2.1 Singularities of map-germs

We deal with germs $F\colon (M,p)\to (N,F(p))$ of holomorphic maps between complex manifolds. Taking coordinate charts, this is the same as dealing with map-germs $(\mathbb {C}^n,0)\to (\mathbb {C}^p,0)$.

Let ${\mathcal {O}}_n$ be the local ring of germs of holomorphic functions $(\mathbb {C}^n,0 ) \to \mathbb {C}$ and ${\mathfrak m}_n$ its maximal ideal (which is the subset of germs that vanish at the origin). Denote by $\mathcal {O}(n,p)$ the $\mathcal {O}_n$-module of holomorphic map-germs $(\mathbb {C}^n,0)\to \mathbb {C}^p$, so $\mathcal {O}(n,p)=\bigoplus _p\mathcal {O}_n$.

Let ${\mathcal {R}}$ (resp. ${\mathcal {L}}$) be the group of bi-holomorphic germs $(\mathbb {C}^n,0)\to (\mathbb {C}^n,0)$ (resp. $(\mathbb {C}^p,0)\to (\mathbb {C}^p,0)$). The group ${\mathcal {A}}={\mathcal {R}}\times {\mathcal {L}}$ of right-left equivalence acts on ${\mathfrak {m}}_n\cdot {\mathcal {O}}(n,p)$ by $(h_1,h_2)\cdot G=h_2\circ G\circ h_1^{-1}$. Two germs $H,G$ are said to be $\mathcal {A}$-equivalent, and write $H\sim _{\mathcal {A}} G$, if $H=(h_1,h_2)\cdot G$ for some $(h_1,h_2)\in \mathcal {A}$.

The $l$-jet space of map-germs in $\mathfrak {m}_n\cdot \mathcal {O}(n,p)$ is by definition

\[ J^l(n,p)={\mathfrak{m}_n\cdot\mathcal{O}}(n,p)/{\mathfrak{m}}_n^{l+1}\cdot{}{\mathcal{O}}(n,p). \]

Given a germ $G\in \mathfrak {m}_n\cdot {\mathcal {O}}(n,p)$, we identified its $l$-jet $j^lG$ with its Taylor polynomial of degree $l$ at the origin. Let $\mathcal {A}_{l}$ be the subgroup of $\mathcal {A}$ whose elements have $l$-jets the germ of the identity. The group $\mathcal {A}_l$ is a normal subgroup of $\mathcal {A}$. Define $\mathcal {A}^{(l)}=\mathcal {A}/\mathcal {A}_l$. The elements of $\mathcal {A}^{(l)}$ are the $l$-jets of the elements of $\mathcal {A}$. The action of $\mathcal {A}$ on ${\mathfrak {m}}_n\cdot {\mathcal {O}}(n,p)$ induces an action of the jet group $\mathcal {A}^{(l)}$ on $J^l(n,p)$ as follows. For $j^lG\in J^l(n,p)$ and $j^l(h_1,h_2)\in \mathcal {A}^{(l)}$, $j^l(h_1,h_2)\cdot j^rG=j^l((h_1,h_2)\cdot G).$

A germ $G$ is said to be finitely $\mathcal {A}$-determined if there exist an integer $l$ such that $G\sim _{\mathcal {A}}H$ for any $H$ with $j^lH=j^lG$; $j^lG$ is then said to be a sufficient jet of $G$. The germ $G$ is then said to be $l$-$\mathcal {A}$-determined. The least $l$ satisfying this property is called the degree of determinacy of $G$.

There are classifications of finitely determined map-germs for various pairs $(n,p)$. When $p=1$, there is Arnold's extensive list of the $\mathcal {R}$-classification of germs of functions [Reference Arnold, Guseĭn-Zade and Varchenko1]. For $(n,p)=(2,2)$, classifications were carried out by several authors, the most extensive ones are given in [Reference Goryunov16, Reference Rieger29]. Here we need only the singularities of $\mathcal {A}_e$-codimension $\le \! 2$, which we reproduce in table 9. For $(n,p)=(2,3)$, Mond [Reference Mond22] produced an extensive list of finitely $\mathcal {A}$-determined map-germs. We use in this paper the following singularities from [Reference Mond22]:

\[ \begin{array}{cl} \text{Immersion} & (x,y,0) \\ \text{Cross-cap} & (x,y^2,xy) \\ S_{k} & (x, y^2, y^3 + x^{k+1}y), k\ge 1 \\ B_{k} & (x, y^2, x^2y + y^{2k+1}), k\ge 2\\ C_{3} & (x, y^2, xy^3 + x^{3}y)\\ H_k & (x,xy+y^{3k-1},y^3), k\ge 2\\ X_4 & (x,y^3,x^2y+xy^2+y^{4}) \end{array} \]

The notion of a simple germ is defined in [Reference Arnold, Guseĭn-Zade and Varchenko1] as follows. Let $X$ be a manifold and $\mathcal {G}$ a Lie group acting on $X$. The modality of a point $g\in X$ under the action of $\mathcal {G}$ on $X$ is the least number $m$ such that a sufficiently small neighbourhood of $g$ may be covered by a finite number of $m$-parameter families of orbits. The point $g$ is said to be simple if its modality is $0$, that is, a sufficiently small neighbourhood intersects only a finite number of orbits. The modality of a finitely $\mathcal {A}$-determined map-germ is the modality of a sufficient jet in the jet-space under the action of the jet-group.

We also need the notion of topological equivalence. We say that two germs $H, G\in {\mathfrak {m}}_n\cdot {\mathcal {O}(n,p)}$ are topologically equivalent if $H=h_2\circ G\circ h_1^{-1}$ for some germs of homeomorphisms $h_1$ and $h_2$ of, respectively, the source and target.

2.2 Reflections and $k$-folding maps

In all this paper, we fix the inner product $\langle a,b\rangle =\sum _ia_i\overline {b_i}$ in $\mathbb {C}^3$.

Let $\pi$ be an element of the affine Grassmannian ${\rm {Graff}}(2,3)$ of planes in $\mathbb {C}^3$. A plane $\pi$ has equation $\left \langle q,v\right \rangle =d$, where $v$ is a fixed non-zero vector orthogonal to $\pi$ and $d$ is a fixed scalar. However, any non-zero scalar multiple of $(d,v)$ gives an equation of $\pi$, so $\pi$ is identified with the class $\overline {(d,v)}\in \mathbb {C} P^3$ of $(d,v)\in \mathbb {C}^4$.

