}=(\min \{\alpha _1,\beta _1\},\ldots,\min \{\alpha _r,\beta _r\})$, that we call the value semiring associated to $\mathcal {C}$. More precisely, there exists a minimal set of generators $\{v_j,\ j=1,\ldots,g\}\subset \Gamma$ such that any $\gamma \in \Gamma$ can be written as

(1)\begin{equation} \gamma=\inf\bigg\{\sum_{j=1}^{g}a_{1j}v_j,\ldots,\sum_{j=1}^{g}a_{rj}v_j\bigg\}, \end{equation}

where $a_{kj}\in \mathbb {N}$, $k\in I$, $1\leqslant j\leqslant g$. The elements in $\{v_j,\ j=1,\ldots,g\}\cap \mathbb {N}^r$ are precisely the valuation of branches that achieve maximal contact with some $\mathcal {C}_i$ for $i\in I$ and the subset of generators with some coordinate equal to $\infty$ is precisely $\{\nu (f_i),\ i\in I\}$ (see [Mat87, CH20a]).

In a more general situation, given any (regular) fractional ideal $\mathcal {I}\subseteq \mathcal {Q}=\bigoplus _{i\in I}\mathcal {Q}_i$ of $\mathcal {O}$ we can consider the set of values

\[ \nu(\mathcal{I})=\big\{\nu(z):=(\nu_1(z_1),\ldots ,\nu_r(z_r));\ z=(z_1,\ldots ,z_r)\in\mathcal{I}\subseteq\mathcal{Q}\big\}\subseteq \overline{\mathbb{Z}}^r. \]

The set $(\nu (\mathcal {I}),\inf )$ is a $\Gamma$-semimodule, that is, $\Gamma + \nu (\mathcal {I}) \subseteq \nu (\mathcal {I})$. Moreover, there exists (unique) $(\zeta _1,\ldots,\zeta _r)=:\inf (\nu (\mathcal {I}))\in \nu (\mathcal {I})$ such that $\zeta _i\leqslant \alpha _i$ for every $(\alpha _1,\ldots,\alpha _r)\in \nu (\mathcal {I})$ and $\nu (\mathcal {I})$ admits a ‘conductor’ $\varrho \in \nu (\mathcal {I})$, that is, $\varrho + \overline {\mathbb {N}}^r \subseteq \nu (\mathcal {I})$ and $\varrho -e_i\not \in \nu (\mathcal {I})$ for all $e_i$ in the canonical $\mathbb {Q}$-basis of $\mathbb {Q}^r$ (see [GH20] for more properties concerning values set of fractional ideals of $\mathcal {O}$).

The next definition generalizes the corresponding concept introduced by Garcia in [Gar82] and Mata in [Mat87] for $\Gamma \cap \mathbb {N}^r$.

Definition 2.1 Let $\mathcal {I}$ be a fractional ideal of $\mathcal {O}$ with set of values $\Delta =\nu (\mathcal {I})$. Given $J\subseteq I$ the $J$-fiber of $\gamma =(\gamma _1,\ldots,\gamma _r)\in \overline {\mathbb {N}}^r$ with respect to $\Delta$ is the set

\[ F^{\Delta}_{J}(\gamma)=\{(\delta_1,\ldots ,\delta_r)\in\Delta;\ \delta_j=\gamma_j\ \mbox{for every}\ j\in J\ \mbox{and}\ \delta_i>\gamma_i\ \mbox{for all}\ i\in I\setminus J\}. \]

We say that $\gamma \in \overline {\mathbb {N}}^r$ is

1. (i) a maximal element of $\Delta$, if $F^{\Delta }_I(\gamma )\neq \emptyset$ (i.e. $\gamma \in \Delta$) and $F^{\Delta }_{\{i\}}(\gamma )= \emptyset$ for all $i\in I$;

2. (ii) a relative maximal element of $\Delta$, if $\gamma$ is maximal and $F^{\Delta }_J(\gamma ) \neq \emptyset$, for all $J \subseteq I$ with $\sharp J \geqslant 2$;

3. (iii) an absolute maximal element of $\Delta$, if $\gamma$ is maximal and $F^{\Delta }_J(\gamma ) = \emptyset$, for all $J \subsetneq I$, $J \neq \emptyset$.

Remark 2.2 Let $\mathcal {I}$ be a (regular) fractional ideal of $\mathcal {O}$ such that $\inf (\nu (\mathcal {I}))=(\zeta _1, \ldots, \zeta _r)$ and $\varrho =(\varrho _1, \ldots, \varrho _r)$ is the conductor of $\nu (\mathcal {I})$. In [CH20b, Theorem 16], the authors presented an algorithm to compute a finite set of generators of $\nu (\mathcal {I})$ (as $\Gamma$-semimodule) and showed that $\nu (\mathcal {I})$ is characterized by its elements in the box $R=[\zeta _1,\varrho _1]\times \cdots \times [\zeta _r,\varrho _r]$. In particular, we have that all maximal elements of $\nu (\mathcal {I})$ belong to $R$.

For $\mathcal {I}=\mathcal {O}$ we obtain $\nu (\mathcal {O})=\Gamma$ and $\Gamma$ is determined by its elements in $[0,\kappa _1]\times \cdots \times [0,\kappa _r]$.

Other $\mathcal {A}$-invariants that play an important role in this work are related with set of orders of differential 1-forms.

Let $\Omega ^1=\mathbb {C}\{X, Y\}\,dX+\mathbb {C}\{X, Y\}\,dY$ be the $\mathbb {C}\{X,Y\}$-module of differential 1-forms. If $\varphi _i=(x_i,y_i)\in \mathbb {C}\{t_i\}\times \mathbb {C}\{t_i\}$ is a parameterization of a plane branch $\mathcal {C}_i$ defined by $f_i$, we extend the homomorphism $\varphi ^*_i$ to $\Omega ^1$ in the following way.

Given $\omega =a(X,Y)\,dX+b(X,Y)\,dY \in \Omega ^1$, we define

\[ \varphi^*_i(\omega):=t_i\cdot (\varphi^*_i(a)\cdot x'_i+\varphi^*_i(b)\cdot y'_i)\in\mathcal{Q}_i, \]

where $x'_i$ and $y'_i$ denote, respectively, the derivative of $x_i,y_i\in \mathbb {C}\{t_i\}$ with respect to $t_i$.

Note that the kernel of $\varphi _i^*$ is

\[ \{\omega\in\Omega^1;\ \varphi^*_i(\omega)=0\}=\bigg\{\omega\in\Omega^1;\ \frac{\omega\wedge df_i}{dX\wedge dY}\in \langle f_i\rangle\bigg\}=f_i\cdot\Omega(\log \mathcal{C}_i), \]

where $\Omega (\log \mathcal {C}_i)$ is the module of logarithmic differential forms along $\mathcal {C}_i$ which is the dual module of the logarithmic vector field $Der(-\log \mathcal {C}_i)$ along $\mathcal {C}_i$ (see [Sai80]).

In addition, if $\mathcal {F}_i:=f_i\cdot \Omega ^1+\mathbb {C}\{X,Y\}\cdot df_i$, then the Kähler differential module of $\mathcal {O}_i$ is $\Omega _{i}\approx {\Omega ^1}/{\mathcal {F}_i}$ and its torsion submodule is given by $\mathcal {T}_{i}\approx {(f_i\cdot \Omega (\log \mathcal {C}_i))}/{\mathcal {F}_i}$. Moreover,

\[ \frac{\Omega_i}{\mathcal{T}_{i}}\approx\frac{\Omega^1}{f_i\cdot\Omega(\log \mathcal{C}_i)}\approx \varphi^*_i(\Omega^1).\]

The set $\varphi ^*_i(\Omega ^1)\subset \mathcal {Q}_i$ is a fractional ideal of $\mathcal {O}_i$ and

\[ \Lambda_i=\{\nu_i(\omega):=\nu_i(\varphi^*_i(\omega));\ \omega\in\Omega^1\} \]

is an $\mathcal {A}$-invariant of $\mathcal {C}_i$. As $\nu _i(dh)=\nu _i(h)$ for any $h\in \mathcal {M}$ we have $\Gamma _i\setminus \{0\}\subseteq \Lambda _i$. The set $\Lambda _i$ is one of the main tools considered in [HH11] in order to classify plane branches up to analytical equivalence.

Similarly for a multigerm $\varphi =[\varphi _1,\ldots,\varphi _r]$ we get the $\mathcal {A}$-invariant

(2)\begin{equation} \Lambda =\nu(\varphi^*(\Omega^1))=\{\nu(\omega):=(\nu_1(\omega),\ldots ,\nu_r(\omega));\ \omega\in\Omega^1\}\subset\Lambda_1\times\cdots\times\Lambda_r \subset \overline{\mathbb{N}}^r. \end{equation}

Remark that $\varphi ^*(\Omega ^1)\approx {\Omega ^1}/{f\cdot \Omega (\log \mathcal {C})}$ is a fractional ideal of $\mathcal {O}$ with $\Gamma \setminus \{\underline {0}\}\subseteq \Lambda$ and conductor $\varrho =(\varrho _1,\ldots, \varrho _r)$ satisfying $\varrho _i\leqslant \kappa _i$ for $i=1,\ldots,r$.

In particular, by Remark 2.2, the set $\Lambda$ is characterized by its points in the box $[0,\varrho _1]\times \cdots \times [0,\varrho _r]\subseteq [0,\kappa _1]\times \cdots \times [0,\kappa _r]$. Consequently, for each topological class determined by the semiring $\Gamma$ there is a finite number of possible $\Lambda$ sets.

The set $\Lambda$ is related to the Tjurina number of $\mathcal {C}$ (see [BGHH20]), the set $\nu (J(f))=\{\nu (h);\ h\in J(f):=\langle f,f_X,f_Y\rangle \subseteq \mathbb {C}\{X,Y\}\}$ and the values of residues of elements in $f\cdot \Omega (log\ \mathcal {C})$. More explicitly, if $\omega \in f\cdot \Omega (\log \mathcal {C})$, then there exist $\eta \in \Omega ^1$, $g,h\in \mathbb {C}\{X,Y\},\ g\not \in \bigcup _{i=1}^{r}\langle f_i\rangle$ such that $g\cdot \omega =h\cdot df+f\cdot \eta$. The class $res(\omega )$ of $ {h}/{g}$ in $\mathcal {Q}$ is called the residue of $\omega$ and we put $Res(f)=\{res(\omega ),\ \omega \in f\cdot \Omega (\log \mathcal {C})\}$ (see [Sai80]). In [Pol18], Pol showed that $\nu (J(f))=\Lambda +\kappa -(1,\ldots,1)$ and $\lambda \in \Lambda$ if and only if $-\lambda \not \in \nu (Res(f))$.

Using the value semiring $\Gamma$ or more specifically the subset $\nu (\mathcal {M}^2)$ of values of $\mathcal {M}^2$ we can obtain a distinguished presentation for a multigerm as we show in § 4. We consider particular fractional ideals of $\mathcal {O}$ determined by differential $1$-forms that are related to the tangent space to the orbit of a multigerm according to the action of a Lie group.

3. Group actions

Let $\varphi =[\varphi _1, \ldots, \varphi _r]\in \mathcal {P}$ such that $\varphi _i(t_i)=(a_it_i^{n_i}+\cdots, b_it_i^{n_i}+\cdots )$ with $a_i \neq 0$ or $b_i \neq 0$. Given $(\rho _1, \ldots, \rho _r, \sigma )\in \mathcal {A}$ with $\rho _i^{-1}(t_i)=u_it_i+\cdots$ and $\sigma (X,Y) =(\alpha X + \beta Y+\cdots, \gamma X + \delta Y+\cdots )$ such that $u_i, \alpha \delta - \beta \gamma \in \mathbb {C}^*:=\mathbb {C}\setminus \{0\}$ we have

\[ \psi_i(t_i):=\sigma \circ \varphi_i\circ\rho_i^{-1}(t_i) =((\alpha a_i+\beta b_i)u_i^{n_i}t_i^{n_i}+\cdots, (\gamma a_i+\delta b_i)u_i^{n_i}t_i^{n_i}+\cdots). \]

Denoting by $j^k\xi$ the $k$-jet of a map-germ $\xi$ at the origin we put $j^k\varphi :=[j^k\varphi _1, \ldots, j^k\varphi _r]$.

The invariance of the multiplicity of the branches gives us a one-to-one correspondence between $j^1 \sigma$ and a Möbius transformation on $\overline {\mathbb {C}}:=\mathbb {C} \cup \{\infty \}$ given by $T(\theta )= ({\gamma +\delta \theta })/({\alpha +\beta \theta })$ such that if $\theta _i:= {b_i}/{a_i}\in \overline {\mathbb {C}}$ is the slope of the tangent line $a_iY-b_iX=0$ of $\mathcal {C}_i$ at the origin, then $T(\theta _i)$ is the slope of the tangent line to the curve corresponding to $\psi _i$ for $1\leqslant i\leqslant r$. In this way, up to change of coordinates, we can consider the tangent lines of the branches in a such way that up to three chosen directions are fixed.

If all branches have the same tangent line at the origin with slope $\theta$, considering $\gamma =-\delta \theta$, then $T(\theta )=0$. If the branches have exactly two tangent lines with distinct slopes $\theta _1$ and $\theta _2$, then taking $\gamma =-\delta \theta _1$ and $\alpha =-\beta \theta _2$ we obtain $T(\theta _1)=0$ and $T(\theta _2)=\infty$. On the other hand, if there exist at least three branches with distinct tangent lines of slopes $\theta _1$, $\theta _2$, and $\theta _3$, we can consider the unique Möbius transformation $T$ such that $T(\theta _1)=0$, $T(\theta _2)=\infty$, and $T(\theta _3)=1$.

By the above description and an $\mathcal {S}\times \mathcal {A}$-action, we may adjust the components of $\varphi \in \mathcal {P}$ in a particular way.

Definition 3.1 Given $\varphi =[\varphi _1, \ldots, \varphi _r]\in \mathcal {P}$ we say that $\varphi$ is in block form if there are sets $B_1=\{\varphi _{k_1}=\varphi _1, \ldots, \varphi _{k_2-1}\}, B_2=\{\varphi _{k_2}, \ldots, \varphi _{k_3-1}\}, \ldots,B_{s}=\{\varphi _{k_{s}}, \ldots, \varphi _{r}\}$ satisfying the following.

1. (i) The tangent line of every branch in $B_i$ has the same slope $\theta _i$ and, if $i\neq j$ then $\theta _i\neq \theta _j$ for $1\leqslant i,j\leqslant s$. In addition, $\theta _1=0, \theta _2=\infty$ and $\theta _3=1$.

2. (ii) If $\varphi _j$ and $\varphi _k$ are elements of $B_i$ with $j< k$, then $n_j\leqslant n_k$.

3. (iii) We have $n_{k_1}\leqslant n_{k_2}\leqslant \cdots \leqslant n_{k_{s}}$.

A set $B_i$ as above is called a block of $\varphi$.

It is clear that any element in $\mathcal {P}$ is $\mathcal {S}\times \mathcal {A}$-equivalent to a multigerm in block form.

From now on we consider $\mathcal {A}$-action on block form multigerms in $\mathcal {P}$ and without loss of generality we can suppose $j^{n_i}\varphi _i=(a_it_i^{n_i}, b_it_i^{n_i})$ such that

\[ \begin{array}{lll} a_i=1, & b_i=0 & \mbox{if}\ \varphi_i\in B_1, \\ a_i=0, & b_i=1 & \mbox{if}\ \varphi_i\in B_2, \\ a_i=1, & b_i=1 & \mbox{if}\ \varphi_i\in B_3\ \mbox{and}\\ a_i=1, & b_i=\theta_j & \mbox{if}\ \varphi_i\in B_j\ \mbox{for}\ j>3. \end{array} \]

The next lemma is an immediate consequence of the above explanation and gives us the subgroups of $\mathcal {A}$ that preserve the tangent cone of the curve $\mathcal {C}$ and the properties of the block form.

In what follows, we consider the subgroups of $\mathcal {A}$:

\[ \begin{aligned} \mathcal{H}&=\{(\rho_1,\ldots, \rho_r,\sigma)\in \mathcal{A}; \ \rho_i=u_it_i,\ \sigma =(\alpha X, \delta Y)\ \mbox{and}\ u_i,\alpha,\delta \in \mathbb{C}^{*}\},\\ \mathcal{H}'&=\{(\rho_1,\ldots, \rho_r,\sigma)\in \mathcal{A}; \ \rho_i=u_it_i,\ \sigma =(\alpha X, \alpha Y)\ \mbox{and}\ u_i,\alpha \in \mathbb{C}^{*}\},\\ \tilde{\mathcal{A}}_1&=\{(\rho_1,\ldots, \rho_r,\sigma)\in \mathcal{A};\ j^1\rho_i=t_i,\ j^1\sigma =(X+\beta Y, Y)\ \mbox{and}\ \beta \in \mathbb{C}\}, \\ \mathcal{A}_1&=\{(\rho_1,\ldots, \rho_r,\sigma)\in \mathcal{A};\ j^1\rho_i=t_i\ \mbox{and}\ j^1\sigma =(X, Y)\}. \end{aligned} \]

Lemma 3.2 The subgroups of $\mathcal {A}$ that preserve the elements of $\mathcal {P}$ as in Definition 3.1, according to number $s$ of blocks, are as follows.

Proof. The description follows from the correspondence between the Möbius transformation $T(\theta )= ({\gamma +\delta \theta })/({\alpha + \beta \theta })$ and $j^1 \sigma =(\alpha X + \beta Y, \gamma X + \delta Y)$ with $\sigma \in {\rm Diff}(\mathbb {C}^2,0)$.

For $s=1$, all branches have tangent line with slope $0$ and to preserve it we must consider $\gamma =0$. Thus, the corresponding element in $\mathcal {A}$ is a composition of elements of $\mathcal {H}$ and $\tilde {\mathcal {A}}_1$.

If $\varphi \in \mathcal {P}$ has just two blocks with $\theta _1=0$ and $\theta _2=\infty$, then we must consider $T(\theta )= {\delta \theta }/{\alpha }$ to preserve the slopes of the tangent lines and consequently the corresponding elements in $\mathcal {A}$ are obtained as a composition of elements of $\mathcal {H}$ and $\mathcal {A}_1$.

On the other hand, if $s\geqslant 3$ the branches of the first three blocks have tangent line with slope $0$, $\infty$, and $1$, respectively, and the only Möbius transformation that preserve them is the identity ($\gamma =\beta =0$ and $\alpha = \delta$). Consequently, the associated diffeomorphisms belong to $\mathcal {H}' \circ \mathcal {A}_1$.

The standard strategy to solve the analytic equivalence problem is to find a representative element in each orbit (a normal form), for instance, with a few number of parameters, in such a way that it is manageable to decide whether two normal forms are equivalent or not.

Note that the action of any subgroup of $\mathcal {S}\times \mathcal {H}$ on an element $\varphi \in \mathcal {P}$ does not introduce or eliminate terms in the multigerm components. Thus, to obtain equivalent multigerms in the block form with short parameterizations it is natural to consider the $\mathcal {A}_1$-action or $\tilde {\mathcal {A}}_1$-action according to the number $s$ of blocks described in the previous lemma.

In a more general situation, concerning a Lie group action $\mathfrak {G}$ on an affine space $\mathbb {A}$, the complete transversal theorem (CTT; see [BKP97]) provides us a method for obtaining representative elements in the orbit $\mathfrak {G}(v)$ of $v\in \mathbb {A}$, if we have the description of the tangent space $T\mathfrak {G}(v)$ to orbit $\mathfrak {G}(v)$ at $v$. More precisely, the following version will be useful for our purposes.

Theorem 3.3 (Complete transversal theorem)

Let $\mathfrak {G}$ be a Lie group acting on an affine space $\mathbb {A}$ with underlying vector space $V$ and let $W$ be a subspace of $V$. Suppose that $T\mathfrak {G}(v+w)=T\mathfrak {G}(v)$ for all $v \in \mathbb {A}$ and $w\in W$. If $v \in \mathbb {A}$ and $W \subseteq T\mathfrak {G}(v)$, then $\mathfrak {G}(v+w)=\mathfrak {G}(v)$, for all $w\in W$, that is, the $\mathfrak {G}$-orbits of $v+w$ and $v$ coincide.

Let us apply the above theorem in our context. To simplify the notation let $\mathcal {G}$ be one of the groups $\mathcal {A}_1$ or $\tilde {\mathcal {A}}_1$ and $I=\{1,\ldots,r\}$.

The description of $T\mathcal {A}_1(\varphi )$ for $\varphi \in \mathcal {P}$ is classical (see, for instance, [Wal81]) and we do not need any effort to extend it to the group $\tilde {\mathcal {A}}_1$. Explicitly, we get

(3)\begin{equation} \begin{array}{c} T{\mathcal{G}}(\varphi)=\left\{\left [\begin{array}{ccc} x_1' \cdot \epsilon_1+ \varphi^*_1(\eta_1) & \ldots & x_r' \cdot\epsilon_r+ \varphi^*_r(\eta_1)\\ y_1' \cdot \epsilon_1+ \varphi^*_1(\eta_2) & \ldots & y_r' \cdot\epsilon_r+ \varphi^*_r(\eta_2) \end{array}\right ]; \ \epsilon_i \in \langle t^2_i \rangle, i \in I \right\},\\ \mbox{where} \ \begin{cases} \eta_1,\eta_2 \in \mathcal{M}^2, & \mbox{if}\ \ \mathcal{G}=\mathcal{A}_1,\\ \eta_1 \in \langle X^2,Y \rangle,\ \eta_2 \in \mathcal{M}^2, & \mbox{if}\ \ \mathcal{G}=\tilde{\mathcal{A}}_1. \end{cases} \end{array} \end{equation}

Denoting by $B^k$ the set of $k$-jets of elements of a set $B$, we know that $\mathcal {G}^k$ is a unipotent Lie group that acts on $\mathcal {P}^k$ in the natural way:

\[ j^{k}(\rho_1,\ldots ,\rho_r,\sigma)\cdot j^{k}\varphi:=[j^{k}(\sigma\circ\varphi_1\circ\rho^{-1}_1),\ldots ,j^{k}(\sigma\circ\varphi_r\circ\rho^{-1}_r)]. \]

The corresponding tangent space $T{\mathcal {G}}^k(j^k\varphi )$ is the set of $k$-jets of elements in $T{\mathcal {G}}(\varphi )$.

If $H^k$ denotes the $\mathbb {C}$-vector space

(4)\begin{equation} \left\{ \left[\begin{array}{ccc} c_1t_1^k & \cdots & c_rt_r^k \\ d_1t_1^k & \cdots & d_rt_r^k \\ \end{array}\right] ; \ c_i,\ d_i\in\mathbb{C}\ \mbox{with}\ c_i=d_i=0\ \mbox{if}\ k \leqslant n_i,\ i\in I \right\} \end{equation}

then, we can prove, similarly as in Proposition 2 of [HHR15], that $T{\mathcal {G}}^k(j^k\varphi +\zeta )=T{\mathcal {G}}^k(j^k\varphi )$ for any $j^k\varphi \in \mathcal {P}^k$ and $\zeta \in H^k$.

In this way, $\mathfrak {G}=\mathcal {G}^k$, $\mathbb {A}=\mathcal {P}^k$, and $W\subseteq H^k\cap T{\mathcal {G}}^k(j^k\varphi )$ fulfill the hypothesis of Theorem 3.3, that is, for any $w\in W$ we have that $j^k\varphi +w$ is $\mathcal {G}^k$-equivalent to $j^k\varphi$ and, consequently, there exists $\psi \in \mathcal {P}$ which is $\mathcal {A}$-equivalent to $\varphi$ with $j^k\psi =j^k\varphi +w$.

In the next two sections we recognize elements in $H^k\cap T{\mathcal {G}}^k(j^k\varphi )$ using $\mathcal {G}$-invariants.

4. Puiseux block form

In this section, we present a convenient parameterization for the components of the multigerm $\varphi =[\varphi _1,\ldots,\varphi _r]\in \mathcal {P}$ preserving the block form, as in Definition 3.1. In addition, we exhibit some elimination criteria for terms of $\varphi$.

For simplicity, we denote by $E_{ji}$ for $j=1,2$, $i\in I$, an element of the canonical basis of the complex matrices of order $2\times r$ and $\mathcal {G}$ denotes one of the groups $\mathcal {A}_1$ or $\tilde {\mathcal {A}}_1$.

Given a multigerm $\varphi =[\varphi _1,\ldots,\varphi _r]\in \mathcal {P}$ with $\varphi _i(t_i)=(x_i(t_i),y_i(t_i))$, an element

\[ \left[\begin{array}{ccc} u_1 & \cdots & u_r \\ v_1 & \cdots & v_r \end{array}\right]:=\sum_{i=1}^r (u_iE_{1i}+v_iE_{2i}) \]

with $u_i,v_i\in \mathbb {C}\{t_i\}$ belongs to $T\mathcal {G}(\varphi )$ if and only if there exist $\epsilon _i$ for $i=1,\ldots, r$ and $\eta =(\eta _1, \eta _2)$ satisfying (3) such that

(5)\begin{equation} \begin{cases} u_i=x_i' \cdot \epsilon_i + \varphi^*_i(\eta_1), \\ v_i= y_i' \cdot \epsilon_i + \varphi^*_i(\eta_2). \end{cases} \end{equation}

As we remarked in § 3 describing $H^k\cap T\mathcal {G}^k(j^k\varphi )$ we can apply Theorem 3.3 in order to obtain representative elements in a same $\mathcal {G}^k$-orbit.

Note that our analysis, up to this point, has taken into account a few $\mathcal {A}$-invariants, just the multiplicities of the branches and the behavior of the tangent cone. It is time to consider finer invariants as defined in § 2.

Let $\varphi =[\varphi _1,\ldots,\varphi _r]\in \mathcal {P}$ be a multigerm of plane curve in block form, with local ring $\mathcal {O}$. The (fractional) ideal $\varphi ^*(\mathcal {M}^2)$ of $\mathcal {O}$ plays a relevant role and provides us an elimination criterion for terms in $\varphi$.

Lemma 4.1 With the standard notation, if $k=\gamma +\sum _{{j\in I\atop j\neq i}}\nu _i(f_j)\in \nu _i(\mathcal {M}^2)$ with $\gamma \in \Gamma _i$, then there exists $\psi =[\psi _1,\ldots,\psi _r]$ $\mathcal {G}$-equivalent to $\varphi$ such that $j^{k}\varphi _j=j^{k}\psi _j$ for all $j\in I\setminus \{i\}$ and $j^k\psi _i=j^{k-1}\varphi _i$.

Proof. By hypothesis, there exists an element $h\in \mathbb {C}\{X,Y\}$ with $h\cdot \prod _{{j\in I\atop j\neq i}}f_j\in \mathcal {M}^2$ and $j^k(\varphi _i^*(h\cdot \prod _{{j\in I\atop j\neq i}}f_j))=t_i^k$.

Taking $\epsilon _j=0$ for $j\in I$ and $\eta _l= \alpha _lh\cdot \prod _{{j\in I\atop j\neq i}}f_j$ for $l=1,2$ in (5), we get

\[ \alpha_1 t_i^kE_{1i}+\alpha_2 t_i^kE_{2i}=\left[\begin{array}{ccccc} 0 & \cdots & \alpha_1t_i^{k} & \cdots & 0 \\ 0 & \cdots & \alpha_2t_i^{k} & \cdots & 0 \end{array}\right] \in H^k\cap T\mathcal{G}^k(j^k \varphi). \]

By Theorem 3.3, we obtain $\psi \in \mathcal {G}(\varphi )$ with $j^k\psi =j^k\varphi +\alpha _1 t_i^kE_{1i}+\alpha _2 t_i^kE_{2i}$ and for a convenient choice of $\alpha _1,\alpha _2\in \mathbb {C}$ we have $j^{k}\psi _j=j^{k}\varphi _j$ for all $j\in I\setminus \{i\}$ and $j^k\psi _i=j^{k-1}\varphi _i$.

Remark 4.2 Note that the description of tangent vector $\alpha _1 t_i^kE_{1i}+\alpha _2 t_i^kE_{2i}\in H^k\cap T\mathcal {G}^k(j^k \varphi ),$ $k=\gamma +\sum _{{j\in I\atop j\neq i}}\nu _i(f_j)\in \nu _i(\mathcal {M}^2)$ with $\gamma \in \Gamma _i$ in the previous lemma gives us a clue to obtain elements $(\rho _1,\ldots,\rho _r,\sigma )\in \mathcal {G}$ such that $(\rho _1,\ldots,\rho _r,\sigma )\cdot \varphi =\psi$ with $j^{k}\psi _j=j^{k}\varphi _j$ for all $j\in I\setminus \{i\}$ and $j^k\psi _i=j^{k-1}\varphi _i$. In fact, with the above notation, it is sufficient to consider $\rho _j(t_j)=t_j$ for $j\in I$ and $\sigma (X,Y)=(X-\alpha _1\cdot h\cdot \prod _{{j\in I\atop j\neq i}}f_j,\ Y-\alpha _2\cdot h\cdot \prod _{{j\in I\atop j\neq i}}f_j)$ with appropriate $\alpha _1,\alpha _2\in \mathbb {C}$.

As a consequence of the above lemma we obtain an estimate for the finite determinacy of a given $\varphi \in \mathcal {P}$.

Denoting $(\mathcal {O}:\overline {\mathcal {O}})^c$ the contraction of $(\mathcal {O}:\overline {\mathcal {O}})$ by $\varphi ^*$ we have the following result.

Proposition 4.3 If $d_i$ is the conductor of the set $\nu _i\big (\mathcal {M}^2\cap (\mathcal {O}:\overline {\mathcal {O}})^c\big )$ for $i=1, \ldots, r$, then $\varphi \underset {\mathcal {G}}{\sim }[j^{d_1-1}\varphi _1,\ldots,j^{d_r-1}\varphi _r].$

Proof. For each $i\in I$ we put $\varphi _i-j^{d_i-1}\varphi _i=(t_i^{d_i}u_{i1},t_i^{d_i}u_{i2})$ with $u_{il}\in \mathbb {C}\{t_i\}$ for $l=1,2$. As $d_i$ is the conductor of $\nu _i\big (\mathcal {M}^2\cap (\mathcal {O}:\overline {\mathcal {O}})^c\big )$ there exists $h_{il}\cdot \prod _{j\in I\atop j\neq i}f_j\in \mathcal {M}^2\cap (\mathcal {O}:\overline {\mathcal {O}})^c$ such that $\varphi _i^*(h_{il}\cdot \prod _{j\in I\atop j\neq i}f_j)=t_i^{d_i}u_{il}$ for $l=1,2$.

Now, taking $(\rho _1,\ldots, \rho _r,\sigma )\in \mathcal {G}$, where $\rho _j(t_j)=t_j$ for $j\in I$ and

\[ \sigma (X,Y)=\Bigg(X-\sum_{i\in I} h_{i1}\cdot\prod_{j\in I\atop j\neq i}f_j,\ Y-\sum_{i\in I} h_{i2}\cdot\prod_{{j\in I\atop j\neq i}}f_j\Bigg), \]

we obtain $\psi =(\rho _1,\ldots,\rho _r,\sigma )\cdot \varphi =[j^{d_1-1}\varphi _1,\ldots,j^{d_r-1}\varphi _r]$.

Since $\kappa =(\kappa _1, \ldots, \kappa _r)$ is the conductor of the semiring $\Gamma$, the above-mentioned integer $d_i$ satisfies $d_i\geqslant \kappa _i$ with equality if $(\mathcal {O}:\overline {\mathcal {O}})^c\subseteq \mathcal {M}^2$. More precisely, we have the following.

Proposition 4.4 Let $\varphi =[\varphi _1, \ldots, \varphi _r]\in \mathcal {P}$ be a block form multigerm and $s$ the number of blocks. For each $i \in I$, the integer $d_i$ in the last proposition satisfies

\[ d_i=\begin{cases} \kappa_i+2 & \mbox{if}\ r=1\ \mbox{and}\ n_i\leqslant 2; \\ \kappa_i+1 & \mbox{if}\ r=s=2\ \mbox{and}\ n_1=n_2=1; \\ \kappa_i & \mbox{otherwise}. \end{cases} \]

Proof. Let us consider the cases $r=1$, $r=2$, and $r\geqslant 3$ separately.

Case $r=1$. Let $\varphi _1$ be a parameterization of a branch with multiplicity $n_1$. As we remarked in § 2, $\kappa _1$ is the Milnor number $\mu _1$.

If $n_1=1$, then $\mu _1=0$ and $\mathcal {O}=\overline {\mathcal {O}}$. Thus, $d_1=\min \nu _i\big (\mathcal {M}^2\cap (\mathcal {O}:\overline {\mathcal {O}})^c\big )=2=\mu _1+2=\kappa _1+2$.

For $n_1>1$ we have $\overline {\mathcal {O}}\neq \mathcal {O}$ and $(\mathcal {O}:\overline {\mathcal {O}})^c \subseteq \mathcal {M}$. Setting $m=\min (\Gamma \setminus n_1\mathbb {N})$, we have

\[ \nu_1 (\mathcal{M}\setminus\mathcal{M}^2)=\bigg\{n_1, 2n_1,\ldots , \bigg[\frac{m}{n_1}\bigg]n_1, m\bigg\}. \]

If $n_1=2$, then $\Gamma =\langle 2,m\rangle =2\mathbb {N}+m\mathbb {N}$ and $\mu _1=m-1$, which implies that $\nu _1((\mathcal {O}:\overline {\mathcal {O}})^c)=\{\gamma \in \mathbb {N};\ \gamma \geqslant m-1\}$. Thus, the conductor $d_1$ of $\nu _1\big (\mathcal {M}^2\cap (\mathcal {O}:\overline {\mathcal {O}})^c\big )$ satisfies $d_1=m+1=\mu _1+2$. On the other hand, for $n_1>2$ we have $\mu _1>m$, $(\mathcal {O}:\overline {\mathcal {O}})^c\subseteq \mathcal {M}^2$ and consequently $d_1=\kappa _1$.

Case $r=2$. Consider a plane curve $\mathcal {C}$ given by $f=f_1\cdot f_2$ with corresponding multigerm $[\varphi _1,\varphi _2]$.

As $(\mathcal {O}:\overline {\mathcal {O}})=(\mathcal {O}_1:\overline {\mathcal {O}_1})\cdot \varphi _1^*(f_2)\oplus (\mathcal {O}_2:\overline {\mathcal {O}_2})\cdot \varphi _2^*(f_1)$ and $f_1,f_2\in \mathcal {M}$, it follows that if $n_i>1$ for some $i\in \{1,2\}$, then $(\mathcal {O}:\overline {\mathcal {O}})^c\subseteq \mathcal {M}^2$ and $d_i=\kappa _i$.