Let $\pi :\left \langle q,v\right \rangle =d$ be a plane in $\mathbb {C}^3$. The orthogonal projection of a point $p\in \mathbb {C}^3$ to $\pi$ along the vector $v$ is the point $q= p+\lambda v\in \pi$ with $\lambda =(d-\left \langle p,v\right \rangle )/\left \langle v,v\right \rangle$.

Consider the map $\omega _k^{\pi }\colon \mathbb {C}^3\to \mathbb {C}^3$ given by

\[ \omega_k^{\pi}(p)=q+\lambda^k v=p+\lambda v+\lambda^k v. \]

If we take $(d',v')=(\alpha d,\alpha v)$, $\alpha \in \mathbb {C}\setminus 0$, as another representative of $\pi =\overline {(d,v)}$, then

\begin{align*} p+\lambda' v'+\lambda'^k v'& =p+\frac{\alpha(d-\left\langle p,v\right\rangle))} {\alpha^2\left\langle v,v\right\rangle} \alpha v+ \frac{\alpha^k(d-\left\langle p,v\right\rangle)^k)}{\alpha^{2k}\left\langle v,v\right\rangle}\alpha v\\ & =p+\lambda v+\lambda^k \alpha^{1-k} v\\ & =q+\lambda^k \alpha^{1-k} v. \end{align*}

Clearly, the map $\omega _k^{\pi }$ depends on the points on the line $(\alpha d, \alpha v)\in \mathbb {C}^4$ and not merely on the class of the line $\overline {(d,v)}\in \mathbb {C}P^3$. However, all these maps are $\mathcal {L}$-equivalent: the bi-holomorphic map $q-\lambda v\mapsto q-\lambda \alpha ^{\frac {1-k}{k}}v$ composed (on the left) with the map $\omega _k^{\pi }$ with $\pi$ represented by $(d,v)$ gives the map $\omega _k^{\pi }$ with $\pi$ represented by $(\alpha d, \alpha v)$. Therefore, the $\mathcal {L}$-class of $\omega _k^{\pi }$ depends only on $\pi$.

A $k$-fold may be viewed as generalization of the Whitney fold $(x,y^2,z)\mapsto (x,y^2,z)$. While the Whitney fold folds the space along the plane $\{y=0\}$ and identifies the points $(x,y,z)$ and $(x,-y,z)$, the Whitney fold of order $k$ with respect to a plane $\pi$ represented by $(d,v)$ is a generically a $k$-to-1 branched cover, ramified along $\pi$, and identifies $k$-tuples of points $q-\lambda v, q-\xi \lambda v, \dots, q-\xi ^{k-1}\lambda v,$ where $\xi ={\rm {e}}^{2\pi {{\rm i}}/k}$ is a primitive $k{\rm {th}}$-root of unity and $q\in \pi$.

The map $\omega _k^{\pi }$ can also be viewed as the quotient map associated to the action of the cyclic group $\mathbb {Z}/k\mathbb {Z}$, regarded as a complex reflection group whose hyperplane arrangement consist of the single plane $\pi$. We regard $\mathbb {Z}/k\mathbb {Z}$ as the group generated by the order $k$ complex reflection $q-\lambda v\mapsto q-\xi \lambda v$. Observe that, even though the plane $\pi$ does not determine $\omega _k^{\pi }$ uniquely (it depends on the choice of a representative of $\pi =\overline {(d,v)}\in \mathbb {C}P^3$), the action of $\mathbb {Z}/k\mathbb {Z}$ on $\mathbb {C}^3$ is determined uniquely by $\pi$.

Given any subset $X\subseteq \mathbb {C}^3$, $\omega _k^{\pi }(X)$ encodes the order $k$ reflectional symmetries of $X$ with respect to $\pi$. See [Reference Peñafort Sanchis17] for a recent work on singular maps related to reflection groups.

3.1 The double-point curve

We start by recalling the definition of the double- and triple-point spaces of a corank one map-germ $F\colon (\mathbb {C}^n,0)\to (\mathbb {C}^{n+1},0)$ from [Reference Marar and Mond19]. Any such germ can be written in a suitable coordinate system in the form $F(x,y)=(x,f_{n}(x,y),f_{n+1}(x,y)),$ with $x=(x_1,\dots,x_{n-1})\in (\mathbb {C}^{n-1},0)$ and $y\in (\mathbb {C},0)$.

Given $h\in \mathcal {O}_n$, the iterated divided differences of $h$ are defined as

\begin{align*} h[x,y,y']& =\dfrac{h(x,y')-h(x,y)}{y'-y}\in \mathcal{O}_{n+1},\\ h[x,y,y',y'']& =\dfrac{h[x,y,y'']-h[x,y,y']}{y''-y'}\in\mathcal{O}_{n+2}. \end{align*}

The multiple point ideals of a map-germ $F$ as above are defined as

\begin{align*} I^2(F)& =\langle f_{n}[x,y,y'],f_{n+1}[x,y,y']\rangle\subseteq \mathcal{O}_{n+1},\\ I^3(F)& =\langle f_{n}[x,y,y'],f_{n}[x,y,y',y''],f_{n+1}[x,y,y'],f_{n+1}[x,y,y',y'']\rangle\subseteq \mathcal{O}_{n+2}. \end{align*}

The double- and triple-point spaces of $F$ are, respectively,

\begin{align*} {\mathcal{D}}^2(F)& =V(I^2(F)),\\ {\mathcal{D}}^3(F)& =V(I^3(F)). \end{align*}

By counting variables and generators, it follows that ${\mathcal {D}}^2(F)$ (resp. ${\mathcal {D}}^3(F)$) is a complete intersection whenever it has dimension $n-1$ (resp. $n-2$).

The double-point space ${\mathcal {D}}^2(F)$, as a subset of $(\mathbb {C}^{n-1}\times \mathbb {C}\times \mathbb {C},0)$, consists of points $(x,y,y')$ such that either $y\neq y'$ and $F(x,y)=F(x,y')$ or $y'=y$ and $F$ is singular at $(x,y)$.

To define the source double-point space, we assume that $F$ is finite. Then, the projection $\pi \colon {\mathcal {D}}^2(F)\to \mathbb {C}^{n-1}\times \mathbb {C}$ given by $(x,y,y')\mapsto (x,y)$ is also finite. As a consequence, the image of $\pi$ can be given a complex structure as the $0$-th Fitting ideal $\mathcal {F}_0(\pi _*\mathcal {O}_{{\mathcal {D}}^2(F)})$ of the push forward module $\pi _*\mathcal {O}_{{\mathcal {D}}^2(F)}$ (see [Reference Mond and Pellikaan26] for details). The source double-point space ${\mathcal {D}}(F)$ is defined as the projection $\pi ({\mathcal {D}}^2(F))$ endowed with this complex space structure, that is,

\[ {\mathcal{D}}(F)=V(\mathcal{F}_0(\pi_*\mathcal{O}_{{\mathcal{D}}^2(F)}))\subseteq \mathbb{C}^n. \]

To compute the source double-point space of a $k$-folding map-germ, we need the following result.