Let us consider $n_1=n_2=1$. In this case we have $\kappa =(\kappa _1,\kappa _2)=(\nu _1(f_2),\nu _2(f_1))$. If $s=1$, then $\varphi _1,\varphi _2\in B_1$ and $\nu _1(f_2)=\nu _2(f_1)>1$. Hence, $(\mathcal {O}:\overline {\mathcal {O}})^c\subseteq \mathcal {M}^2$ and $d_i=\kappa _i$ for $i=1,2$. If $s=2$, then $\varphi _1\in B_1$ and $\varphi _2\in B_2$. Thus, $\nu _1(f_2)=\nu _2(f_1)=1$ and $d_i=\min \ \nu _i\big (\mathcal {M}^2\cap (\mathcal {O}:\overline {\mathcal {O}})^c\big )=2=\kappa _i+1$ for $i=1,2$.

Case $r\geqslant 3$. Now, we consider a plane curve $\mathcal {C}$ given by $f=f_1\cdots f_r$ with $f_i\in \mathcal {M}$.

As $(\mathcal {O}:\overline {\mathcal {O}})=\bigoplus _{i\in I}(\mathcal {O}_i:\overline {\mathcal {O}_i})\cdot \prod _{j\in I\atop j\neq i}\varphi _i^*(f_j)$, we get $(\mathcal {O}:\overline {\mathcal {O}})^c\subseteq \mathcal {M}^2$ and $d_i=\kappa _i$ for $i\in I$.

Another direct application of Theorem 3.3 is obtained as a consequence of the following lemma.

Lemma 4.5 Let $\varphi =[\varphi _1, \ldots, \varphi _r]\in \mathcal {P}$ be a block form multigerm associated to a curve $\mathcal {C}$ and $\theta _i= {b_{i}}/{a_{i}}\in \overline {\mathbb {C}}$ the slope of the tangent line to $\mathcal {C}_i$ at the origin. For each $i\in I$, if $k> n_i$, then

\[ t_i^k(a_{i}E_{1i}+b_{i}E_{2i})\in H^k\cap T\mathcal{G}^k(j^k\varphi). \]

Proof. As $\varphi _i(t_i)=(a_{i}t_i^{n_i}+\cdots, b_{i}t_i^{n_i}+\cdots )$ and $k>n_i$, taking $\epsilon _i= ({1}/{n_i})t_i^{k-n_i+1}\in \langle t_i^2 \rangle$, $\epsilon _l=0$ for $l\in I\setminus \{i\}$, and $\eta _1=\eta _2=0$ in (5), we obtain

\[ t_i^k(a_{i}E_{1i}+b_{i}E_{2i})=\left[\begin{array}{ccccc} 0 & \cdots & a_{i}t_i^{k} & \cdots & 0 \\ 0 & \cdots & b_{i}t_i^{k} & \cdots & 0 \end{array}\right] \in H^k\cap T\mathcal{G}^k(j^k \varphi).\]

By the above lemma and Theorem 3.3, for every $k>n_i$ there exists a block form multigerm $\psi \in \mathcal {G} (\varphi )$ with $j^k\psi =j^k\varphi +\alpha t_i^k(a_{i}E_{1i}+b_{i}E_{2i})$ for any $\alpha \in \mathbb {C}$. Thus, if $\varphi _i=(x_i,y_i)\in B_1$, that is, $a_i=1$ and $b_i=0$, then we can choose $\alpha$ in such way that $j^k\psi _i=(j^{k-1}x_i,j^{k}y_i)$, that is, we eliminate the $k$-order term of the first component of $\varphi _i$. On the other hand, for $\varphi _i\in B_2$ there exists $\alpha$ such that $j^k\psi _i=(j^kx_i,j^{k-1}y_i)$. Finally, for $\varphi _i\in B_j$ with $j>2$, as $a_i\cdot b_i\neq 0$, we can choose $\alpha$ to obtain $j^k\psi _i=(j^{k-1}x_i,j^{k}y_i)$ or $j^k\psi _i=(j^kx_i,j^{k-1}y_i)$.

Similarly to Remark 4.2, as consequence of the previous lemma, we can exhibit an element of $\mathcal {G}$ to perform the proposed action. In fact, with the above notation, it is sufficient to consider $\rho _i^{-1}(t_i)=t_i+\alpha \epsilon _i$ with $\alpha \in \mathbb {C}$, $\epsilon _i\in \langle t_i^2\rangle$ given as in the proof of Lemma 4.5, $\epsilon _j=0$ for $j\in I\setminus \{i\}$ and $\sigma (X,Y)=(X,Y)$.

Thus, we recover the classical Puiseux expansion for plane curves.

Proposition 4.6 Any multigerm $\varphi =[\varphi _1, \ldots, \varphi _r]\in \mathcal {P}$ is $\mathcal {G}$-equivalent to a block form multigerm $\psi =[\psi _1,\ldots,\psi _r]$ with

(6)\begin{equation} \psi_i(t_i)=\begin{cases} \bigg(t_i^{n_i},\ \theta_lt_i^{n_i}+\displaystyle\sum_{j>n_i}a_{ij}t_i^j\bigg) & \mbox{for}\ \psi_i\in B_l,\ l\neq 2;\\ \bigg(\displaystyle\sum_{j>n_i}a_{ij}t_i^j,\ t_i^{n_i}\bigg) & \mbox{for}\ \psi_i\in B_2; \end{cases} \end{equation}

with $\theta _1=0$, $\theta _3=1$, and $\theta _k \neq \theta _l$ for $1\leqslant k$, $l\leqslant r$, and $k\neq l$.

Proof. We may suppose that $\varphi$ is given by a block form multigerm. For each $i\in I$ and for every $k>n_i$ we consider elements in $\mathcal {G}$ to apply the process described after Lemma 4.5. If $\varphi _i=(x_i,y_i)\in B_l$ with $l\geqslant 3$, then we choose $j^k\psi _i=(j^{k-1}x_i,j^{k}y_i)$.

Repeating this process for all $k$ satisfying $n_i< k< d_i$ we obtain $j^{d_i}\psi _i=(t_i^{n_i},j^{d_i}y_i)$ for $\psi _i\not \in B_2$ and $j^{d_i}\psi _i=(j^{d_i}x_i,t_i^{n_i})$ for $\psi _i\in B_2$. The result follows from Proposition 4.3.

We call the multigerm as highlighted in the above result the Puiseux block form of $\varphi$.

From the above results, we obtain the well-known representative in the analytical class for the particular cases of Proposition 4.4. More explicitly, if $\varphi$ is a Puiseux block form, then:

\[ \varphi\underset{\mathcal{A}}{\sim} (t,0)\quad \mbox{if}\ r=n_1=1; \]

\[ \varphi\underset{\mathcal{A}}{\sim} (t^2,t^{m})\quad \mbox{if}\ r=1,\ n_1=2\ \mbox{and}\ m=\min(\Gamma\setminus 2\mathbb{N}); \]

\[ \varphi\underset{\mathcal{A}}{\sim} [\psi_1,\psi_2];\quad \psi_1=(t_1,0),\quad \psi_2=(0,t_2)\quad \mbox{if}\ r=s=2,\ n_1=n_2=1. \]

Note that if $r=1$, $n_1=2$, and $m=\min (\Gamma \setminus 2\mathbb {N})$ the previous results provide us with the result $\varphi \underset {\mathcal {A}}{\sim } (t^2,at^{m})$ with $a\neq 0$. However, considering $\rho (t)=t$ and $\sigma (X,Y)=(X,a^{-1}Y)$ we obtain the above equivalence.

In what follows, we consider the non-exceptional cases and consequently we can suppose that the finite determinacy order for each $\varphi _i$ of a Puiseux block form multigerm is $\kappa _i$.

5. $\mathcal {G}$-Normal forms and analytic equivalence

In § 4, we apply the CTT (Theorem 3.3) to reduce a multigerm $\varphi \in \mathcal {P}$ to its Puiseux block form taking into account only a few topological invariants. Although such forms are a shorter way of presenting a multigerm, it is not easy to decide whether two Puiseux block form multigerms correspond to analytically equivalent plane curves.

Our goal in this section is to partition a topological class into a finite number of strata so that in each stratum we have a constant $\mathcal {A}$-invariant and every element belonging to it admits a particular Puiseux block form, that we call a normal form, which will allow us to distinguish them from the analytical equivalence viewpoint.

As before, according to Lemma 3.2, $\mathcal {G}$ denotes $\mathcal {A}_1$ if the Puiseux block form of $\varphi \in \mathcal {P}$ has at least two blocks and $\tilde {\mathcal {A}}_1$ if there is a single block.

Recall that as a consequence of Lemma 4.5, $\{\sum _{\varphi _i\not \in B_2}\mathbb {C}\cdot t_i^kE_{1i}+\sum _{\varphi _i\in B_2}\mathbb {C}\cdot t_i^kE_{2i}\}\cap H^k \subseteq T\mathcal {G}^k(j^k \varphi )$ for every $k$. Thus, in order to obtain a Puiseux block form multigerm with a smaller number of terms, it is sufficient to describe elements in the set $D^k:=\{\sum _{\varphi _i\not \in B_2}\mathbb {C}\cdot t_i^kE_{2i}+\sum _{\varphi _i\in B_2}\mathbb {C}\cdot t_i^kE_{1i}\}\cap H^k$ that belong to $T\mathcal {G}^k(j^k \varphi )$. In fact, if $\zeta \in D^k\cap T\mathcal {G}^k(j^k\varphi )$, then $\varphi$ is $\mathcal {G}$-equivalent to a Puiseux block form multigerm such that its $k$-jet is $j^k\varphi +\zeta$, that is, there exists a multigerm $\psi$, $\mathcal {A}$-equivalent to $\varphi$, with $j^{k-1}\psi =j^{k-1}\varphi$ and, by adjusting coefficients in $\zeta$, the element $j^k\psi$ has a smaller number of non-zero terms than $j^k\varphi$.

The key to obtain our normal form is the connection between $D^k\cap T\mathcal {G}^k(j^k\varphi )$ and the set of values of a particular fractional ideal as we describe in the following.

Considering $\varphi \in \mathcal {P}$ and

\[ \Omega_{\mathcal{A}_1}=\big\{\eta_2\,{d}X-\eta_1\,dY; \ \eta_1,\eta_2 \in \mathcal{M}^2\big\}, \quad \Omega_{\tilde{\mathcal{A}}_1}=\big\{\eta_2\,dX-\eta_1\,dY; \ \eta_1\in\langle X^2,Y\rangle \mbox{and}\ \eta_2\in \mathcal{M}^2\big\}, \]

we get the fractional ideal

\[ \mathcal{I}_{\mathcal{G}}=\bigg\{\bigg(\frac{\varphi^{*}_{1}(\omega)}{n_1t_1^{n_1}},\ldots , \frac{\varphi^{*}_{r}(\omega)}{n_rt_r^{n_r}}\bigg);\ \omega\in\Omega_{\mathcal{G}}\bigg\} \]

of $\mathcal {O}$ according to the group $\mathcal {G}$.

We define $\Lambda _{\mathcal {G}}:=\{\nu ({\rm w});\ {\rm w} \in \mathcal {I}_{\mathcal {G}}\}\subset \Lambda -n \subseteq \overline {\mathbb {N}}^r$, where $\Lambda$ is the $\mathcal {A}$-invariant defined in (2) and $n:=(n_1,\ldots, n_r)$ with $n_i$ the multiplicity of the branch $\varphi _i$.

Note that the conductor $\varrho '=(\varrho _1',\ldots, \varrho _r')$ of $\Lambda _{\mathcal {G}}$ satisfies $\varrho _i' \leqslant \varrho _i-n_i < \kappa _i$, where $(\varrho _1, \ldots, \varrho _r)$ and $(\kappa _1, \ldots, \kappa _r)$ are the conductors of $\Lambda$ and $\Gamma$, respectively. As a consequence of Remark 2.2, for each fixed value semiring $\Gamma$ there are only finitely many $\mathcal {A}$-invariants $\Lambda _{\mathcal {G}}$ that can be computed by the algorithm presented in Theorem 16 of [CH20b].

In what follows, we consider the set $\Sigma _{\Gamma,\Lambda _{\mathcal {G}}}$ of all Puiseux block form multigerms in $\mathcal {P}$, as given in (6), with semiring $\Gamma$ and $\Lambda _{\mathcal {G}}$ fixed.

Proposition 5.1 Given $\varphi =[\varphi _1, \ldots, \varphi _r]\in \Sigma _{\Gamma,\Lambda _{\mathcal {G}}}$ and $k\in \mathbb {N}^*$ there exists ${\rm w}=({\rm w}_1,\ldots,{\rm w}_r) \in \mathcal {I}_{\mathcal {G}}$ with $j^k{\rm w}_i=\alpha _it_i^{k}$, $\alpha \in \mathbb {C}$ if and only if $\sum _{\varphi _i\not \in B_2} \alpha _it_i^{k}E_{2i}-\sum _{\varphi _i\in B_{2}} \alpha _it_i^{k}E_{1i} \in T{\mathcal {G}}^k(j^k\varphi )$.

Proof. Let ${\rm w}=({\rm w}_1,\ldots,{\rm w}_r)\in \mathcal {I}_{\mathcal {G}}$ with $j^k{\rm w}_i=j^k\big ( {\varphi ^{*}_{i}(\omega )}/{n_it_i^{n_i}}\big )=\alpha _it_i^{k}$ for $i\in I$ and some $\omega =\eta _2\,dX-\eta _1\,dY\in \Omega _{\mathcal {G}}$.

As $\epsilon _i = - ({\varphi ^*_i(\eta _1)}/{n_it_i^{n_i-1}})\in \langle t_i^2\rangle$ for $\varphi _i \not \in B_2$, $\epsilon _i =- ({\varphi ^*_i(\eta _2)}/{n_it_i^{n_i-1}})\in \langle t_i^2\rangle$ for $\varphi _i \in B_{2}$, $\eta _1$ and $\eta _2$ satisfying (5) we have the desired element in $T{\mathcal {G}}^k(j^k\varphi )$.

Conversely, if $\sum _{\varphi _i\not \in B_2} \alpha _it_i^{k}E_{2i}-\sum _{\varphi _i\in B_{2}} \alpha _it_i^{k}E_{1i} \in T{\mathcal {G}}^k(j^k\varphi )$, then there exist $\epsilon _i \in \langle t_i^2 \rangle,$ $i\in I$, $\eta _1$, and $\eta _2$ as given in (3) such that

(7)\begin{equation} \begin{cases} x_i' \cdot \epsilon_i + \varphi^*_i(\eta_1)=P_i \\ y_i' \cdot \epsilon_i + \varphi^*_i(\eta_2)=\alpha_it_i^k+Q_i \end{cases}\ \mbox{if}\ \varphi_i\not\in B_2,\quad \begin{cases} x_i' \cdot \epsilon_i + \varphi^*_i(\eta_1)=-\alpha_it_i^k+P_i \\ y_i' \cdot \epsilon_i + \varphi^*_i(\eta_2)=Q_i \end{cases}\ \mbox{if}\ \varphi_i\in B_2, \end{equation}

with $\nu _i(P_i)>k$ and $\nu _i(Q_i)>k$ for $i\in I$. Considering $\omega = \eta _2 \,dX - \eta _1 \,dY \in \Omega _{\mathcal {G}}$ we provide the element ${\rm w}\in \mathcal {I}_{\mathcal {G}}$ satisfying the proposition.

Note that if $k\leqslant n_i$, then, by the definition of $\mathcal {I}_{\mathcal {G}}$, $\alpha _i$ should be zero in (7).

In order to simplify the notation, from now on given a non-empty subset $J\subseteq I$, we denote the $J$-fiber of ${\underline k}=(k, \ldots, k)\in \mathbb {N}^r$ with respect to $\Lambda _{\mathcal {G}}$ by $F_J(\underline {k})$. The following corollary is immediate.

Corollary 5.2 We have $F_J(\underline {k})\neq \emptyset$ if and only if there exist $\alpha _i\neq 0$ for every $i\in J$ and $\alpha _i= 0$ for every $i\in I\setminus J$ such that $\sum _{\varphi _i\not \in B_2} \alpha _it_i^kE_{2i}-\sum _{\varphi _i\in B_{2}} \alpha _it_i^kE_{1i} \in T{\mathcal {G}}^k(j^k\varphi )$.

The next proposition describes subspaces of $D^k\cap T{\mathcal {G}}^k(j^k\varphi )$ in order to choose terms which can be eliminated by $\mathcal {G}$-action. Such subspaces are related to the fibers of $\underline {k}$ with respect to $\Lambda _{\mathcal {G}}$.

Proposition 5.3 Given $k\in \mathbb {N}^*$ there exists a $d$-dimensional subspace in $D^k\cap T{\mathcal {G}}^k(j^k\varphi )$ if and only if there exists $L=\{l_1, \ldots, l_{d}\}\subseteq I$ satisfying the following condition:

\begin{align} \textit{for each}\ l\in L,\ \textit{there exists}\ J_l \subseteq I\ \textit{with}\ l \in J_l \setminus \mathop{\bigcup\nolimits_{i\in L\setminus \{l\}}}J_i,\ \textit{and}\ F_{J_l}(\underline {k})\neq \emptyset. \end{align}

Proof. We show that there exist $v_1,\ldots,v_d\in D^k\cap T{\mathcal {G}}^k(j^k\varphi )$ linearly independent if and only if there exists $L=\{l_1, \ldots, l_{d}\}\subseteq I$ satisfying (✶).

Let $\{v_i:=\sum _{\varphi _j\not \in B_2} \alpha _{ij}t_j^kE_{2j}+\sum _{\varphi _j\in B_{2}} \alpha _{ij}t_j^kE_{1j}\in D^k\cap T{\mathcal {G}}^k(j^k\varphi );\ 1\leqslant i\leqslant d\}$ be a set of linearly independent vectors. Then there exists $L=\{l_1, \ldots, l_{d}\}\subseteq I$ with $[\alpha _{il_j}]_{1\leqslant i,j\leqslant d}\in Gl_d(\mathbb {C})$.