Proof. It is enough to prove the statement for the case $r=2$. We can assume that $X_1$ and $X_2$ have no common irreducible component as topological spaces. Indeed, consider the two subspaces $\mathcal {X}_1=V(h_1-t)$ and $\mathcal {X}_2=V(h_2)$ of $Z\times (\mathbb {C},0)$ and the map $\phi \times \operatorname {Id}\colon Z\times (\mathbb {C},0)\to (\mathbb {C}^{n+1},0)$. The spaces $\mathcal {X}_1$ and $\mathcal {X}_2$ have no common irreducible component. Moreover, if the statement holds for $\phi \times \operatorname {Id}$, then it holds for $\phi$. This is a consequence of the fact that Fitting ideals commute with base change (see lemma 1.2 in [Reference Mond and Pellikaan26]). Now consider the disjoint union $X_1\sqcup X_2\subseteq Z\sqcup Z$ and the commutative diagram

The map $\alpha$ is generically a local isomorphism (i.e. a local isomorphism on a Zariski open and dense subset) because $X_1$ and $X_2$ are assumed to have no common component. Moreover, both $X$ and $X_1\sqcup X_2$ are Cohen–Macaulay spaces, which implies that the ideals $\mathcal {F}_0(\psi _*\mathcal {O}_{X_1\sqcup X_2})$ and $\mathcal {F}_0((\phi \vert _{X})_*\mathcal {O}_{X})$ are principal. Since $\alpha$ is generically a local isomorphism, they are necessarily equal. The statement then follows from the equalities

\begin{align*} \mathcal{F}_0(\psi_*\mathcal{O}_{X_1\sqcup X_2})& =\mathcal{F}_0((\phi\vert_{X_1})_*\mathcal{O}_{X_1}\oplus (\phi\vert_{X_2})_*\mathcal{O}_{X_2}))\\ & =\mathcal{F}_0((\phi\vert_{X_1})_*\mathcal{O}_{X_1})\cdot \mathcal{F}_0((\phi\vert_{X_2})_*\mathcal{O}_{X_2}). \end{align*}

Theorem 3.2 For a $k$-folding map-germ $F_k(x,y)=(x,y^k,f(x,y))$, the double-point space ${\mathcal {D}}$ is the zero locus $V(\lambda )$, where $\lambda =\prod _{j=1}^{k-1}\lambda _j$ and

\[ \lambda_j=\frac{f(x,y)-f(x,\xi^j y)}{(1-\xi^j)y}, \]

for $1\le j\le k-1.$ We have thus a decomposition ${\mathcal {D}}=\bigcup _{j=1}^{k-1}{\mathcal {D}}_j$, with ${\mathcal {D}}_j=V(\lambda _j)$.

Proof. The double-point space ${\mathcal {D}}^2(F_k)$ is the intersection of the zero loci of the divided differences

\[ \frac{((y')^k-y^k)}{y'-y}\quad {\rm{and}}\quad \frac{f(x,y)-f(x,y')}{y-y'}. \]

Since $((y')^k-y^k)/{(y'-y)}=\prod _{j=1}^{k-1}(y'-\xi ^j y)$, we conclude that, as a set, the space ${\mathcal {D}}^2(F_k)$ is the union of the spaces

\[ {\mathcal{D}}^2_j=\left\{(x,y,\xi^jy)\in (\mathbb{C}^{n-1}\times \mathbb{C}\times\mathbb{C},0) \mid \frac{f(x,y)-f(x,\xi^j y)}{(1-\xi^j)y}=0\right\}, \]

for $j=1,\ldots, k-1$. Each of the sets ${\mathcal {D}}^2_j$ projects to $V(\lambda _j)$, which shows that ${\mathcal {D}}(F_k)=V(\lambda )$ as sets.

To show the equality as complex spaces, observe that the possible dimension of ${\mathcal {D}}^2(F_k)$ is one or two. If ${\mathcal {D}}^2(F_k)$ has dimension two, then some branch ${\mathcal {D}}^2_j$ has dimension two. Therefore, the corresponding function $\lambda _j$ is identically zero, which in turn implies $\lambda =0$. Since $F_k$ is finite by construction, the projection ${\mathcal {D}}^2(F_k)$ is finite, hence the image of ${\mathcal {D}}^2_j$ is a germ of a 2-dimensional analytic closed subset of $(\mathbb {C}^2,0)$, so is equal to $(\mathbb {C}^2,0)$. This implies that ${\mathcal {D}}(F_k)= (\mathbb {C}^2,0)=V(0)$.

Suppose now that ${\mathcal {D}}^2(F_k)$ has dimension one. This implies that the functions $\prod _{j=1}^{k-1}(y'-\xi ^j y)$ and $f[x,y,y']$ form a regular sequence. Applying lemma 3.1 with $Z=V(f[x,y,y'])$ gives ${\mathcal {D}}(F_k)=V(\prod _{j=1}^{k-1}\mathcal {F}_0((\pi \vert _{{\mathcal {D}}^2_j})_*\mathcal {O}_{{\mathcal {D}}^2_j})))$, where the ${\mathcal {D}}^2_j$, $j=1,\ldots, k-1$, are given the natural complex space structure. Each of the morphisms $\pi \vert _{{\mathcal {D}}^2_j}\colon {\mathcal {D}}^2_j\to \mathbb {C}^2$ consists of forgetting the third coordinate of the tuple $(x,y,\xi ^j y)$, and this implies $\mathcal {F}_0((\pi \vert _{{\mathcal {D}}^2_j})_*\mathcal {O}_{{\mathcal {D}}^2_j}))=\langle \lambda _j\rangle$.

Now we introduce some results we use to check finite $\mathcal {A}$-determinacy of a $k$-folding map-germ $F_k$ and topological triviality in families of such germs. The first of these results was proven in [Reference Marar and Mond19] for corank one map-germs, then extended to arbitrary corank in [Reference Marar, Nuño-Ballesteros and Peñafort-Sanchis20].