In this way, for each $1\leqslant i\leqslant d$ there exists $w_i=\sum _{\varphi _j\not \in B_2} \beta _{ij}t_j^kE_{2j}+\sum _{\varphi _j\in B_{2}} \beta _{ij}t_j^kE_{1j}\in span\{v_1,\ldots,v_d\}\subseteq D^k\cap T{\mathcal {G}}^k(j^k\varphi )$ with $\beta _{il_i}=1$ and $\beta _{ij}=0$ for $j\in L\setminus \{l_i\}$. Taking the set $J_{l_i}=\{j\in I;\ \beta _{ij}\neq 0\}$, the previous corollary ensures the condition (✶).

On the other hand, suppose that there is $L=\{l_1, \ldots, l_{d}\}\subseteq I$ satisfying the condition (✶). Thus, by Corollary 5.2, for each $i=1,\ldots,d$, there exists $v_i=\sum _{\varphi _j\not \in B_2} \alpha _{ij}t_j^kE_{2j}+\sum _{\varphi _j\in B_{2}} \alpha _{ij}t_j^kE_{1j}\in D^k\cap T{\mathcal {G}}^k(j^k\varphi )$ such that $\alpha _{ij}\neq 0$ if $j\in J_{l_i}$ and $\alpha _{ij}=0$ for $j\in I\setminus J_{l_i}$. As the matrix $[\alpha _{il_j}]_{1\leqslant i,j\leqslant d}$ is invertible, it follows that $v_1,\ldots,v_d$ are linearly independent.

Remark 5.4 Note that, by the previous result, if $L=\{l_1, \ldots, l_{d}\}\subseteq I$ satisfies the condition (✶), then there exist $v_1,\ldots,v_d\in D^k\cap T{\mathcal {G}}^k(j^k\varphi )$ linearly independent with $v_i=\sum _{\varphi _j\not \in B_2} \alpha _{ij}t_j^kE_{2j}+\sum _{\varphi _j\in B_{2}} \alpha _{ij}t_j^kE_{1j}$ such that $\alpha _{ij}\neq 0$ if $j\in J_{l_i}$ and $\alpha _{ij}=0$ for $j\in I\setminus J_{l_i}$. In particular, for any $b_{l_1},\ldots,b_{l_d}\in \mathbb {C}$ there exists $v\in span\{v_1,\ldots,v_d\}$ with $v=\sum _{\varphi _j\not \in B_2} b_jt_j^kE_{2j}+\sum _{\varphi _j\in B_{2}} b_jt_j^kE_{1j}\in D^k\cap T{\mathcal {G}}^k(j^k\varphi )$ or, equivalently, by Proposition 5.1, there exists ${\rm w}=({\rm w}_1,\ldots, {\rm w}_r)\in \mathcal {I}_{\mathcal {G}}$ with $j^k{\rm w}_j=b_jt_j^k$ for any $b_{l_1},\ldots,b_{l_d}\in \mathbb {C}$.

As an immediate consequence we can establish an elimination criterion based on the set $\Lambda _{\mathcal {G}}$.

Corollary 5.5 Let $k\in \mathbb {N}^*$, $\varphi \in \mathcal {P}$ as in (6), and $L=\{l_1, \ldots, l_{d}\}\subseteq I$ satisfying the condition (✶). Then there exists $\psi \in \mathcal {P}$ such that $\psi \underset {\mathcal {G}}{\sim }\varphi$, $j^{k-1}\psi =j^{k-1}\varphi$, and $j^{k}\psi _{l}=j^{k-1}\varphi _{l}$ for all $l\in L$.

Proof. Since $\varphi$ is given as in (6) we have that $j^k\varphi =j^{k-1}\varphi + w$, with $w=\sum _{\varphi _j\not \in B_2} \beta _{j}t_j^kE_{2j}+\sum _{\varphi _j\in B_{2}} \beta _{j}t_j^kE_{1j}\in D^k$. By the above remark, there exists $v=\sum _{\varphi _j\not \in B_2} b_jt_j^kE_{2j}+\sum _{\varphi _j\in B_{2}} b_jt_j^kE_{1j} \in D^k\cap T{\mathcal {G}}^k(j^k\varphi )$ with $b_l=-\beta _l$ for all $l\in L$. By Theorem 3.3, it follows that $\varphi$ is $\mathcal {G}$-equivalent to some $\psi \in \mathcal {P}$ with $j^{k}\psi =j^{k}\varphi +v=j^{k-1}\varphi +w+v$, that is, $j^{k-1}\psi =j^{k-1}\varphi$ and $j^{k}\psi _{l}=j^{k-1}\varphi _{l}$ for all $l\in L$.

Given $k>0$ the parameter elimination method established in the above result depends upon a set $L$ satisfying the condition (✶) in Proposition 5.3. The aim is to eliminate terms of order $k$ from the greater number of branches prioritizing the first ones. We can formalize this by defining the injective map $S:\{L;\ L\subseteq I\}\to \mathbb {N}^r$ given by $S(L)=(z_1, \ldots, z_r)$, where

\[ z_i=\begin{cases} 1 & \mbox{if}\ i \in L,\\ 0 & \mbox{if}\ i \not\in L. \end{cases} \]

Definition 5.6 For each fixed $k$, we consider the set of indices $L_k\subseteq \{1,\ldots,r\}$ such that $S(L_k)=\max _{GLex}\{S(L);\ L\ \mbox {satisfying}\ (\mbox *)\ \mbox {in Proposition 5.3}\}$, where $\max _{GLex}$ means the maximum with respect to graded lexicographic order.

Remark 5.7 Let $\varrho$ be the conductor of $\Lambda _{\mathcal {G}}$ and $\underline {k}\in \varrho +\mathbb {N}^r$. We have that $F_{\{i\}}(\underline {k})\neq \emptyset$ for all $i\in I$, consequently $L_k=I$.

If $\underline {k}$ is an absolute maximal of $\Lambda _{\mathcal {G}}$, then $F_I(\underline {k})\neq \emptyset$ and $F_{J}(\underline {k})=\emptyset$ for all $J\subsetneq I$. In this case, $L_k=\{1\}$.

If $\underline {k}$ is a relative maximal of $\Lambda _{\mathcal {G}}$, then $F_{\{i\}}(\underline {k})=\emptyset$ for all $i\in I$ and $F_J(\underline {k})\neq \emptyset$ for all $J\subseteq I$ with $\sharp J>1$. Note that for any $i_0\in I$, the set $L=I\setminus \{i_0\}$ and $J_l=\{l,i_0\}$ for all $l\in L$ satisfy the condition (✶). Then, in this case we have $L_k=I\setminus \{r\}$.

Now we can establish one of the main results of this paper:

Theorem 5.8 ($\mathcal {G}$-normal form)

With the above notation any $\varphi =[\varphi _1,\ldots, \varphi _r]\in \Sigma _{\Gamma,\Lambda _{\mathcal {G}}}$ is $\mathcal {G}$-equivalent to $\psi =[\psi _1, \ldots, \psi _r]\in \mathcal {P}$ such that

(8) \begin{equation} \psi_i(t_i)=\begin{cases} \Bigg(t_i^{n_i},\ \theta_lt_i^{n_i}+\displaystyle\sum\limits_{\substack{ j>n_i\\ i\not\in L_j}}a_{ij}t_i^j\Bigg) & \mbox{for}\ \psi_i\in B_l,\ l\neq 2;\\ \Bigg(\displaystyle\sum_{\substack{j>n_i\\ i\not\in L_j}}a_{ij}t_i^j,\ t_i^{n_i}\Bigg) & \mbox{for}\ \psi_i\in B_2, \end{cases} \end{equation}

where $L_j$ is given in Definition 5.6, $\theta _1=0$, $\theta _3=1$, and $\theta _k \neq \theta _l$ for $1\leqslant k$, $l\leqslant r$ with $k \neq l$. Moreover, if $\psi '=[\psi '_1, \ldots, \psi '_r]\in \mathcal {P}$ is another multigerm $\mathcal {G}$-equivalent to $\varphi$ as (8), then $\psi '=\psi$, that is, the $\mathcal {G}$-normal form of $\varphi$ is unique.

Proof. By Proposition 4.6 we can consider $\varphi \in \mathcal {P}$ in its Puiseux block form. For all $k>0$ taking $L_k$ as defined previously and applying the Corollary 5.5 we obtain a $\mathcal {G}$-normal form $\psi$ as (8). Remark that by Proposition 4.3 it is sufficient to consider $k<\max \{d_i;\ i\in I\}$ where $d_i$ is given in Proposition 4.4.

Now suppose $\psi '=[\psi '_1, \ldots, \psi '_r]\in \mathcal {P}$ as (8), $\mathcal {G}$-equivalent to $\varphi$ and $\psi '\neq \psi$. Let us consider $k=\min \{l;\ j^{l}\psi \neq j^{l}\psi '\}$.

By Proposition 5.3 there exists $\{v_i\in D^k\cap T\mathcal {G}^k(j^k\psi );\ l_i\in L_k\}$ linearly independent (possibly empty) and subsets $J_{l_i}\subseteq I$ satisfying the condition (✶) with $v_i=\sum _{\psi _j\not \in B_2} \alpha _{ij}t_j^kE_{2j}+\sum _{\psi _j\in B_{2}} \alpha _{ij}t_j^kE_{1j}$ such that $\alpha _{ij}\neq 0$ if $j\in J_{l_i}$ and $\alpha _{ij}=0$ for $j\in I\setminus J_{l_i}$.

Let $\mathcal {F}^k\subseteq \mathcal {G}^k$ be the subgroup that leave invariant the affine space $N^k=j^{k-1}\psi +V$ where

\[ V=\bigg\{\sum_{\psi_j\not\in B_2} \beta_{j}t_j^kE_{2j}+\sum_{\psi_j\in B_{2}} \beta_{j}t_j^kE_{1j}\in D^k;\ \beta_j\in\mathbb{C}\ \mbox{and}\ \beta_l=0\ \mbox{for all}\ l\in L_k\bigg\}. \]

As the ground field is $\mathbb {C}$ and $\mathcal {F}^k$ is unipotent it follows that $\mathcal {F}^k$ is a connected group. Thus, the orbit $\mathcal {F}^k(j^k\psi )\subseteq N^k\subseteq \mathcal {G}^k(j^k\psi )$ is connected and it contains $j^k\psi$ and $j^k\psi '$. Let $w:=\sum _{\psi _j\not \in B_2} \gamma _{j}t_j^kE_{2j}+\sum _{\psi _j\in B_{2}} \gamma _{j}t_j^kE_{1j}\in T\mathcal {F}^k(j^k\psi )\subseteq V\subseteq T\mathcal {G}^k(j^k\psi )$ be a non-zero vector and use the notation $J'=\{j;\ \gamma _{j}\neq 0\}\subseteq I$. Remark that $l\not \in J'$ for all $l\in L_k$.

Choosing $j_0\in J'=:J'_{j_0}$, taking the element $w_l:=v_l- ({\alpha _{lj_0}}/{\gamma _{j_0}})w \in T\mathcal {G}^k(j^k\psi )$ and setting $J'_l=\{j\in J_l\setminus \{j_0\};\ \alpha _{lj}\gamma _{j_0}\neq \alpha _{lj_0}\gamma _{j}\}$ for all $l\in L_k$, it follows by Corollary 5.2 that $F_{J'_l}(\underline {k})\neq \emptyset$. In addition, for each $l\in L':=L_k\cup \{j_0\}$ we have that $l \in J'_l \setminus \bigcup _{i\in L'\setminus \{l\}}J'_i$ and $F_{J'_l}(\underline {k})\neq \emptyset$. In this way, $L'\supsetneq L_k$ satisfies the condition (✶) in Proposition 5.3, which contradicts the maximality of the set $L_k$.

Hence, $\psi =\psi '$ and the $\mathcal {G}$-normal form of $\varphi$, as (8), is unique.

Remark 5.9 Note that the parameters $a_{ij}$ in the $\mathcal {G}$-normal form (8) are not necessarily free because, in order to guarantee that the multigerm is in $\Sigma _{\Gamma,\Lambda _{\mathcal {G}}}$, some algebraic conditions must be imposed on the coefficients. The conditions imposed to have the semiring $\Gamma$ are easily determined by the non-zero coefficients of the terms with characteristic exponents and some conditions to get the intersection multiplicity of each pair of branches. On the other hand, the algebraic conditions related with $\Lambda _{\mathcal {G}}$ can be obtained applying the algorithm presented in [CH20b].

As we remarked in § 2, the analytic equivalence of plane curves with $r$ irreducible components can be translated by the $\mathcal {S}\times \mathcal {A}$-action on $\mathcal {P}$. Any element in $\mathcal {P}$ is $\mathcal {S}\times \mathcal {A}$-equivalent to a multigerm given in a Puiseux block form and the $\mathcal {A}$-action on such elements splits according to the action of subgroups given in Lemma 3.2, namely $\mathcal {H}\circ \tilde {\mathcal {A}_1}$, $\mathcal {H}\circ \mathcal {A}_1$, or $\mathcal {H}'\circ \mathcal {A}_1$.

In Theorem 5.8 we presented $\mathcal {G}$-normal forms for Puiseux block form multigerms with $\mathcal {G}=\tilde {\mathcal {A}_1}$ or $\mathcal {G}=\mathcal {A}_1$. In this way, to present $\mathcal {A}$-normal forms of such multigerms we have to consider the action of $\mathcal {H}$ or $\mathcal {H}'$, that we call homothety actions, on the respective $\mathcal {G}$-normal forms. Recall that an action by a subgroup of $\mathcal {H}$ does not introduce nor eliminate terms in a multigerm.

Let us consider $\varphi =[\varphi _1,\ldots,\varphi _r]\in \mathcal {P}$ given as (8) and the parameter vector

(9)\begin{equation} p=(a_{1,n_1+1}, \ldots, a_{1,k_0},a_{2,n_2+1}, \ldots, a_{2,k_0},\ldots ,a_{r,n_r+1}, \ldots, a_{r,k_0}), \end{equation}

where $k_0=\min \{z\in \mathbb {N};\ \underline {z}\in \varrho +\mathbb {N}^r\}$ (see Remark 5.7).

Given an element $(\rho _1,\ldots,\rho _r,\sigma )\in \mathcal {H}$ where $\rho _i(t_i)=u_it_i$ and $\sigma (X,Y)=(\alpha X,\gamma Y)$ with $\alpha,\gamma,u_i\in \mathbb {C}^*$ for $i\in I$, in order to keep $\psi =\sigma \circ \varphi \circ \rho ^{-1}$ as in (8), that is, $j^{n_i}(\varphi _i)=j^{n_i}(\psi _i)$ for all $i\in I$, we must have

\[ u_i=\alpha^{-({1}/{n_i})}\quad \mbox{if}\ \varphi_i\not\in B_2,\quad u_i=\gamma^{-({1}/{n_i})}\quad \mbox{if}\ \varphi_i\in B_2 \]

and $\theta _l\gamma \alpha ^{-1}=\theta _l$ for all $\varphi _i\in B_l$ with $l\neq 2$. Remark that $\alpha =\gamma$ for $(\rho _1,\ldots,\rho _r,\sigma )\in \mathcal {H}'\subset \mathcal {H}$, that is, if we have $s>2$ blocks.

The corresponding parameter vector of $\psi$ is $(b_{1,n_1+1}, \ldots, b_{1,k_0},\ldots,b_{r,n_r+1}, \ldots, b_{r,k_0})$ with

(10)\begin{equation} b_{ij}=\begin{cases} \gamma\alpha^{-({j}/{n_i})}a_{ij} & \mbox{if} \ \psi_i\in B_1 \\ \alpha\gamma^{-({j}/{n_i})}a_{ij} & \mbox{if} \ \psi_i\in B_2 \end{cases}\quad \mbox{for}\ s\leqslant 2,\quad \mbox{and} \quad b_{ij}=\alpha^{{(n_i-j)/n_i}}a_{ij}\ \mbox{for all}\ i\in I,\ \mbox{if}\ s\geqslant 3. \end{equation}

Note that the parameter vector $p$ is null if and only if

(11)\begin{equation} \varphi_1=\psi_1=(t_1,0),\quad \varphi_2=\psi_2=(0,t_2),\quad\varphi_i=\psi_i=(t_i,\theta_it_i)\quad \mbox{for all}\ 3\leqslant i\leqslant r, \end{equation}

with $\theta _i\neq \theta _j$ for $i\neq j$.

On the other hand, if $p$ is a non-zero vector and $a_{i_0j_0}$ is the first non-zero coordinate, then $\psi _i$ (and $\varphi _i$) for all $i< i_0$ is given as (11) and we can choose $\alpha \in \mathbb {C}^*$ in (10) corresponding to $a_{i_0j_0}$ in order to normalize the corresponding coefficient in $\psi$, that is, to obtain $b_{i_0j_0}=1$.

Eventually (10) allow us to normalize one more coefficient, but for this purpose a finer analysis would be necessary. However, the above description it is sufficient to present the following theorem that gives a solution (up to $\mathcal {S}$-action) for the analytical equivalence problem for plane curves with $r$ irreducible components.

Theorem 5.10 ($\mathcal {A}$-normal form)

Let $\varphi =[\varphi _1,\ldots, \varphi _r]\in \Sigma _{\Gamma,\Lambda _{\mathcal {G}}}$ be a multigerm associated to the curve $\mathcal {C}$. Then $\varphi$ is $\mathcal {A}$-equivalent to $\psi =[\psi _1,\ldots, \psi _r]$ as (8) with parameter vector $p=(b_{1,n_1+1}, \ldots, b_{r,k_0})$ such that the following hold.