The decomposition ${\mathcal {D}}=\bigcup _{j=1}^{k-1}{\mathcal {D}}_j$ in theorem 3.2 can be used to compute $\mu ({\mathcal {D}})$, making it easier to apply theorem 3.3 (and theorem 3.10 below). We denote by ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}$ the intersection multiplicity of two distinct branches of the double-point curve. Clearly, ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}={\mathcal {D}}_{j'}\cdot {\mathcal {D}}_{j}$, and

\[ {\mathcal{D}}_j\cdot {\mathcal{D}}_{j'}=\operatorname{dim}_\mathbb{C}\frac{\mathcal{O}_2}{\langle \lambda_j,\lambda_{j'}\rangle}. \]

Proposition 3.4 A $k$-folding map-germ $F_k$ is finitely $\mathcal {A}$-determined if and only if the Milnor numbers $\mu ({\mathcal {D}}_j)$, $j=1,\ldots,k-1$, and the intersection multiplicities ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}$ of all pairs ${\mathcal {D}}_j$ and ${\mathcal {D}}_j'$, with $j'\neq j$, are finite. In that case,

\[ \mu({\mathcal{D}}(F_k))=\sum_{j=1}^{k-1}\mu({\mathcal{D}}_j)+2 \sum_{\substack{j,j'=1\\ j< j'}}^{k-1}{\mathcal{D}}_j\cdot {\mathcal{D}}_{j'} -k+2. \]

Proof. By theorem 3.3, $F_k$ is finitely $\mathcal {A}$-determined if and only if $\mu ({\mathcal {D}}(F_k))$ is finite, equivalently, ${\mathcal {D}}(F_k)$ has an isolated singularity. This occurs if and only if every branch ${\mathcal {D}}_j$ has an isolated singularity and no pair of branches ${\mathcal {D}}_j$ and ${\mathcal {D}}_{j'}$, with $j\ne j'$, have a common component. Using the formula $\mu =2\delta -r+1$ for plane curves (see [Reference Milnor21]) and the property $\delta (X\cup Y)=\delta (X)+\delta (Y)+X\cdot Y$, we get

\begin{align*} \mu({\mathcal{D}}(F_k)) & =2\delta({\mathcal{D}}(F_k))-r({\mathcal{D}}(F_k))+1\\ & =\sum_{j=1}^{k-1}(2\delta({\mathcal{D}_j})-r(D_j)+1)-k+2+2\sum_{j,j'=1, j< j'}^{k-1}{\mathcal{D}}_j\cdot {\mathcal{D}}_{j'}\\ & =\sum_{j=1}^{k-1}\mu({\mathcal{D}}_j)+2\sum_{j,j'=1, j< j'}^{k-1}{\mathcal{D}}_j\cdot {\mathcal{D}}_{j'}-k+2. \end{align*}

The defining functions $\lambda _j$ of the branches ${\mathcal {D}}_j$ of the double-point curve play a major role in our study of finite $\mathcal {A}$-determinacy and topological equivalence of $k$-folding map-germs. We take $F_k(x,y)=(x,y^k,f(x,y))$ and write, for any given integer $p\ge 1$,

(3.1)\begin{equation} j^pf(x,y)=\sum_{q=1}^p\sum_{s=0}^qa_{qs}x^{q-s}y^s. \end{equation}

Then,

(3.2)\begin{equation} j^{p-1}\lambda_j=\sum_{q=1}^{p}\sum_{s=1}^{q}\vartheta_{sj} a_{qs}x^{q-s}y^{s-1}, \end{equation}

with

\[ \vartheta_{sj}=\frac{1-\xi^{sj}}{1-\xi^j}=1+\xi^j+\dots+\xi^{(s-1)j}. \]

The constants $\vartheta _{sj}$ play a significant role in determining the singularity type of the germs $\lambda _j$ and in computing ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}$. The following properties are needed in § 4.

Proof. We observe that $\vartheta _{0j}=0$, $\vartheta _{1j}=1$ and, for any integers $m,n$, we have $\vartheta _{mj}=\vartheta _{(m+n)j}$ if and only if $k\mid nj$. For (1) we take $m=0$ and $n=s$, and for (2) we take $m=1$ and $n=s-1$.

For (3), we observe that the constants $\vartheta _{sj}$ lie in the images of the curves $\gamma _n: S^1\subset \mathbb {C}\to \mathbb {C}$ given by $\gamma _n(z)={(1-z^{n+1})}/{(1-z)}=1+z+\cdots +z^n.$ We show that the self intersection points of the curves $\gamma _n$ are $0$ and $1$ (for $n\ge 3$).

Write $z={\rm e}^{{\rm i}\theta }$, with $\theta \in [0,2\pi )$. Then $\gamma _n(\theta )=x+{\rm i}y$, with $(x,y)\in \mathbb {R}^2$, gives $1-{\rm e}^{{\rm i}(n+1)\theta }=(x+{\rm i}y)(1-{\rm e}^{{\rm i}\theta }).$ Therefore,

\begin{align*} \cos((n+1)\theta)& =1-x+x\cos(\theta)-y\sin(\theta),\\ \sin((n+1)\theta)& ={-}y+y\cos(\theta)+x\sin(\theta). \end{align*}

Now the identity $\cos ((n+1)\theta )^2+\sin ((n+1)\theta )^2=1$ gives

(3.3)\begin{equation} (1-\cos(\theta))(x^2+y^2-x)-y\sin(\theta)=0. \end{equation}

Suppose that $y=0$. Then equation (3.3) becomes $(1-\cos (\theta ))x(x-1)=0$, so $x=0$ or $1$ or $\theta =0$.

When $x=0$, we have $1-{\rm {e}}^{{{\rm i}}\theta }\ne 0$, so $1-{\rm {e}}^{{{\rm i}}(n+1)\theta }=0$. That gives $\theta = {2\pi j}/{(n+1)}, j=1,\ldots, n$. Therefore, $\gamma _n$ passes $n$-times through the origin.

When $x=1$, we get ${\rm e}^{{\rm i}n\theta }=1$, so $\theta ={2\pi j}/{n}, j=1,\ldots, n-1$. Therefore, $\gamma _n$ passes $(n-1)$-times through the point $1$.

When $\theta =0$, we have $\gamma _n(0)=n+1$ and the curve has no self-intersections at that point. See figure 2 for the cases $n=5,6,7.$

FIGURE 2. Maple plots of the curves $\gamma _n$ for $n=5,6,7$ (from left to right).