• – If $p=(0,\ldots,0)$, then $\psi _1=(t_1,0),\ \psi _2=(0,t_2),\ \psi _i=(t_i,\theta _it_i)\ \mbox {for all}\ 3\leqslant i\leqslant r$.

• – If $p\neq (0,\ldots,0)$, then $b_{i_0j_0}=1$ is the first non-zero coordinate of $p$. In addition, $\psi '\in \Sigma _{\Gamma,\Lambda _{\mathcal {G}}}$ as (8) with parameter vector $p'=(b'_{1,n_1+1}, \ldots, b'_{r,k_0})$ and first non-zero coordinate $b'_{i_0j_0}=1$ is $\mathcal {A}$-equivalent to $\psi$ if and only if we have the following conditions according to the number $s$ of blocks.

  

Proof. By Theorem 5.8 we obtain a unique $\mathcal {G}$-normal form for $\varphi$ as (8) and, as described previously, if the parameter vector is non-zero, then the $\mathcal {G}$-normal form is $\mathcal {H}$-equivalent (or $\mathcal {H}'$-equivalent if $s\geqslant 3$) to $\psi =[\psi _1,\ldots, \psi _r]$ as (8) with parameter vector $p=(b_{1,n_1+1}, \ldots, b_{r,k_0})$ such that its first non-zero coordinate is $b_{i_0j_0}=1$.

Now it is sufficient to describe the parameter vector $p'=(b'_{1,n_1+1}, \ldots, b'_{r,k_0})$ of a multigerm $\psi '\in \Sigma _{\Gamma,\Lambda _{\mathcal {G}}}$ as (8) with the first non-zero coordinate $b'_{i_0j_0}=1$ in the orbit $\mathcal {H}(\psi )$ if $s\leqslant 2$ or in $\mathcal {H}'(\psi )$ if $s\geqslant 3$. This will be done using (10).

If $s=1$, by (10) we have $\gamma \alpha ^{- ({j_0}/{n_{i_0}})}=1$ and $\gamma \alpha ^{- ({j}/{n_{i}})}b_{ij}=b'_{ij}$. Consequently, $\gamma =\alpha ^{ {j_0}/{n_{i_0}}}$ and $b_{ij}=\alpha ^{ ({jn_{i_0}-j_0n_i})/{n_in_{i_0}}}b'_{ij}$ for all coordinates of $p$ and $p'$.

For multigerms with two blocks it follows, by (10), that $b'_{ij}=\gamma \alpha ^{- ({j}/{n_{i}})}b_{ij}$ if $\psi _i\in B_1$ and $b'_{ij}=\alpha \gamma ^{- ({j}/{n_{i}})}b_{ij}$ if $\psi _i\in B_2$. If $\psi _{i_0}\in B_1$, then $\gamma =\alpha ^{ {j_0}/{n_{i_0}}}$ and $b_{ij}=\alpha ^{ ({jn_{i_0}-j_0n_i})/{n_in_{i_0}}}b'_{ij}$ for coordinates of $p$ and $p'$ with $\psi _i\in B_1$ and $b_{ij}=\alpha ^{{(jj_0-n_{i_0}n_i)/n_in_{i_0}}}b'_{ij}$ for all coordinates of $p$ and $p'$ with $\psi _i\in B_2$. If $\psi _{i_0}\in B_2$, then $\alpha =\gamma ^{ {j_0}/{n_{i_0}}}$ and $b_{ij}=\gamma ^{{(jn_{i_0}-j_0n_i)/n_in_{i_0}}}b'_{ij}$ for all entries of $p$ and $p'$.

Finally, for $s\geqslant 3$ we have $\alpha ^{{(n_{i_0}-j_0)/n_{i_0}}}=1$ and $\alpha ^{{(n_i-j)/n_i}}b_{ij}=b'_{ij}$ for all $i\in I$ and $n_i+1\leqslant j\leqslant k_0$.

The above theorem provides a method to decide whether two plane curves are analytically equivalent. In fact, considering an associated multigerm to a plane curve $\mathcal {C}$ with $r$ irreducible components we compute their $\mathcal {A}$-invariants $\Gamma$ and $\Lambda _{\mathcal {G}}$ using the appropriate group $\mathcal {G}$ and the algorithms in [CH20b] for instance. Applying the results in this section, we obtain an $\mathcal {A}$-normal form $\varphi =[\varphi _1,\ldots,\varphi _r]$ for the multigerm of $\mathcal {C}$.

Given another plane curve $\mathcal {C}'$ with multigerm associated $\psi =[\psi _1,\ldots,\psi _r]$, if the $\mathcal {A}$-invariants $\Gamma ^{\pi }$ and $\Lambda ^{\pi }_{\mathcal {G}}$ of $\pi (\psi )$ are distinct to the corresponding ones of $\varphi$ for every $\pi \in \mathcal {S}$, then $\mathcal {C}$ and $\mathcal {C}'$ are not analytically equivalent. If it is not the case, we consider the subgroup $\mathcal {R}$ of $\mathcal {S}$ such $\Gamma ^{\pi }=\Gamma$ and $\Lambda ^{\pi }_{\mathcal {G}}=\Lambda _{\mathcal {G}}$ for all $\pi \in \mathcal {R}$. The curves $\mathcal {C}$ and $\mathcal {C}'$ are analytically equivalent if and only if there exists $\pi \in \mathcal {R}$ such that $\varphi$ and an $\mathcal {A}$-normal form for $\pi (\psi )$ satisfy the corresponding condition in Theorem 5.10.

6. Particular cases

In this final section, we recover some known results regarding the analytic classification of plane curves using Theorems 5.8 and 5.10 and we illustrate how we can apply them in particular topological classes.

6.1 Irreducible case and bigerms

For the irreducible case, i.e. $s=r=1$, we must consider $\mathcal {G}=\tilde {\mathcal {A}}_1$ and the normal form described in Theorem 5.8 is rewritten as $\varphi =(t^n,\sum _{j>n\atop L_j=\emptyset }c_jt^j).$

Note that $L_j=\emptyset$ if and only if $F_{\{1\}}(j)=\emptyset$, that is, $j\not \in \Lambda _{\mathcal {G}}$.

If $n=1$, then $\Lambda _{\mathcal {G}}=\overline {\mathbb {N}}\setminus \{0,1\}$ and $\varphi$ is $\mathcal {A}$-equivalent to $(t,0)$.

Let us suppose that $n\geqslant 2$ and $\Gamma \cap \mathbb {N} =\langle n,v_1,\ldots,v_g\rangle$. Thus, $\varphi =(t^n,c_{v_1}t^{v_1}+\sum _{j\not \in \Lambda _{{\mathcal {G}}}}c_jt^j)$ with $c_{v_1}\neq 0$ and, by Theorem 5.10, we have $\varphi \underset {\mathcal {A}}{\sim }(t^n,t^{v_1}+ \sum _{j\not \in \Lambda _{{\mathcal {G}}}}b_jt^j)$.

If $j\in \Lambda _{\mathcal {G}}$ for every $j>v_1$, then $\varphi \underset {\mathcal {A}}{\sim }(t^n,t^{v_1})$. By [Zar66], this is equivalent to claim that $\Lambda \setminus \Gamma =\emptyset$.

If there exists $j\not \in \Lambda _{\mathcal {G}}$ for some $j>v_1$, then setting $\lambda =\min \{j>v_1;\ j\not \in \Lambda _{\mathcal {G}}\}$ we have

(12)\begin{equation} \varphi\ \underset{\mathcal{A}}{\sim} \bigg(t^n,t^{v_1}+b_{\lambda}t^{\lambda}+ \sum_{\lambda< j\not\in \Lambda_{{\mathcal{G}}}}b_jt^j\bigg) \end{equation}

with $b_{\lambda }\neq 0$ and $\lambda +n=\nu (\varphi ^*(\omega _0))$ where $\omega _0:=v_1Y\,dX-nX\,dY$.

Remark 6.1 Note that $\{j>\lambda ;\ j\not \in \Lambda _{\mathcal {G}}\}=\{j>\lambda ;\ j\not \in \Lambda -n\}$. In fact, by definition we have $\Lambda _{\mathcal {G}}\subset \Lambda -n$. On the other hand, given $\omega =(\alpha _0+\alpha _1X+\alpha _2Y+h_1)\,dX+(\alpha _3+\alpha _4X+h_2)\,dY\in \Omega ^1$ with $\alpha _i\in \mathbb {C}$ and $\omega _1=h_1\,dX+h_2\,dY\in \Omega _{\mathcal {G}}$ in order to have $\nu (\varphi ^*(\omega ))-n>\lambda$ we must have $\alpha _0=\alpha _1=\alpha _3=0$ and $n\alpha _2+v_1\alpha _4=0$. In this way, $\omega = ({\alpha _2}/{v_1})\omega _0+\omega _1$, as $\nu (\varphi ^*(\omega _0))-n=\lambda \neq \nu (\varphi ^*(\omega _1))-n\in \Lambda _{\mathcal {G}}$ (by (12)), we have $\nu (\varphi ^*(\omega ))-n=\nu (\varphi ^*(\omega _1))-n\in \Lambda _{\mathcal {G}}$.

By the above remark, if $\Lambda \setminus \Gamma \neq \emptyset$, that is, there exists $\lambda =\min \{j>v_1;\ j\not \in \Lambda _{\mathcal {G}}\}$, then $\varphi \underset {\mathcal {A}}{\sim }(t^n,t^{v_1}+b_{\lambda }t^{\lambda }+ \sum _{\lambda < j\not \in \Lambda -n}b_jt^j)$ with $b_{\lambda }\neq 0$.

Taking $c\in \mathbb {C}^*$ with $c^{\lambda -v_1}=b_{\lambda }$, $\rho (t)=ct$ and $\sigma (X,Y)=(c^nX,c^{v_1}Y)$ we obtain $(\rho,\sigma )\cdot \varphi =(t^n,t^{v_1}+t^{\lambda }+ \sum _{\lambda < j\not \in \Lambda -n}a_jt^j)$, that is, the normal form presented in Theorem 2.1 in [HH11].

For bigerms, we have two distinct cases according to the number $s\in \{1,2\}$ of Puiseux blocks.

By Theorem 5.8 we get

\[ \varphi_1=\Bigg(t_1^{n_1},\sum_{\substack{j>n_1\\ 1\not\in L_j}}a_{1j}t_1^j\Bigg)\quad \mbox{and}\quad \varphi_2= \begin{cases} \bigg( t_2^{n_2},\displaystyle\sum_{\substack{j>n_2\\ 2\not\in L_j}}a_{2j}t_2^j\bigg) & \mbox{if}\ s=1;\\[10pt] \bigg( \displaystyle\sum_{\substack{j>n_2\\ 2\not\in L_j}}a_{2j}t_2^j, t_2^{n_2}\bigg) & \mbox{if}\ s=2. \end{cases} \]

Note that $1\not \in L_j$ if and only if $F_{\{1\}}(\underline {j})=F_{\{1,2\}}(\underline {j})=\emptyset$, that is, $F_{\{1\}}(\underline {j})=\emptyset$ and $\underline {j}$ is not a maximal point of $\Lambda _{\mathcal {G}}$. On the other hand, $2\not \in L_j$ means that $F_{\{2\}}(\underline {j})=\emptyset$.

Hence, for curves with two branches, Theorem 5.8 recovers Theorem 6 of [HHR15].

6.2 Singular ordinary point

Let us consider a plane curve $\mathcal {C}$ with a singular ordinary point at the origin of multiplicity $r>1$, that is, $\mathcal {C}$ has $r$ irreducible regular components given by $f_i\in \mathcal {M}\setminus \mathcal {M}^2$ such that the slopes $\theta _i$ of their tangent lines satisfy $\theta _i\neq \theta _j$ for all $i,j\in I,\ i\neq j$.

Applying Proposition 3.9 of [CH20a] for this case we obtain that the value semiring $\Gamma$ is minimally generated by

\[ v_i=\nu(f_i)=(1,\ldots ,1,\infty ,1,\ldots ,1);\quad i\in I \]

where the symbol ‘$\infty$’ occupies the $i{{\rm th}}$ coordinate. In particular, any element $\gamma \in \Gamma$ can be written as (1), i.e. $\gamma =\inf \big \{\sum _{i\in I}a_{1i}v_i,\ldots,\sum _{i\in I}a_{ri}v_i\big \}$. If $a_{li}\neq 0$ for all $i\in I$, then $\sum _{i\in I}a_{li}v_i=(\infty,\ldots,\infty )$ which is irrelevant for the computation of $\gamma$. Thus, we can assume $a_{li}=0$ for every $l\in I$ and some $i\in I$ which depends on $l$.

Note that $\mu _i=0$ and $\nu _j(f_i)=1$ for all $i,j\in I$ with $i\neq j$, then the conductor of $\Gamma$ is $\kappa =(r-1,\ldots,r-1)$. In particular, $F^{\Gamma }_{J}(\underline {k})\neq \emptyset$ for every $k\geqslant r-1$ and all $\emptyset \neq J\subseteq I$.

By Propositions 4.3 and 4.4, for $r\in \{1,2,3\}$ the plane curve $\mathcal {C}$ admits a multigerm $\varphi$ that is $\mathcal {A}$-equivalent to $[(t_1,0)]$, $[(t_1,0),(0,t_2)]$, or $[(t_1,0),(0,t_2),(t_3,t_3)]$, respectively.

In what follows we consider $r\geqslant 4$. Note that the appropriated subgroup to apply Theorem 5.8 is $\mathcal {G}=\mathcal {A}_1$.

Using the generators of the value semiring $\Gamma$ we can describe the non-empty fibers $F^{\Gamma }_J(\underline {k})$ for $0\leqslant k< r$.

Lemma 6.2 For a singular ordinary point with multiplicity $r$ the $J$-fiber of $\underline {k}\in \mathbb {N}^r$ of $\Gamma$ is non-empty if and only if $\sharp J\geqslant r-k$. Moreover, for any $J\subseteq I$ with $\sharp J\geqslant r-k$ we have

\[ F_{J}^{\Gamma}(\underline{k})=\{\underline{k}+\alpha;\ \alpha=(\alpha_1,\ldots ,\alpha_r)\in\overline{\mathbb{N}}^r\ \mbox{with}\ \alpha_j=0\ \mbox{if}\ j\in J\ \mbox{and}\ \alpha_j\neq 0\ \mbox{if}\ j\in I\setminus J\}. \]

Proof. As the conductor of $\Gamma$ is $\kappa =(r-1,\ldots,r-1)$ it is sufficient to describe the fiber $F^{\Gamma }_{J}(\underline {k})$ for every $\underline {k}=(k,\ldots,k)\in \mathbb {N}^r,\ 0\leqslant k\leqslant r-1$ and every $J\subseteq I$ with $J\neq \emptyset$.

First, note that if $F^{\Gamma }_{J}(\underline {k})\neq \emptyset$, then $\sharp J\geqslant r-k$. In fact, let $\gamma \in F^{\Gamma }_{J}(\underline {k})$. Since $\gamma \in \Gamma$, there exists $h\in \mathbb {C}\{X,Y\}$, namely of multiplicity $n_h$, such that $\gamma =\nu (h)$. If $\sharp J< r-k$, as $n_i=1$, $\theta _i\neq \theta _l$ for all $i,l\in I$ with $i\neq l$, we must have $m_h\leqslant n_h\leqslant \nu _j(h)=k< r-\sharp J$, for all $j \in J$, where $m_h$ is the number of distinct tangent lines of $h$. However, in this way, there exists $i\in I\setminus J$ such that $f_i$ and $h$ do not share the same tangent line. Thus, $\nu _i(h)=n_h\leqslant k$ and, therefore, $\gamma \not \in F^{\Gamma }_{J}(\underline {k})$ which is a contradiction.

Now we describe $F^{\Gamma }_{J}(\underline {k})$ with $\sharp J\geqslant r-k$.

For $0\leqslant k\leqslant r-1$ we take any $J_0\subseteq I$ with $\sharp J_0=r-k\ (\geqslant 1)$ and let us consider the sets $I\setminus J_0=\{i_1,\ldots,i_k\}$ and $\{i_1,\ldots, \widehat {i_m},\ldots,i_k\}:=\{i_1,\ldots,i_k\}\setminus \{i_m\}$.

Fixing $i_0\in J_0$ we set

\[ \zeta_0=\sum_{l\in \{i_1,\ldots ,i_k\}}\nu(f_l),\quad \zeta_m=\sum_{l\in \{i_1,\ldots ,\widehat{i_m},\ldots, i_k\}}\nu(f_l)+(1+\alpha_{m})\nu(f_{i_0})\in\overline{\mathbb{N}}^r, \]

with $1\leqslant m\leqslant k$ and $\alpha _1,\ldots,\alpha _k\in \mathbb {N}$. Note that the coordinates of $\zeta _0$ and $\zeta _m$, respectively, satisfy

\[ \zeta_{0j}=\begin{cases} k & \mbox{if} \ j\in J_0, \\ \infty & \mbox{if} \ j\in I\setminus J_0,\end{cases}\quad \zeta_{mj}=\begin{cases} \infty & \mbox{if}\ j\in \{i_0,i_1,\ldots ,\widehat{i_m},\ldots,i_k\}, \\ k+\alpha_{m} & \mbox{if} \ j\not\in \{i_0,i_1,\ldots ,\widehat{i_m},\ldots,i_k\}. \end{cases} \]

Thus, $\gamma =(\gamma _1,\ldots,\gamma _r):=\inf \left \{\zeta _0,\zeta _1,\ldots,\zeta _k\right \}\in \Gamma$ is such that $\gamma _j=k$ if $j\in J_0$, $\gamma _j=k+\alpha _j$ with $\alpha _j\in \mathbb {N}$ for $j\in I\setminus J_0$.