Suppose now that $y\ne 0$. Then equation (3.3) can be written as $\cot ({\theta }/{2})={(x^2+y^2-x)}/{y}.$ This shows that, for any $(x,y)\in \mathbb {R}^2$ with $y\ne 0$, there is at most one $\theta \in [0,2\pi )$ satisfying $\gamma _n(\theta )=x+{\rm i}y$. Therefore, the only self-intersection points of $\gamma _n$ are $0$ and $1$.

3.2 Cross-caps and triple points

The number of cross-caps and triple points are invariants of finitely $\mathcal {A}$-determined map-germs $F\colon (\mathbb {C}^2,0)\to (\mathbb {C}^3,0)$ that can be described using stable deformations. A stable mapping $U\to (\mathbb {C}^3,0)$ with $U$ an open neighbourhood of the origin in $\mathbb {C}^2$ exhibits only regular points, transverse double points along curves, cross-caps and isolated transverse triple points. Every stable deformation $F_t$ of $F$ exhibits the same number $C$ of cross-caps and $T$ of triple points [Reference Mond23]. For a corank one map-germ, these are given by the formulae

\[ C=\operatorname{dim}_\mathbb{C}\frac{\mathcal{O}_2}{JF},\quad T=\frac{1}{6}\operatorname{dim}_\mathbb{C}\frac{\mathcal{O}_4}{I^3(F)}, \]

where $JF$ is the ideal generated by the $2\times 2$ minors of the differential matrix of $F$ [Reference Mond23] (the formula for $C$ holds without the corank one assumption).

For $j,j'\in \{1,\dots,k-1\}$, with $j\neq j'$, we set

\[ \lambda_{j,j' }=\frac{\lambda_j-\lambda_{j'}}{y} \]

and define

\[ T_{j,j'}=\operatorname{dim}_\mathbb{C}\frac{\mathcal{O}_2}{\langle \lambda_j,\lambda_{j,j'}\rangle}. \]

Observe that $T_{j,j'}=T_{j',j}$ for all $j\neq j'$.

Proposition 3.8 The number of cross-caps and of triple points of a finitely $\mathcal {A}$-determined $k$-folding map-germ $F_k$ are given by

\[ C=\operatorname{dim}_{\mathbb{C}}\dfrac{\mathcal{O}_{2}}{\langle y^{k-1},{(\partial{f}}/{\partial y)}(x,y)\rangle} \quad \mbox{and} \quad T=\frac{1}{3}\sum_{1\leq j< j'\leq k-1}T_{j,j'}. \]

Proof. The formula for $C$ is trivial. For $T$, we claim that

\[ J=\left\langle \frac{y'^k-y^k}{y'-y},\frac{\frac{y''^k-y^k}{y''-y}-\frac{y'^k-y^k}{y'-y}}{y''-y'}\right\rangle\subseteq I^3(F_k) \]

is a radical ideal of $\mathcal {O}_{4}$ whose zero locus is

\[ Z=\{(x,y,y',y'')\mid y'=\xi^j y, y''=\xi^{j'}y \text{ for some }j,j'\in \{1,\dots,k-1\} \text{ with } j\neq j'\}. \]

Clearly, $Z$ is contained in the zero locus $V(J)$. The converse inclusion follows by considering the degree of the generators of $J$ and by the fact that $V(J)$ is generically reduced. Since $V(J)$ is also a complete intersection, it is reduced. Thus $J$ is a radical ideal.

The irreducible decomposition of $Z$ consists of the branches

\[ Z_{j,j'}=\{(x,y,\xi^j y,\xi^{j'}y)\mid x,y\in \mathbb{C}\}. \]

Thus, we may decompose ${\mathcal {D}}^3(F_k)$, as a set, as the union of the (not necessarily irreducible) branches

\[ {\mathcal{D}}^3_{j,j'}(F_k)=V(y'-\xi^jy,y''-\xi^{j'}y,f[x,y,y'],f[x,y,y',y'']). \]

Eliminating $y'$ and $y''$, gives ${\mathcal {D}}^3_{j,j'}(F_k)=\langle \lambda _j,\lambda _{j,j'}\rangle$. Therefore, we only need to show that $\operatorname {dim}_\mathbb {C} ({\mathcal {O}_4}/{I^3(F_k)})=\sum _{j\neq j'}\operatorname {dim}_\mathbb {C} ({\mathcal {O}_2}/({\langle \lambda _{j},\lambda _{j,j'}\rangle })$. It is clear that the same decomposition of ${\mathcal {D}}^3(F_k)$ applies to an unfolding $\tilde {F}_k(x,y,t)=(x,y^k,f_t(x,y),t)$ of $F_k$, and fixing a representative and a nonzero parameter $\delta$, the same holds for $(\tilde {F}_k)_{\delta }$. This gives a decomposition of ${\mathcal {D}}^3(({\tilde {F}_k})_{\delta })$ into the branches ${\mathcal {D}}^3_{j,j'}(({\tilde {F}_k})_{\delta })$.

It is possible to choose an unfolding where the branches ${\mathcal {D}}^3_{j,j'}(({\tilde {F}_k})_{\delta })$ are pairwise disjoint. Indeed, it follows from their defining equations that two spaces ${\mathcal {D}}^3_{j,j'}(({\tilde {F}_k})_{\delta })$ and ${\mathcal {D}}^3_{s,s'}(({\tilde {F}_k})_{\delta })$ can only intersect on $\{y=0\}$, so it is enough to find an unfolding where ${\mathcal {D}}^3_{j,j'}(({\tilde {F}_k})_{\delta })\cap \{y=0\}$ is empty. Again, it follows from the defining equations that a point $(x,0,0,0)\in {\mathcal {D}}^3_{j,j'}(({\tilde {F}_k})_{\delta })$ if $({\partial f_{\delta }}/{\partial x})(x,0)=({\partial f^2_{\delta }}/{\partial x^2})(x,0)=0$, a condition that can be avoided by choosing a suitable deformation $f_t$ of $f$.

We take now an unfolding as above. Since $Z$ is reduced, ${\mathcal {D}}^3(({\tilde {F}_k})_{\delta })$ is isomorphic to the union of the ${\mathcal {D}}^3_{j,j'}(({\tilde {F}_k})_{\delta })$ as complex spaces. The equality we need to show follows now from the constancy of the numbers involved under continuous deformations. Defining $\lambda _{j}^{\delta }$ and $\lambda _{j,j'}^{\delta }$ in the obvious way, we obtain

\begin{align*} \operatorname{dim}_\mathbb{C}\frac{\mathcal{O}_4}{I^3(F_k)}& =\sum_{(x,y,y',y'')}\operatorname{dim}_\mathbb{C}\frac{\mathcal{O}_4}{I^3(({\tilde{F}_k})_{\delta})_{(x,y,y',y'')}}\\ & =\sum_{(x,y)}\sum_{j\neq j'}\operatorname{dim}_\mathbb{C}\frac{\mathcal{O}_2}{\langle\lambda_{j}^{\delta},\lambda_{j,j'}^{\delta}\rangle} =\sum_{j\neq j'}\operatorname{dim}_\mathbb{C}\frac{\mathcal{O}_2}{\langle\lambda_{j},\lambda_{j,j'}\rangle}. \end{align*}

Since $T_{j,j'}=T_{j',j}$, we add only the numbers $T_{j,j'}$, with $j< j'$, and replace $1/6$ by $1/3$.