Hence, the above construction ensures that for any $J\subseteq I$ with $\sharp J\geqslant r-k$ we have

\[ F_{J}^{\Gamma}(\underline{k})=\{\underline{k}+\alpha;\ \alpha=(\alpha_1,\ldots ,\alpha_r)\in\overline{\mathbb{N}}^r\ \mbox{with}\ \alpha_j=0\ \mbox{if}\ j\in J\ \mbox{and}\ \alpha_j\neq 0\ \mbox{if}\ j\in I\setminus J\}. \]

Remark that $\nu (\mathcal {M}^2)-\underline {1}\subseteq \Lambda _{\mathcal {A}_1}$ which implies that its corresponding fibers satisfy $\emptyset \neq F_J^{\Gamma }(\underline {k+1})-\underline {1}\subseteq F_J(\underline {k})$ for all $k\geqslant 2$ and $\sharp J \geqslant r-(k+1)$.

Taking $L=\{1,\ldots,\min \{k+2,r\}\}$ and $J_i=\{i,k+3,\ldots,r\}$ for $i\in L$ in Proposition 5.3, then Corollary 5.5 allows us to eliminate the term corresponding to $t_i^k$ of the component $\varphi _i$ for all $i=1, \ldots, \min \{k+2,r\}$ preserving the analytic class of $\mathcal {C}$. With this approach we obtain Theorem 1.1 in [GP11].

Proposition 6.3 Any curve with a singular ordinary point with $r$ branches is analytically equivalent to a curve defined by a multigerm $\psi =[\psi _1, \ldots, \psi _r]$ with

(13)\begin{equation} \psi_1=(t_1,0),\quad \psi_2=(0,t_2), \quad \psi_i=\Bigg(t_i, \ \theta_{i}t_i + \displaystyle\sum_{j=2}^{i-3}a_{ij}t_i^j\Bigg),\quad i=3, \ldots, r, \end{equation}

where $\theta _3=1$ and $\theta _i\neq \theta _l$ for $i\neq l$.

For $r\geqslant 4$, Granger [Gra79] considers the generic parameter vector to obtain $\mathcal {A}$-normal forms and to compute the dimension of the moduli space. The same result was obtained with other methods by Genzmer and Paul [GP11]. From now on, we consider the generality hypothesis and we apply our techniques with the purpose to compute the dimension of the moduli space.

The generality of the coefficients give us a symmetry on the coordinates of $\Lambda _{\mathcal {A}_1}$. In this way, for $k\geqslant 2$ the set $L_k\subseteq I$ in Theorem 5.8 is $L_k=\{1,2,\ldots,e(k)\}$ for some $4\leqslant e(k)\leqslant r$. By Remark 5.4, $e(k)$ is the maximum integer such that for any $b_{ik}\in \mathbb {C}$ with $1\leqslant i\leqslant e(k)$ there exists ${\rm w}\in \mathcal {I}_{\mathcal {A}_1}$ with $j^k{\rm w}=(b_{1k}t_1^{k},\ldots,b_{rk}t_r^{k})$.

Recall that ${\rm w}=\big (t_1^{-1}\varphi ^*_1(\omega ),\ldots,t_r^{-1}\varphi ^*_r(\omega )\big )\in \mathcal {I}_{\mathcal {A}_1}$ and $t_i^{-1}\varphi ^*_i(\omega )=\varphi ^*_i(p)\cdot x'_i+\varphi ^*_i(q)\cdot y'_i\in \mathbb {C}\{t_i\}$ for $i\in I$ where $\omega =p\,dX+q\,dY\in \Omega ^1$ and $p,q\in \langle X,Y\rangle ^2$.

In what follows, we use the notation

(14)\begin{equation} \omega_h=\bigg(\displaystyle\sum_{m=0}^{h}\alpha_{h,m}X^{h-m}Y^m\bigg)\,dX+\bigg(\displaystyle \sum_{m=0}^{h}\beta_{h,m}X^{h-m}Y^m\bigg)\,dY \in \Omega^1 \end{equation}

with $\alpha _{h,m},\beta _{h,m}\in \mathbb {C}$. Note that given $\varphi =[\varphi _1,\ldots,\varphi _r]$ as (13) and $\omega \in \Omega ^1$, if we want to obtain $\nu _i(\omega )=k$ and consequently to determine $e(k)$, it is sufficient to consider $w=\sum _{h=2}^{k}w_h$.

Lemma 6.4 With the above notation, if $r\geqslant 4$, we have $e(2)=4$ and $e(k)=\min \{2k+1,r\}$ for every $k\geqslant 3$.

Proof. Considering a generic multigerm $\varphi$ given as in (13) if $\omega =\omega _2$, that is $k=2$, then

\begin{gather} j^2(t_1^{-1}\varphi_1^{*}(\omega))=\alpha_{2,0}t_1^2,\quad j^2(t_2^{-1}\varphi_2^{*}(\omega))=\beta_{2,2}t_2^2,\nonumber \end{gather}

\begin{gather} j^2(t_i^{-1}\varphi_i^{*}(\omega))=\big( \alpha_{2,0}+(\alpha_{2,1}+\beta_{2,0})\theta_i+(\alpha_{2,2}+\beta_{2,1})\theta_i^2+\beta_{2,2}\theta_i^3\big)t_i^2,\quad \mbox{for all}\ 3\leqslant i\leqslant r.\nonumber \end{gather}

Now, it is immediate to verify that the maximum integer $e(2)$ such that the system $j^2(t_i^{-1}\varphi _i^{*}(\omega ))=b_{i2}t_i^2$ admits solution for any $b_{i2}\in \mathbb {C}$ with $1\leqslant i\leqslant e(2)$ is precisely $e(2)=4$.

We show that for any $3\leqslant l\leqslant [ {(r-1)/2} ]$ if $j^{l}(t_i^{-1}\varphi _i^{*}(\omega _{l-1}+\omega _{l}))=0$ for every $i\in I$, then $w_{l-1}=0$.

Note that for $w_{h}$ as (14) with $h\geqslant 2$, we get

(15)\begin{align} j^{h+1}(t_1^{-1}\varphi_1^{*}(\omega_h))&=\alpha_{h,0}t_1^h,\nonumber\\ j^{h+1}(t_2^{-1}\varphi_2^{*}(\omega_h))&=\beta_{h,h}t_2^h, \nonumber\\ j^{h+1}(t_i^{-1}\varphi_i^{*}(\omega_h))&=\bigg(\alpha_{h,0}+\sum_{m=1}^h(\alpha_{h,m}+\beta_{h,m-1})\theta_{i}^m +\beta_{h,h} \theta_i^{h+1}\bigg)t_i^h\nonumber\\ &\quad +\bigg(\sum_{m=0}^{h-1}((m+1)\alpha_{h,m+1}+(m+2)\beta_{h,m})\theta_{i}^{m}+(h+2)\beta_{h,h}\theta_i^{h}\bigg) a_{i2}t_i^{h+1}\nonumber\\ &\quad\qquad \mbox{for}\ 3\leqslant i\leqslant r. \end{align}

The condition $j^{l}(t_i^{-1}\varphi _i^{*}(\omega _{l-1}+\omega _{l}))=0$ for every $i\in I$ implies that $\alpha _{l-1,0}=\alpha _{l,0}=\beta _{l-1,l-1}=\beta _{l,l}=0$ and the coefficient of $t^{l-1}$ in $t_i^{-1}\varphi _i^{*}(\omega _{l-1}+\omega _{l})$ vanishes for every $3\leqslant i\leqslant r$, that is,

\[ \left(\begin{array}{cccc} \theta_{3} & \theta_{3}^2 & \cdots & \theta_{3}^{l} \\ \vdots & \vdots & & \vdots \\ \theta_{r} & \theta_{r}^2 & \cdots & \theta_{r}^{l} \\ \end{array}\right) \cdot \left(\begin{array}{c} \alpha_{l-1,1}+\beta_{l-1,0}\\ \vdots \\ \alpha_{l-1,l-1}+\beta_{l-1,l-2} \end{array}\right)=\left(\begin{array}{c} 0\\ \vdots \\ 0 \end{array}\right). \]

Thus, $\alpha _{l-1,m}=-\beta _{l-1,m-1}$ for $1\leqslant m\leqslant l-1$. In addition, as the coefficient of $t^{l}$ in $t_i^{-1}\varphi _i^{*}(\omega _{l-1}+\omega _{l})$ is null, by (15) we must have

(16)\begin{equation} \left(\begin{array}{ccccccc} a_{32} & \theta_{3}a_{32} & \cdots & \theta_{3}^{l-2}a_{32} & \theta_{3} & \cdots & \theta_{3}^{l} \\ \vdots & \vdots & & \vdots & \vdots & & \vdots \\ a_{r2} & \theta_{3}a_{32} & \cdots & \theta_{r}^{l-2}a_{r2} & \theta_{r} & \cdots & \theta_{r}^{l} \\ \end{array}\right) \cdot \left(\begin{array}{c} \beta_{l-1,0}\\ \vdots \\ \beta_{l-1,l-2}\\ \alpha_{l,1}+\beta_{l,0}\\ \vdots \\ \alpha_{l,l}+\beta_{l,l-1} \end{array}\right)=\left(\begin{array}{c} 0\\ \vdots \\ 0 \end{array}\right). \end{equation}

As we suppose $\varphi$ generic, the coefficient matrix has maximal rank $\min \{2l-1,r-2\}$ and we get the solution $\beta _{l-1,m}=0$ for $0\leqslant m\leqslant l-2$ and $\alpha _{l,m}=-\beta _{l,m-1}$ for $1\leqslant m\leqslant l$. Consequently, $w_{l-1}=0$.

In order to obtain $j^{k}(t_i^{-1}\varphi _i^{*}(\sum _{h=2}^{k}\omega _h))=b_{ik}t_i^{k}$ it is necessary that $j^{l}(t_i^{-1}\varphi _i^{*}(\omega _h))=0$ for each $h=2,\ldots, k-1$ and for all $l=h,\ldots, k-1$. From this and the above discussion, it follows by induction that $\omega _h=0$ for $2\leqslant h\leqslant k-2$. Thus, it is sufficient to consider $\omega =\omega _{k-1}+\omega _{k}$.

By a similar computation as in (15) using $l=k$ we get the system

\[ \begin{cases} \alpha_{k,0}=b_{1k},\\ \beta_{k,k}=b_{2k},\\ \alpha_{k,0}+\sum\limits_{m=0}^{k-2}\beta_{k-1,m}\theta_i^{m}a_{i2} +\sum\limits_{m=1}^{k}(\alpha_{k,m}+\beta_{k,m-1})\theta_i^m+\beta_{k,k}\theta_i^{k+1}=b_{ik} \quad \mbox{for}\ 3\leqslant i\leqslant r. \end{cases} \]

Substituting the two first solutions into the other equations, we obtain a system with the same coefficient matrix as (16). Thus, generically we can solve at most $e(k)=2+\min \{2k-1,r-2\}=\min \{2k+1,r\}$ such equations.

As a consequence, we recover the analytical normal form for regular transversal branches in the generic case, as given in [Gra79, Proposition 2] and [GP11].

Proposition 6.5 Generically, any plane curve with a singular ordinary point and $r \geqslant 4$ is analytically equivalent to a curve defined by a multigerm $\psi =[\psi _1, \ldots, \psi _r]$ with

\[ \begin{array}{c} \psi_1=(t_1,0),\quad \psi_2=(0,t_2), \quad \psi_3=(t_3,t_3),\quad \psi_4=(t_4,\theta_4t_4),\\ \psi_i=\Bigg(t_i, \ \theta_{i}t_i + a_{i2}t_i^2+ \displaystyle\sum_{k=3}^{[ {(i-2)/2}]}a_{ik}t_i^k\Bigg),\quad 5\leqslant i\leqslant r. \end{array} \]

Proof. The previous lemma gives us $L_2=\{1,2,3,4\}$ and $L_k=\{1,\ldots,\min \{2k+1,r\}\}$ for $k\geqslant 3$. Thus, by Theorem 5.8, we have the above normal form for $1\leqslant i\leqslant 4$ and

\[ \psi_i=\Bigg(t_i, \theta_{i}t_i + a_{i2}t_i^2+ \displaystyle\sum_{\substack{k\geqslant 3\\ i\not\in L_k}}a_{ik}t_i^k\Bigg)\quad\mbox{for}\ 5\leqslant i\leqslant r. \]

The result follows since that $i\not \in L_k$ is equivalent to $k\leqslant [ {(i-2)/2}]$ for $k\geqslant 3$.

Note that for $r\leqslant 3$ the moduli space corresponding is a single point and for $r=4$ is a one-dimensional space.

For $r\geqslant 5$ the parameter vector associated to the normal form $\psi$ presented in the previous result is $\big (a_{5,2},a_{6,2},a_{6,3},\ldots,a_{r,2},\ldots,a_{r,[ {(r-2)/2}]}\big )$. Generically, $a_{5,2}\neq 0$ and by Theorem 5.10 we get a multigerm $\mathcal {A}$-equivalent to $\psi$ with parameter vector $p=\big (1,a'_{6,2},a'_{6,3},\ldots,a'_{r,2},\ldots,a'_{r,[ {(r-2)/2}]}\big )$.

In particular, the dimension of the moduli space $\mathbb {M}_r$ for multigerm of curves with singular ordinary point and $r\geqslant 5$ branches is the number of parameters in $p$ plus $r-3$ which corresponds to $\sharp \{\theta _i;\ 4\leqslant i\leqslant r\}$. If we denote $L_1=\{1,2,3\}$ just to unify the notation, we have that $\sharp I\setminus L_1 = \sharp \{\theta _i;\ 4\leqslant i\leqslant r\}$. Thus,

\[ \dim_{\mathbb{C}}\mathbb{M}_r=\sum_{k\geqslant 1}(\sharp I\setminus L_k)-1=\sum_{k=1}^{[ {(r-2)/2}]} (r-2k-1)=\begin{cases} \dfrac{(r-2)^2}{4} & \mbox{for}\ r\ even,\\[8pt] \dfrac{(r-1)(r-3)}{4} & \mbox{for}\ r\ odd, \end{cases} \]

as originally obtained by Granger [Gra79].

6.3 Multigerms with $\Gamma _i\cap \mathbb {N}=\langle n,m\rangle$ and $\nu _i(f_l)=nm$

We have just studied a class of multigerms in which each block has just one element. In this subsection, we consider the opposite situation, that is, curves such that we have only one block.

Let $\mathcal {C}$ be a plane curve defined by a reduced element $f=\prod _{i=1}^{r}f_i\in \mathbb {C}\{X,Y\}$ such that each branch $\mathcal {C}_i$ admits $\Gamma _i\cap \mathbb {N}=\langle n,m\rangle$, $GCD(n,m)=1< n< m$, and $\nu _i(f_l)=nm$ for all $i,l\in I$ with $i\neq l$. We present a pre-normal form considering just the topological data and we express the dimension of the generic component $\mathbb {M}_r(n,m)$ of the moduli space by means of the sets $L_k$. For the specific case of $n=2$ and $m=3$ we present a closed formula for $\dim _{\mathbb {C}}\mathbb {M}_r(2,3)$.

The curve $\mathcal {C}$ admits a multigerm with Puiseux block form $\varphi =[\varphi _1,\ldots,\varphi _r]$ where

\[ \varphi_i=\bigg(t_i^n,a_{im}t_i^m+\sum_{j>m}a_{ij}t_i^j\bigg) \]

with $a^n_{im}\neq a^n_{lm}$, $a_{im}\neq 0\neq a_{lm}$ for every $1\leqslant i$, $l\leqslant r$, and $i\neq l$. In this situation we have a single block and consequently the corresponding group to consider in Theorem 5.8 is $\mathcal {G}=\tilde {\mathcal {A}}_1$.

The topological class of $\mathcal {C}$ can be characterized by the value semiring $\Gamma$ that, according to Proposition 3.9 of [CH20a], is $\Gamma =\langle v_1,v_2,\ldots,v_{r+2}\rangle$ with

(17)\begin{equation} v_1=(n,\ldots ,n),\quad v_2=(m,\ldots ,m),\quad v_{i+2}=(nm,\ldots ,\infty,\ldots ,nm)\quad i\in I, \end{equation}

where the symbol ‘$\infty$’ occupies the $i{\rm th}$ coordinate in $v_{i+2}$.

The conductor of $\Gamma$ is $\kappa =(\kappa _1,\ldots,\kappa _r)$ with $\kappa _i=rnm-n-m+1$ for all $i\in I$. In particular, if $k\geqslant rnm$, then $F^{\Gamma }_{J}(\underline {k})\neq \emptyset$ for every $\emptyset \neq J\subseteq I$ and as $\Gamma \subset \Gamma _1\times \cdots \times \Gamma _r$, if $k\not \in \langle n,m\rangle$, then $F^{\Gamma }_{J}(\underline {k})=\emptyset$.

The next lemma characterizes the non-empty fibers of $\underline {k}\in \mathbb {N}^r$ with respect to $\Gamma$ for $k< rnm$.