3.3 Topological triviality

We say that two subspaces $S$ and $S'$ of a topological space $X$ have the same topological type if there is a homeomorphism $X\to X$ restricting to an homeomorphism $S\to S'$. Milnor showed that two isolated hypersurface singularities with same topological type have the same Milnor number. In the case of our invariants, a similar result can be obtained using the results in [Reference Nemethi and Pinter28, Reference Siersma30] (see [Reference Bobadilla, Peñafort and Sampaio4] for details).

The Milnor number of the double-point curve is enough to determine the topological triviality of families of finitely $\mathcal {A}$-determined map-germs.

Using the upper semi-continuity of the numbers involved, proposition 3.10 can be combined with proposition 3.4 to yield the following result.

4.1 The case $k= 3$

When $k=3$ we get several $\mathcal {A}$-simple map-germs. For this reason, we treat this case separately.

Proof. We can write $f(x,y)=f_0(x,y^3)+yf_1(x,y^3)+y^2f_2(x,y^3)$, for some germs of holomorphic functions $f_i$, $i=0,1,2$. Then $F_3\sim _{\mathcal {A}} (x,y^3,yf_1(x,y^3)+y^2f_2(x,y^3)).$

Suppose that $a_{10}=0$ ($F_3$ is singular) and $a_{22}\ne 0$. Then

\[ F_3\sim_{\mathcal{A}}(x,y^3,y(g(x)+y^3h(x,y^3))+y^2(a_{22}+k(x,y^3))) \]

for some germs of holomorphic functions $g\in \mathfrak {m}_1$, $h\in \mathcal {O}_2$ and $k\in \mathfrak {m}_2$. We can make successive changes of coordinates in the target so that $j^pF_3\sim _{\mathcal {A}^{(p)}}(x,y^3,yL(x)+a_{22}y^2)$ for any $p\ge 3$. It is not difficult to show that $F_3$ is finitely $\mathcal {A}$-determined if and only if $ord(L)=ord(g)=ord( f_y (x,0))$ is finite. Suppose that this is the case and denote by $l$ that order. Then $j^lL(x)=a_{l1}x^l$, $a_{l1}\ne 0$, and the change of coordinates $y\mapsto y-(a_{l1}/2a_{22})x^l$ in the source yields $j^{2l+1}F_3\sim _{\mathcal {A}^{(2l+1)}}(x,y^3-3(a_{l1}/2a_{22})^2yx^{2l},a_{22}y^2)$. This is an $S_{2l-1}$-singularity, and since it is $(2l+1)$-$\mathcal {A}$-determined, we have $F_3\sim _{\mathcal {A}} (x,y^3-yx^{2l},y^2)$. In the real case, this is an $S_{2l-1}^{-}$-singularity.

The above calculations show, in particular, that we do not get the simple singularities $B_l^{\pm }$, $C_l^{\pm }$ and $F_4$ whose 2-jets are $\mathcal {A}^{(2)}$-equivalent to $(x,0,y^2)$.

Similar calculations show that when $a_{11}=a_{22}=0$ and $a_{21}\ne 0$, we get an $H_l$-singularity when the singularity of $F_3$ is finitely $\mathcal {A}$-determined.

The remaining cases are studied in the same way as in the case $k\ge 4$ (table 6). We get three strata with topological normal forms ${\bf U}_4^{3}, {\bf X}_4^{3}$ and ${\bf W}_4^{3,1}$ (the germs in these strata are not $\mathcal {A}$-simple) together with the stratum represented by the topological normal form ${\bf U}^3_3$, which is topologically equivalent to Mond's singularity $X_4$ (see § 2 and [Reference Mond22]).

Remark 4.5 The invariants associated to the simple singularities can be found in [Reference Mond23]; those associated to ${\bf U}_4^{3}, {\bf X}_4^{3}, {\bf W}_4^{3,1}$ are given in table 7.

TABLE 7. Topological invariants of germs in strata in table 6

4.2 The case $k\ge 4$

We consider here the case when $k\ge 4$, which we divide into the four branches according to the $\mathcal {A}^{(2)}$-orbits in lemma 4.1. In all that follows, $j\in \{1,\ldots,k-1\}$, and subindices of singularities indicate the codimension of the stratum.

4.2.1 Branch 1: $a_{11}=0$, $a_{21}a_{22}\neq 0$

Theorem 4.6 Any germ $F_k$ of a $k$-folding map satisfying $a_{11}=0$ and $a_{21}a_{22}\neq 0$ is finitely $\mathcal {A}$-determined and is topologically equivalent to

\[ {\bf M}^k_1\colon(x,y)\mapsto(x,y^k,xy+y^2). \]

The invariants take the values $\mu ({\mathcal {D}})=(k-2)^2$, $C=k-1$, $T=0$ and $r({\mathcal {D}})=k-1$ and the double-point curve of $F_k$ is the union of $k-1$ regular curves intersecting transversally.

Proof. The functions defining the branches ${\mathcal {D}}_j$ of the double-point curve (see theorem 3.2) are given by $\lambda _j=a_{21}x+(1+\xi ^j)a_{22}y+O(2),$ where $O(l)$ denotes a remainder of order $l$. Clearly, all of the branches ${\mathcal {D}}_j$ are regular curves. As the scalars $1+\xi ^j$ are pairwise distinct, the space ${\mathcal {D}}={\mathcal {D}}(F_k)=\bigcup _j{\mathcal {D}}_j$ consists of $k-1$ regular curves intersecting transversally at the origin.

We have $\mu ({\mathcal {D}}_j)=0$ and ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=1$, and from proposition 3.4 we obtain $\mu ({\mathcal {D}})=(k-2)^2$. By theorem 3.3, any germ $F_k$ satisfying the conditions in the statement of the theorem is finitely $\mathcal {A}$-determined. These germs form a stratum of codimension 1 defined by $\{a_{11}=0,a_{21}a_{22}\neq 0\}$. Since this stratum is path connected, we conclude by theorem 3.10 that it consists of topologically equivalent germs. We choose for a topological model the germ ${\bf M}^k_1$ given in the statement of the theorem.

By proposition 3.11, it is enough to compute $C$ and $T_{j,j'}$ for ${\bf M}^k_1$ as these invariants are constant along the stratum. We have $C=\operatorname {dim}_\mathbb {C} {\mathcal {O}_2}/{\langle y^{k-1},x+2y\rangle }=k-1$ and $T_{j,j'}=\operatorname {dim}_\mathbb {C} {\mathcal {O}_2}/{\langle x+(1+\xi ^j)y,1\rangle }=0.$

Remark 4.7 The singularity ${\bf M}^4_1$ is the $\mathcal {A}$-simple singularity $C_3$ (see the proof of proposition 4.2).

The proofs for the cases in the remaining branches follow by similar arguments used in the proof of theorem 4.6 (except for the calculations of $T$). To avoid repetition, we highlight only key differences in each case. The notation for the conditions that define the strata are those indicated in the tables.

4.2.2 Branch 2: $a_{11}=a_{21}=0$, $a_{22}\ne 0$

Theorem 4.8 The strata of codimension $\le \! 4$ of finitely $\mathcal {A}$-determined $k$-folding map-germs in the branch $a_{11}=a_{21}=0$, $a_{22}\ne 0$ are those given in table 2. The invariants associated to the germs in each stratum are given in table 3.

Proof. The result follows from propositions 4.9, 4.11 and 4.12.

For germs in this branch, the map-germ $(x,y)\mapsto (x,f(x,y))$ is finite and generically two-to-one. Therefore, $T=0$ for any finitely $\mathcal {A}$-determined map-germ in this branch.

The germs of the functions defining the double-point branch ${\mathcal {D}}_j$ is given by

\[ \lambda_j={\vartheta_{2j}a_{22}y}+a_{31}x^2+\vartheta_{2j}a_{32}xy+\vartheta_{3j}a_{33}y^2+O(3). \]

The branch ${\mathcal {D}}_j$ is thus regular if and only if $\vartheta _{2j}\ne 0$. By lemma 3.6, $\vartheta _{2j}=0$ when $k$ is even and $j=k/2$.

Proposition 4.9 Any $k$-folding map-germ $F_k$ satisfying $a_{11}=a_{21}=\cdots = a_{l1}=0$, $a_{22}a_{(l+1)1}\neq 0$, for some $l\ge 2$, and $a_{33}\ne 0$ when $k=2p$, is finitely $\mathcal {A}$-determined and is topologically equivalent to

\[ {\bf M}^{k}_{l}\colon(x,y)\mapsto(x,y^k,y^2+y^3+x^ly). \]

The invariants $C$, $T$, $\mu (\mathcal {D})$ and $r(\mathcal {D})$ are as in table 2. All the double-point branches are regular curves except for the branch $\mathcal {D}_p$, when $k=2p$, which has an $A_{l-1}$-singularity. We have ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=l$, for all $j\ne j'$.

Proof. Fix an index $j$ and assume that $2\nmid k$ or that $k=2p$ but $j\neq p$. Then ${\mathcal {D}}_j$ is a regular curve and can be parametrized by $t\mapsto (t,\gamma _j(t))$, with $\gamma _j(t)=-({a_{l+1,1}}/{((1+\xi ^j)a_{22})})t^{l}+O(l+1).$ Clearly, any two distinct branches have order of contact equal to $l$.

Suppose now that $k=2p$. As $\vartheta ^2_{p}=0$, the coefficients of $x^sy$ in $\lambda _p$ vanish for all $s\ge 1$. Moreover, since $\vartheta ^3_{p}=1$, the function $\lambda _p$ is of the form $\lambda _p=a_{l+1,1}x^l+a_{33}y^2+y^3h(y).$ This implies that $\lambda _p$ is $\mathcal {R}$-equivalent to $y^2+x^{l}$ if and only if $a_{33}\neq 0$, in which case it has an $A_{l-1}$-singularity. As for the contact between branches, $ord(\lambda _p(t,({-a_{l+1,1}}/{((1+\xi ^j)a_{22})})t^{l}+O(l+1))=l,$ hence ${\mathcal {D}}_j\cdot {\mathcal {D}}_{p}=l$. This determines $\mu (\mathcal {D})$, and hence the topological triviality and the constancy of the invariants along the stratum.

Remarks 4.10 1. Branch 1 can be considered as a particular case of the strata ${\bf M}^k_l$ in proposition 4.9 (the condition $a_{33}\ne 0$ is not needed when $l=1$).

2. When $2\nmid k$, the term $y^3$ in ${\bf M}^k_l$ is irrelevant for topological equivalence. We include it to represent both $k$ even and $k$ odd by the same map-germ. We do this for all subsequent topological normal forms.

Proposition 4.11 Suppose that $k=2p$. Any $k$-folding map-germ $F_k$ satisfying $a_{11}=a_{21}=a_{33}=0$, $a_{22}a_{31}\ne 0$ and the additional conditions in ${\rm {(a)}}$ or ${\rm {(b)}}$ below is finitely $\mathcal {A}$-determined and is topologically equivalent to one of the germs

\[ {\bf N}^k_l\colon(x,y)\mapsto(x,y^k,y^2+x^2y+y^{2l-1}), \quad l=3,4. \]

The invariants $\mu ({\mathcal {D}(F_k)}), C,T, r({\mathcal {D}}(F_k))$ are as in table 3. The branches ${\mathcal {D}}_j$ are regular curves for all $j\ne p$ and ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=2$ for all $j\neq j'$.

${\rm {(a)}}$ If ${CndNA}_3\ne 0$, then the branch ${\mathcal {D}}_p$ has an $A_{3}$-singularity and the map-germ is topologically equivalent to ${\bf N}^k_3$.

${\rm {(b)}}$ If ${CndNA}_3=0$ and ${CndNA}_5\ne 0$, then ${\mathcal {D}}_p$ has an $A_{5}$-singularity and the map-germ is topologically equivalent to ${\bf N}^k_4$.

Proof. We have $\lambda _{p}= a_{31}x^2+a_{41}x^3+a_{43}xy^2+a_{51}x^4+ a_{53}x^2y^2+a_{55}y^4 +O(5).$ It has an $A_3$-singularity if and only if ${CndNA}_3\ne 0$. When ${CndNA}_3=0$, we need to consider the $7$-jet of $\lambda _p$. A calculation shows that $\lambda _p$ has an $A_5$-singularity if and only if ${CndNA}_5\ne 0$. In both cases, we have $ord(\lambda _p(t,-{a_{31}}/{((1+\xi ^j)a_{22})}t^{2}+O(3)))=2$ for all $j\ne p$.