Lemma 6.6 Given $k\in \langle n,m\rangle$ with $cnm\leqslant k<(c+1)nm$ and $0\leqslant c\leqslant r-1$ we have

\[ F^{\Gamma}_{J}(\underline{k})\neq\emptyset \Leftrightarrow \begin{cases} \sharp J\geqslant r-c & \mbox{if} \ k-cnm\in\langle n,m\rangle, \\ \sharp J\geqslant r-(c-1) & \mbox{if} \ k-cnm\not\in\langle n,m\rangle. \end{cases} \]

Proof. Remark that $F^{\Gamma }_{J}(\underline {k})\neq \emptyset$ if and only if there exists $\gamma =(\gamma _1,\ldots,\gamma _r)\in \Gamma$ with $\gamma _j=k$ for every $j\in J$ and $\gamma _j>k$ for every $j\in I\setminus J$. By (1), $\gamma \in \Gamma$ can be expressed as

(18)\begin{equation} \gamma=\inf\bigg\{\sum_{s=1}^{r+2}\alpha_{1,s}v_s,\ldots ,\sum_{s=1}^{r+2}\alpha_{r,s}v_s \bigg\}, \end{equation}

where $v_s=(v_{s1},\ldots,v_{sr})$ is given in (17) for $1\leqslant s\leqslant r+2$.

For $j\in J$ there exists $l\in I$ such that

\[ \gamma_j=k=\sum_{s=1}^{r+2}\alpha_{l,s}v_{sj}=\alpha_{l,1}n+\alpha_{l,2}m+\sum_{s=3}^{r+2}\alpha_{l,s}nm \]

with $\alpha _{l,j+2}=0$. In particular, $\sharp J\geqslant \sharp \{i\in I;\ \alpha _{l,i+2}=0\}=r-\sharp \{i\in I;\ \alpha _{l,i+2}\neq 0\}.$

If $k-cnm\not \in \langle n,m\rangle$, then $\sum _{s=3}^{r+2}\alpha _{l,s}\leqslant c-1$ and $\sharp J\geqslant r-(c-1)$. If $k-cnm\in \langle n,m\rangle$, then $\sum _{s=3}^{r+2}\alpha _{l,s}\leqslant c$ and $\sharp J\geqslant r-c$.

On the other hand, if $k-cnm=\alpha _1n+\alpha _2m\in \langle n,m\rangle$, considering any non-negative integers $\alpha _3,\ldots,\alpha _{r+2}$ such that $c=\sum _{s=3}^{r+2}\alpha _s$ we get $\sharp \{i\in J;\ \alpha _{i+2}\neq 0\}\leqslant c$, $\sum _{s=1}^{r+2}\alpha _sv_s\in F_{J}^{\Gamma }(\underline {k})$ for $J=\{i\in I;\ \alpha _{i+2}=0\}$ and $\sharp J\geqslant r-c$. If $k-cnm=g\not \in \langle n,m\rangle$, then $k=g+nm+(c-1)nm=\alpha _1n+\alpha _2m+(c-1)nm$ and considering $c-1=\sum _{s=3}^{r+2}\alpha _s$ we have $\sum _{s=1}^{r+2}\alpha _sv_s\in F_{J}^{\Gamma }(\underline {k})$ for $J=\{i\in I;\ \alpha _{i+2}=0\}$ and $\sharp J\geqslant r-(c-1)$.

Given $k>m$ we have that $\underline {k}\in \nu (\mathcal {M}^2)\subset \Lambda _{\tilde {\mathcal {A}_1}}$. Thus, considering $F_J(\underline {k})$ the $J$-fiber of $\underline {k}$ with respect to $\Lambda _{\tilde {\mathcal {A}_1}}$ we get $F^{\Gamma }_{J}(\underline {k+n})-\underline {n}\subset F_{J}(\underline {k})$ and, by the previous lemma, we obtain

\[ F^{\Gamma}_{J}(\underline{k+n})-\underline{n}\neq \emptyset\Leftrightarrow \begin{cases} \sharp J\geqslant r-c & \mbox{if} \ k-cnm\in\langle n,m\rangle -n,\\ \sharp J\geqslant r-(c-1) & \mbox{if} \ k-cnm\not\in\langle n,m\rangle -n. \end{cases} \]

For each $k>m$ with $cnm-n\leqslant k<(c+1)nm-n$ taking

\[ L=\{1,\ldots ,c+1\},\quad J_i=\{i,c+2, c+3,\ldots ,r\}\quad \mbox{for}\ i\in L\ \mbox{if}\ k-cnm\in\langle n,m\rangle -n; \]

\[ L=\{1,\ldots ,c\},\quad J_i=\{i,c+1, c+2,\ldots ,r\}\quad \mbox{for}\ i\in L\ \mbox{if}\ k-cnm\not\in\langle n,m\rangle -n\]

in Proposition 5.3 and Corollary 5.5 we obtain the following result.

Proposition 6.7 Any curve with value semiring generated by (17) is analytically equivalent to a curve defined by a multigerm $\varphi =[\varphi _1,\ldots,\varphi _r]$ with

\[ \varphi_1=\Bigg(t_1^n,a_{1m}t_1^m+\sum_{\substack{j>m\\ j\not\in\langle n,m\rangle-n}}^{nm-n-1}a_{1j}t_1^j\Bigg),\quad \varphi_i=\Bigg(t_i^n,\sum_{j= m}^{(i-1)nm-n-1}a_{ij}t_i^j+\sum_{\substack{j\geqslant (i-1)nm-n\\ j-(i-1)nm\not\in\langle n,m\rangle-n}}^{inm-n-1}a_{ij}t_i^j\Bigg), \]

for $2\leqslant i\leqslant r$.

As in this topological class the value semiring $\Gamma$ is totally determined by the $m$-jet of the multigerm $\varphi$, we can proceed with the study of $\Lambda _{\tilde {\mathcal {A}_1}}$ for the generic case and consequently we obtain information about the generic component of the moduli space $\mathbb {M}_r(n,m)$ of all plane curves with $\Gamma$ generated by (17) considering the multigerm $\varphi =[\varphi _1,\ldots,\varphi _r]$ with $\varphi _i=(t_i^n,\sum _{j\geqslant m}a_{ij}t_i^j)$ where $a_{ij}\in \mathbb {C}$ are generic, $0\neq a_{im}^n\neq a_{lm}^n\neq 0$ for every $i,l\in I$ and $i\neq l$. The generic component for the moduli space in this topological class was also considered in [GP16] by other methods.

Similarly to the case presented in the last subsection, the generality hypothesis implies that $L_k=\emptyset$, i.e. $e(k)=0$ or $L_k=\{1,\ldots,e(k)\}$ with $e(k)\leqslant r$ for every $k> m$ and, by Remark 5.4, $e(k)$ is the maximum integer such that there exists $\rm {w}\in \mathcal {I}_{\mathcal {G}}$ with $j^k({\rm w})=(b_{1k}t_1^k,\ldots,b_{rk}t_r^{k})$ for any $b_{ik}\in \mathbb {C}$ and $1\leqslant i\leqslant e(k)$.

In this context we have the following proposition.

Proposition 6.8 The dimension of the generic component of the moduli space $\mathbb {M}_r(n,m)$ is zero for $r=1$, $(n,m)\in \{(2,m),(3,4),(3,5)\}$, and

\[ \dim_{\mathbb{C}}\mathbb{M}_r(n,m)=r-2+\sum_{k> m}(r-e(k)) \]

for the other cases.

Proof. Zariski [Zar06] proved that for irreducible plane curves in the topological class determined by $\Gamma \cap \mathbb {N}=\langle n,m\rangle$ with $(n,m)\in \{(2,m),(3,4),(3,5)\}$ the moduli space is a single point.

By Theorem 5.10, we can normalize the coefficient $a_{1m}$ in $\varphi _1$ and, thus, the number of parameters in the $\mathcal {A}$-normal form of $\varphi$ is $r-1+\sum _{k>m}\sharp I\setminus L_k=r-1+\sum _{k> m}(r-e(k))$. Moreover, if $\sum _{k> m}(r-e(k))\geqslant 1$ there exist $j=\min \{k>m;\ L_k\neq I\}$ and $l=\min \{i\in I\setminus L_{j}\}$ such that $a_{lj}\neq 0$. In this way, we can normalize such coefficient taking $c\in \mathbb {C}^*$ with $c^{j-m}=a_{lj}$, $\rho _i(t_i)=ct_i$ for all $i\in I$ and $\sigma (X,Y)=(c^nX,c^mY)$, that is, $(\rho _1,\ldots,\rho _r,\sigma )\cdot \varphi =\psi =[\psi _1,\ldots,\psi _r]$ where the coefficient of $t_{l}^{j}$ of $\psi _{l}$ is equal to $1$. In this case, we obtain

\[ \dim_{\mathbb{C}}\mathbb{M}_r(n,m)=r-2+\sum_{k> m}(r-e(k)). \]

We show that, except for the above particular cases considered by Zariski, we have $\sum _{k> m}(r-e(k))\geqslant 1$. More specifically, we show that $(r-e(m+1))+(r-e(m+2))\geqslant 1$.

Let us compute $e(m+1)$ that, as mentioned previously, it is equivalent to evaluate the maximum number of equations $j^{m+1}\big ( {\varphi ^*_i(\omega )}/{nt_i^{n}}\big )=b_{i,m+1}t_{i}^{m+1}$ that admit solution for any $b_{i,m+1}\in \mathbb {C}$ with $0\leqslant i\leqslant e(m+1)$ and $\omega =p\,dX+q\,dY\in \Omega _{\tilde {\mathcal {A}_1}}$, that is, $p\in \langle X,Y\rangle ^2$ and $q\in \langle X^2, Y\rangle$.

Note that it is sufficient to consider $\nu _i\big ( {\varphi ^*_i(\omega )}/{nt_i^n}\big )= m+1$. Thus, if $m\neq n+1$, then $q=0$, and if $m+1\not \equiv 0 \mod n$, then $p=0$. In particular, for $m\neq n+1$ and $m+1\not \equiv 0 \mod n$ we have $e(m+1)=0$.

For $m=n+1$ and $m+1\not \equiv 0\ {\rm mod}\ n$ we get $\omega =\alpha YdY$, $j^{m+1}\big ( {\varphi ^*_i(\omega )}/{nt_i^{n}}\big )= ({m}/{n})\alpha a_{i,m}^2t_i^{m+1}$, and $e(m+1)=1$. Similarly, if $m\neq n+1$ and $m+1\equiv 0\ {\rm mod}\ n$, then $\omega =\sum _{l=2}^{{(m+1)/n}}\alpha _lX^l\,dX$, $j^{m+1}\big ( {\varphi ^*_i(\omega )}/{nt_i^{n}}\big )=\sum _{l=2}^{{(m+1)/n}}\alpha _lt_i^{ln}$ and $e(m+1)=1$.

If $m=n+1$ and $m+1\equiv 0\ {\rm mod}\ n$, then $n=2, m=3$ and we have $\omega =\alpha _1 X^2\,dX+\alpha _2Y\,dY$, that is, $j^4\big ( {\varphi ^*_i(\omega )}/{2t_i^{2}}\big )=\big ( \alpha _1+\frac {3}{2}a_{i,3}^2\alpha _2\big )t_i^4$. Thus, we get $e(m+1)=\min \{2,r\}$.

Now, we compute $e(m+2)$ for the cases $e(m+1)>0$. As before it is sufficient to consider $\omega =p\,dX+q\,dY\in \Omega _{\tilde {\mathcal {A}_1}}$ with $\nu _i(p)\leqslant m+2$ and $\nu _i(q)\leqslant n+2$.

In the case $m=n+1$ and $m+1\not \equiv 0\ {\rm mod}\ n$, if $n>3$ we have $\omega =\alpha Y\,dY$,

\[ j^{m+2}\bigg(\frac{\varphi^*_i(\omega)}{nt_i^{n}}\bigg)=\frac{m}{n}\alpha a^2_{i,m}t_i^{m+1}+\bigg( \frac{2m+1}{n}\bigg)\alpha a_{i,m}a_{i,m+1}t_i^{m+2} \]

and $e(m+2)=0$. If $n=3$ and, thus, $m=4$, we have $\omega =\alpha _1 X^2\,dX+\alpha _2Y\,dY$,

\[ j^{6}\bigg( \frac{\varphi^*_i(\omega)}{3t_i^{3}}\bigg)=\frac{4}{3}\alpha_2 a^2_{i,4}t_i^{5}+\big( \alpha_1+3\alpha_2a_{i,4}a_{i,5}\big)t_i^{6}\quad \mbox{and}\quad e(m+2)=1. \]

When $m\neq n+1$ and $m+1\equiv 0\ {\rm mod}\ n$ we have the possibilities: if $m\neq n+2$ and $n\neq 2$ then $\omega =\sum _{l=2}^{{(m+1)/n}}\alpha _lX^l\,dX$, $j^{m+2}\big ( {\varphi ^*_i(\omega )}/{nt_i^{n}}\big )=\sum _{l=2}^{{(m+1)/n}}\alpha _lt_i^{ln}$, thus $e(m+2)=0$. If $m\neq n+2$ and $n=2$, then $\omega =\big (\sum _{l=2}^{{(m+1)/2}}\alpha _lX^l+\alpha _1 XY\big )\,dX+\alpha _0X^2\,dY$,

\[ j^{m+2}\bigg( \frac{\varphi^*_i(\omega)}{2t_i^{2}}\bigg)=\sum_{l=2}^{{(m+1)/2}}\alpha_lt_i^{2l}+\bigg( \alpha_1a_{i,m}+\alpha_0\frac{m}{2}a_{i,m}\bigg)t_i^{m+2} \]

and $e(m+2)=1$. If $m=n+2$, then $n=3$, $m=5$, $\omega =\alpha _1 X^2\,dX+\alpha _2Y\,dY$, and

\[ j^{7}\bigg( \frac{\varphi^*_i(\omega)}{3t_i^{3}}\bigg)=\alpha_1t^6_i+\frac{5}{3}\alpha_2a_{i,5}^2t_i^{7}, \]

consequently $e(m+2)=1$.

Finally, if $n=2$ and $m=3$, then $\omega =(\alpha _1X^2+\alpha _2 XY)\,dX+(\alpha _3 Y+\alpha _4 X^2)\,dY$,

\[ j^{5}\bigg( \frac{\varphi^*_i(\omega)}{2t_i^{2}}\bigg)=\bigg(\alpha_1+\frac{3}{2}\alpha_3 a^2_{i,3}\bigg)t^4_i+\bigg(\alpha_2+\frac{3}{2}\alpha_4+\frac{7}{2}a_{i,4}\alpha_3\bigg)a_{i,3}t_i^{5}\quad \mbox{and}\quad e(m+2)=1. \]

These computations give us that, except to the cases $r=1$ and $(n,m)\in \{(2,m),(3,4),(3,5)\}$, we have that $(r-e(m+1))+(r-e(m+2))\geqslant 1$.

To illustrate the above result in a specific case, in what follows $n=2$, $m=3$, and $r\geqslant 2$.

We compute $e(k)$ for any $k>m=3$ which allows us to exhibit normal forms to the generic case and an explicit formula for $\dim _{\mathbb {C}}\mathbb {M}_r(2,3)$.

Our strategy is the same as that of the previous proposition, that is, to evaluate the maximum number of equations $j^{k}\big ( {\varphi ^*_i(\omega )}/{2t_i^{2}}\big )=b_{i,k}t^{k}$ that admit solution for any $b_{i,k}\in \mathbb {C}$ with $\omega =p\,dX+q\,dY\in \Omega _{\tilde {\mathcal {A}_1}}$, that is, $p\in \langle X,Y\rangle ^2$ and $q\in \langle X^2, Y\rangle$. To achieve this goal, for every $h\geqslant 2$ we consider

\[ \omega_{2h}=\Bigg( \sum_{l=0}^{[ {h}/{3}]}\alpha_{h-3l,2l}X^{h-3l}Y^{2l}\Bigg)\,dX+ \Bigg( \sum_{l=0}^{[ {(h-2)/3}]}\beta_{h-2-3l,2l+1}X^{h-2-3l}Y^{2l+1}\Bigg)\,dY, \]

\[ \omega_{2h+1}=\Bigg( \sum_{l=0}^{[ {(h-1)/3}]}\alpha_{h-1-3l,2l+1}X^{h-1-3l}Y^{2l+1}\Bigg)\,dX+\Bigg( \sum_{l=0}^{[ {h}/{3}]}\beta_{h-3l,2l}X^{h-3l}Y^{2l}\Bigg)\,dY. \]

Any $\omega \in \Omega _{\tilde {\mathcal {A}_1}}$ can be uniquely expressed as $\omega =\sum _{l\geqslant 4}\omega _l$. As

\[ \nu_i\bigg(\frac{\varphi_i^*(X^{h-3l}Y^{2l}\,dX)}{2t_i^2} \bigg)=\nu_i \bigg(\frac{\varphi_i^*(X^{h-2-3l}Y^{2l+1}\,dY)}{2t_i^2} \bigg)=2h \]

and

\[ \nu_i\bigg(\frac{\varphi_i^*(X^{h-1-3l}Y^{2l+1}\,dX)}{2t_i^2} \bigg)=\nu_i\bigg(\frac{\varphi_i^*(X^{h-3l}Y^{2l}\,dY)}{2t_i^2} \bigg)=2h+1 \]

for any $i\in I$, in order to evaluate $e(k)$ it is sufficient to consider $\omega =\sum _{l=4}^{k}\omega _l$.

Proposition 6.9 If $\varphi =[\varphi _1,\ldots,\varphi _r]$ is a generic multigerm admitting value semiring $\Gamma$ with generators as in (17), $n=2$, $m=3$, and $r\geqslant 2$, then $e(4)=2$, $e(5)=1$, and for $k\geqslant 6$:

\[ e(k)=\min\bigg\{2\bigg[\frac{k}{6}\bigg]+1,r\bigg\}\quad \mbox{if}\ k\not\equiv 4 \ {\rm mod}\ 6,\quad e(k)=\min\bigg\{2\bigg[\frac{k}{6}\bigg]+3,r\bigg\}\quad \mbox{if}\ k\equiv 4\ {\rm mod}\ 6, \]

and $e( 6[ {(r-2)/2}]+5)=e(3r-1)=r$ if $r$ is even.

Proof. We remark that, by the last proposition, $e(4)=2$ and $e(5)=1$.

First, we consider the system $j^{c+1}\big ( {\varphi ^*_i(\omega _c+\omega _{c+1})}/{2t_i^2} \big )=0$ for every $i\in I$ and $c\geqslant 5$.

Using $\varphi _i=(t_i^2,\sum _{j\geqslant 3}a_{ij}t_i^j)$ we obtain

\begin{align*} j^{2h+1}\Bigg(\frac{\varphi^*_i(\omega_{2h})}{2t_i^2}\Bigg)&= \Bigg(\sum_{l=0}^{[{h}/{3}]}a_{i3}^{2l}\alpha_{h-3l,2l}+ \sum_{l=0}^{[{(h-2)/3}]}\frac{3}{2}a_{i3}^{2l+2}\beta_{h-2-3l,2l+1}\Bigg)t_i^{2h}\\ &\quad + \Bigg(\sum_{l=1}^{[{h}/{3}]}2la_{i3}^{2l-1}a_{i4}\alpha_{h-3l,2l}+ \sum_{l=0}^{[ {(h-2)/3}]}\frac{6l+7}{2}a_{i3}^{2l+1}a_{i4}\beta_{h-2-3l,2l+1}\Bigg)t_i^{2h+1}; \end{align*}

\begin{align*} j^{2h+2}\Bigg(\frac{\varphi^*_i(\omega_{2h+1})}{2t_i^2}\Bigg)&= \Bigg(\sum_{l=0}^{[{(h-1)/3}]}a_{i3}^{2l+1}\alpha_{h-1-3l,2l+1}+ \sum_{l=0}^{[{h}/{3}]}\frac{3}{2}a_{i3}^{2l+1}\beta_{h-3l,2l}\Bigg)t_i^{2h+1}\\ &\quad +\Bigg(\sum_{l=0}^{[{(h-1)/3}]}(2l+1)a_{i3}^{2l}a_{i4}\alpha_{h-1-3l,2l+1}+ \sum_{l=0}^{[{h}/{3}]} (3l+2)a_{i3}^{2l}a_{i4}\beta_{h-3l,2l}\Bigg)t_i^{2h+2}. \end{align*}

If $c=2h$, then to vanish the coefficient of $t_i^{c}$ in $j^{c+1}\big ( {\varphi ^*_i(\omega _{c}+\omega _{c+1})}/{2t_i^2} \big )$ for every $i\in I$ we obtain the system $N\cdot W_p^t=0$ with

\[ N=\left(\begin{array}{ccccc} 1 & a_{13}^2 & a_{13}^4 & \cdots & a_{13}^{2[{h}/{3}]+p} \\ \vdots & \vdots & \vdots & & \vdots \\ 1 & a_{r3}^2 & a_{r3}^4 & \cdots & a_{r3}^{2[{h}/{3}]+p} \\ \end{array}\right)\quad \mbox{with}\ \begin{cases} p=0 & \mbox{if}\ c\equiv0, 2\ {\rm mod}\ 6,\\ p=2 & \mbox{if}\ c\equiv4\ {\rm mod}\ 6, \end{cases} \]

\[ W_0=\bigg(\alpha_{h,0}, \alpha_{h-3,2}+\frac{3}{2}\beta_{h-2,1},\ldots , \alpha_{h-3[{h}/{3}],2[{h}/{3}]}+\frac{3}{2}\beta_{h-3[{h}/{3}]+1,2[ {h}/{3}]-1}\bigg), \]

and $W_2=\big (W_0,\frac {3}{2}\beta _{h-2-3[ {h}/{3}],2[ {h}/{3}]+1}\big )$, where $W_p^t$ denotes the transpose of $W_p$.

Considering the solution of $N\cdot W_p^t=0$, the associated system to vanish the coefficient of $t_i^{c+1}$ in $j^{c+1}\big ( {\varphi _i^*(\omega _c+\omega _{c+1})}/{2t_i^2}\big )$ for every $i\in I$ is $M\cdot Z^t = 0$ where

(19)\begin{equation} M=\left(\begin{array}{cccccccc} a_{14}a_{13} & a_{14}a_{13}^3 & \cdots & a_{14}a_{13}^{2[{h}/{3}]-1} & a_{13} & a_{13}^3 & \cdots & a_{13}^{2[{h}/{3}]+1} \\ \vdots & \vdots & & \vdots & \vdots & \vdots & & \vdots \\ a_{r4}a_{r3} & a_{r4}a_{r3}^3 & \cdots & a_{r4}a_{r3}^{2[ {h}/{3}]-1} & a_{r3} & a_{r3}^3 & \cdots & a_{r3}^{2[{h}/{3}]+1} \end{array}\right), \end{equation}

\[ Z=\bigg(\frac{1}{2}\beta_{h-3l+1,2l-1};\ 1\leqslant l\leqslant \bigg[ \frac{h}{3}\bigg], \alpha_{h-1-3l,2l+1}+\frac{3}{2}\beta_{h-3l,2l};\ 0\leqslant l\leqslant \bigg[\frac{h}{3}\bigg]-1, \frac{3}{2}\beta_{h-3[{h}/{3}],2[{h}/{3}]} \bigg) \]

if $c\equiv 0\ {\rm mod}\ 6$ or

\[ Z=\bigg(\frac{1}{2}\beta_{h-3l+1,2l-1};\ 1\leqslant l\leqslant \bigg[ \frac{h}{3}\bigg], \alpha_{h-1-3l,2l+1}+\frac{3}{2}\beta_{h-3l,2l};\ 0\leqslant l\leqslant \bigg[ \frac{h}{3}\bigg]\bigg)\quad \mbox{if}\ c\equiv2, 4\ {\rm mod}\ 6. \]

Note that the rank of $M$ is $\min \{2[ {h}/{3}]+1,r\}=\min \{2[ {c}/{6}]+1,r\}$. If $2[ {c}/{6}]+1\leqslant r$, then the solution of the above system implies that $\omega _{c}=0$.

For $c=2h+1$ we vanish the coefficient of $t_i^{c}$ in $j^{c+1}\big ( {\varphi ^*_i(\omega _{c}+\omega _{c+1})}/{2t_i^2} \big )$ for every $i\in I$ solving the system $N\cdot W^t=0$ where

\[ N=\left(\begin{array}{cccc} a_{13} & a_{13}^3 & \cdots & a_{13}^{2[{h}/{3}]+1} \\ \vdots & \vdots & & \vdots \\ a_{r3} & a_{r3}^3 & \cdots & a_{r3}^{2[{h}/{3}]+1} \end{array}\right), \]

\[ W=\bigg(\alpha_{h-1-3l,2l+1}+\frac{3}{2}\beta_{h-3l,2l};\ 0\leqslant l\leqslant \bigg[\frac{h}{3}\bigg]\bigg)\quad {\rm if} \ c\equiv3, 5\ {\rm mod}\ 6\]

or

\[ W=\bigg(\alpha_{h-1-3l,2l+1}+\frac{3}{2}\beta_{h-3l,2l};\ 0\leqslant l\leqslant \bigg[\frac{h}{3}\bigg]-1, \frac{3}{2}\beta_{h-3[{h}/{3}],2[{h}/{3}]}\bigg)\quad {\rm if}\ c\equiv1\ {\rm mod}\ 6. \]

Taking the solution of the system $N\cdot W^t=0$ in order to vanish the coefficient of $t_i^{c+1}$ in $j^{c+1}\big ( {\varphi ^*_i(\omega _c+\omega _{c+1})}/{2t_i^2}\big )$ for every $i\in I$ we obtain the system $M\cdot Z^t = 0$ where

\[ M=\left(\begin{array}{cccccccc} a_{14} & a_{14}a_{13}^2 & \cdots & a_{14}a_{13}^{2[{h}/{3}]-2} & 1 & a_{13}^2 & \cdots & a_{13}^{2[{h}/{3}]} \\ \vdots & \vdots & & \vdots & \vdots & \vdots & & \vdots \\ a_{r4} & a_{r4}a_{r3}^2 & \cdots & a_{r4}a_{r3}^{2[ {h}/{3}]-2} & 1 & a_{r3}^2 & \cdots & a_{r3}^{2[{h}/{3}]} \end{array}\right)\quad \mbox{if}\ c\equiv1\ {\rm mod}\ 6, \]

\[ M=\left(\begin{array}{cccccccc} a_{14} & a_{14}a_{13}^2 & \cdots & a_{14}a_{13}^{2[{h}/{3}]} & 1 & a_{13}^2 & \cdots & a_{13}^{2[{h}/{3}]+2} \\ \vdots & \vdots & & \vdots & \vdots & \vdots & & \vdots \\ a_{r4} & a_{r4}a_{r3}^2 & \cdots & a_{r4}a_{r3}^{2[ {h}/{3}]} & 1 & a_{r3}^2 & \cdots & a_{r3}^{2[{h}/{3}]+2} \end{array}\right)\quad \mbox{if}\ c\equiv3, 5\ {\rm mod}\ 6, \]

\begin{align*} Z&=\bigg(\frac{1}{2}\beta_{h-3l,2l}; 0\leqslant l\leqslant \bigg[\frac{h}{3}\bigg]-1, \alpha_{h+1,0},\alpha_{h+1-3l,2l}+\frac{3}{2}\beta_{h+2-3l,2l-1}; \\ &\qquad\ 1\leqslant l \leqslant \bigg[ \frac{h}{3}\bigg]\bigg)\quad \mbox{if}\ c\equiv1\ {\rm mod}\ 6, \end{align*}

\begin{align*} Z&=\bigg(\frac{1}{2}\beta_{h-3l,2l}; 0\leqslant l\leqslant \bigg[ \frac{h}{3}\bigg], \alpha_{h+1,0},\alpha_{h+1-3l,2l}+\frac{3}{2}\beta_{h+2-3l,2l-1};\\ &\qquad\ 1\leqslant l \leqslant \bigg[ \frac{h}{3}\bigg],\frac{3}{2}\beta_{h-1-3[{h}/{3}],2[{h}/{3}]+1} \bigg)\quad \mbox{if}\ c\equiv 3\ {\rm mod}\ 6\end{align*}

and

\[ Z\!=\!\bigg(\frac{1}{2}\beta_{h-3l,2l}; 0\leqslant l\!\leqslant\! \bigg[ \frac{h}{3}\bigg], \alpha_{h+1,0},\alpha_{h-2-3l,2l+2}\!+\!\frac{3}{2}\beta_{h-1-3l,2l+1}; 0\leqslant l\leqslant \bigg[ \frac{h}{3}\bigg]\bigg)\quad \mbox{if}\ c\equiv5\ {\rm mod}\ 6. \]

Remark that the rank of $M$ is $\min \{2[ {h}/{3}]+1,r\}=\min \{2[ {c}/{6}]+1,r\}$ if $c\equiv 1\ {\rm mod}\ 6$ or $\min \{2[ {h}/{3}]+3,r\}=\min \{2[ {c}/{6}]+3,r\}$ if $c\equiv 3, 5\ {\rm mod}\ 6$. In particular, $\omega _{c}=0$ if $2[ {c}/{6}]+1\leqslant r$ for $c\equiv 1\ {\rm mod}\ 6$ or $2[ {c}/{6}]+3\leqslant r$ for $c\equiv 3, 5\ {\rm mod}\ 6$.

In this way, to study the system $j^k\big ( {\varphi ^*_i(\omega )}/{2t_i^2}\big )=b_{i,k}t_i^k$ with $b_{i,k}\in \mathbb {C}$, $i\in I$ for $r\leqslant 2[{(k-1)/6}]+1$ or $2[{(k-1)/6}]+3\leqslant r$ it is sufficient to consider $\omega =\omega _{k-1}+\omega _k\in \Omega _{\tilde {\mathcal {A}_1}}$. For this situation the cases are the same as those considered above, where we obtained the system $M\cdot Z^t=(b_{1,k},\ldots,b_{r,k})^t$ with $M$ and $Z$ described previously taking $k=c+1$. Hence, $e(k)=rank(M)$ for all $k\geqslant 6$.

As $[ {k}/{6}]=[ {c}/{6}]$ for $c\not \equiv 5\ {\rm mod}\ 6$ and $[ {k}/{6}]=[ {c}/{6}]+1$ for $c\equiv 5\ {\rm mod}\ 6$, we get

\[ e(k)=\min\bigg\{2\bigg[\frac{k}{6}\bigg]+1,r\bigg\}\ \mbox{if}\ k\not\equiv 4 \ {\rm mod}\ 6 \quad \mbox{and}\quad e(k)=\min\bigg\{2\bigg[\frac{k}{6}\bigg]+3,r\bigg\}\ \mbox{if}\ k\equiv 4\ {\rm mod}\ 6. \]

We obtain the same conclusion if $r=2[{(k-1)/6}]+2$ and $k\not \equiv 5\ {\rm mod}\ 6$.

For $r=2[{(k-1)/6}]+2$ and $k\equiv 5\ {\rm mod}\ 6$, that is, $k=3r-1=6[ {(r-2)/2}]+5$, the condition $j^{k-1}\big ( {\varphi _i^*(\omega _{k-2}+\omega _{k-1})}/{2t_i^2}\big )=0$ for every $i\in I$ produces a system $M\cdot Z^t=0$ with ${\rm rank}(M)=\min \{2[{(k-2)/6}]+3,r\}=\min \{r+1,r\}=r$. In this way, we have an extra variable for the system $j^{k}\big ( {\varphi _i^*(\omega _{k-2}+\omega _{k-1}+\omega _k)}/{2t_i^2}\big )=b_{i,k}t_i^k$ that can be expressed as $M_1\cdot Z_1^t=(b_{1,k},\ldots,b_{r,k})^t$ where $M_1$ is the matrix $M$ as in (19) with an extra column depending on $a_{i3}, a_{i4}$ and $a_{i5}$. This allows us to conclude that

\[ e\bigg( 6\bigg[ \frac{r-2}{2}\bigg]+5\bigg)=e(3r-1)={\rm rank}(M_1)=\min\bigg\{2\bigg[\frac{k-1}{6}\bigg]+1+1,r\bigg\}=r. \]

As a consequence of Propositions 6.8 and 6.9 we obtain an explicit formula for the dimension of the generic component of $\mathbb {M}_r(2,3)$.

Corollary 6.10 For plane curves that admit value semiring generated by (17) with $n=2$, $m=3$, and $r$ branches, the dimension of the generic component $\mathbb {M}_r(2,3)$ of the moduli space is

\[ \dim_{\mathbb{C}}\mathbb{M}_r(2,3)=\begin{cases} \dfrac{(r-1)(3r-5)}{2} & \mbox{if}\ r\ \mbox{is odd,} \\ \dfrac{(r-1)(3r-5)+1}{2} & \mbox{if}\ r\ \mbox{is even.} \end{cases} \]

Proof. The case $r=1$ is immediate.

By Proposition 6.8 and the above result we have that

(20) \begin{align} \dim_{\mathbb{C}}\mathbb{M}_r(2,3)&=r-2+\displaystyle{\sum_{k\geqslant 4}(r-e(k))}=3r-5+\displaystyle{\sum_{k\geqslant 6}(r-e(k))}\nonumber\\ &=3r-5+5\hspace{-0.3cm}\displaystyle{\sum_{\substack{k\geqslant 6\\ k\not\equiv 4\ {\rm mod}\ 6}}\hspace{-0.4cm}(r-e(k))}+\hspace{-0.4cm}\displaystyle{\sum_{\substack{k\geqslant 6\\ k\equiv 4\ {\rm mod}\ 6}}\hspace{-0.4cm}(r-e(k))}. \end{align}

We remark that for $r=2$ we have $e(k)=2$ for every $k\geqslant 6$, then $\dim _{\mathbb {C}}\mathbb {M}_2(2,3)=1$.

If $k\equiv 4\ {\rm mod}\ 6$, we have $e(k)=\min \{2[ {k}/{6}]+3,r\}$ and

(21) \begin{equation} \displaystyle{\sum_{\substack{k\geqslant 6\\ k\equiv 4\ {\rm mod}\ 6}}\hspace{-0.3cm}(r-e(k))}=\sum_{i=1}^{[{(r-3)/2}]}(r-(2i+3))=\bigg[ \frac{r-3}{2}\bigg]\bigg( r-4- \bigg[\frac{r-3}{2}\bigg]\bigg). \end{equation}

If $k\not \equiv 4\ {\rm mod}\ 6$, then $e(k)=\min \{2[ {k}/{6}]+1,r\}$ and $e\big (6[{(r-2)/2}]+5\big )=e(3r-1)=r$ if $r$ is even. Thus,

(22) \begin{align} &\displaystyle{\sum_{\substack{k\geqslant 6\\ k\not\equiv 4\ {\rm mod}\ 6}}\hspace{-0.3cm}(r-e(k))}\nonumber\\ &\quad =\begin{cases} \sum\limits_{i=1}^{[ {(r-4)/2}]}(r-(2i+1))=\bigg[\dfrac{r-1}{2}\bigg]\bigg( r-2- \bigg[\dfrac{r-1}{2}\bigg]\bigg)-1 & \mbox{if}\ k\equiv 5\ {\rm mod}\ 6\ \mbox{and}\ r\ \mbox{even},\\ \sum\limits_{i=1}^{[{(r-1)/2}]}(r-(2i+1))=\bigg[\dfrac{r-1}{2}\bigg]\bigg( r-2- \bigg[\dfrac{r-1}{2}\bigg]\bigg) & \mbox{otherwise}. \end{cases} \end{align}

For $r\geqslant 3$, considering (21) and (22) in (20) the result follows.