Proposition 4.12 Suppose that $k=2p$. Any $k$-folding map-germ $F_k$ satisfying $a_{11}=a_{21}=a_{31}=a_{33}=0$ and $a_{22}a_{41}a_{43}\ne 0$ is finitely $\mathcal {A}$-determined and is topologically equivalent to

\[ {\bf O}^{k}_4\colon (x,y)\mapsto (x,y^k,y^2+x^3y+xy^3). \]

The codimension of the stratum is $4$ and the invariants $\mu ({\mathcal {D}}(F_k)), C,T,$ $r({\mathcal {D}}(F_k))$ are as in table 3. The branches ${\mathcal {D}}_j,$ $j\ne p$, are regular curves and ${\mathcal {D}}_p$ has a $D_{4}$-singularity. We have ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=2$ for the distinct regular branches and ${\mathcal {D}}_j\cdot {\mathcal {D}}_{p}=3$, for $j\ne p$.

Proof. The result follows from the fact that $\lambda _{p}=a_{41}x^3+a_{43}xy^2+O(4).$

4.2.3 Branch 3: $a_{11}=a_{22}=0$, $a_{21}\ne 0$

Theorem 4.13 The strata of codimension $\le \! 4$ of finitely $\mathcal {A}$-determined $k$-folding map-germs in the branch $a_{11}=a_{22}=0$, $a_{21}\ne 0$ are those given in table 4. The invariants of the germs in each stratum are given in table 5.

Proof. The result follows from propositions 4.14, 4.16 and 4.18.

For any finitely $\mathcal {A}$-determined $k$-folding map-germ in this branch, we have $C=\operatorname {dim}_\mathbb {C} {\mathcal {O}_2}/{\langle y^{k-1},a_{21}x+O(2)\rangle }=k-1.$

The double-point branches are regular curves that can be parametrized by $t\mapsto (\gamma _j(t),t)$, with

\begin{align*} \gamma_j(t)& ={-}\frac{1}{a_{21}}\vartheta_{3j}a_{33}t^2 -\frac{1}{a_{21}^2}(\vartheta_{2j}\vartheta_{3j}a_{32}a_{33}-\vartheta_{4j}a_{21}a_{44})t^3\\ & \quad -\frac{1}{a_{21}^3}(\vartheta_{2j}^2 \vartheta_{3j} a_{32}^2 a_{33}-\vartheta_{2j} \vartheta_{4j} a_{32} a_{44} a_{21} +\vartheta_{3j}^2 a_{33}(a_{31}a_{33}-a_{43} a_{21})\\ & \quad +\vartheta_{5j} a_{55} a_{21}^2) t^4 +O(5). \end{align*}

The strata are determined by the contact between the branches of the double-point curve which depend on $\vartheta _{sj}$ as well as on the coefficients $a_{pq}$. We start with the case $a_{33}\ne 0$, where the strata depend on the divisibility of $k$ by $3$.

Proposition 4.14 Suppose that $a_{11}=a_{22}=0$ and $a_{21}a_{33}\neq 0$. Any $k$-folding map-germ $F_k$ in case ${\rm {(a)}}$ or satisfying the additional conditions in ${\rm {(b)}}$ is finitely $\mathcal {A}$-determined and is topologically equivalent to one of the map-germs

\[ {\bf P}^k_l\colon(x,y)\mapsto(x,y^k,xy+y^3+y^{3l-1}),\quad\text{for }l=2,3,4. \]

The invariants $\mu ({\mathcal {D}}), C,T, r(\mathcal {D})$ are as in table 5. We have contact ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=2$, except for ${\mathcal {D}}_p\cdot {\mathcal {D}}_{2p}$ when $k=3p$ which is given in ${\rm {(b)}}$.

1. (a) If $3\nmid k$, then $F_k$ is topologically equivalent to ${\bf P}^k_2$.

2. (b) If $k=3p$, then the strata are as follows:

   1. If $CndH_2\ne 0$, then ${\mathcal {D}}_p\cdot {\mathcal {D}}_{2p}=4$ and $F_k$ is topologically equivalent to ${\bf P}^k_2$.

   2. If $CndH_2=0$ and $CndH_3\ne 0$, then ${\mathcal {D}}_p\cdot {\mathcal {D}}_{2p}=7$ and $F_k$ is topologically equivalent to ${\bf P}^k_3$.

   3. If $CndH_2=CndH_3=0$ and $CndH_4\ne 0$, then ${\mathcal {D}}_p\cdot {\mathcal {D}}_{2p}=10$ and $F_k$ is topologically equivalent to ${\bf P}^k_4$.

Proof. If $3\nmid k$, then by lemma 3.6 we have $\vartheta _{3j}\ne 0$ and $\vartheta _{3j}\ne \vartheta _{3j'}$ for all $j\neq j'$. This implies ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=2$ for all $j\neq j'$.

If $k=3p$, then by lemma 3.6 the equality $\vartheta _{3j}=\vartheta _{3j'}$ holds only when $\{j,j'\}=\{p,2p\}$. Again, we obtain ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=2$ for all $j\neq j'$ with $\{j,j'\}\neq \{p,2p\}$.

We have $\vartheta _{3p}=\vartheta _{3(2p)}=0$, $\vartheta _{4p}=\vartheta _{4(2p)}=1$ and $\vartheta _{5p}=\vartheta _{2(2p)}$ (lemma 3.6). Using the parametrizations of ${\mathcal {D}}_p$ and ${\mathcal {D}}_{2p}$, we get ${\mathcal {D}}_p\cdot {\mathcal {D}}_{2p}=4$ if and only if $a_{32}a_{44}-a_{21}a_{55}\ne 0$, equivalently, $CndH_2\ne 0$.

When $CndH_2=0$, the exceptional branches ${\mathcal {D}}_p$ and ${\mathcal {D}}_{2p}$ are parametrized by $x=-({a_{44}}/{a_{21}})y^3+\beta _sy^7+O(9)$, $s=1,2$, with $\beta _1-\beta _2\ne 0$ if and only if $CndH_3\ne 0$. Then, ${\mathcal {D}}_p\cdot {\mathcal {D}}_{2p}=7$.

When $CndH_2=CndH_3=0$, the exceptional branches are parametrized by $x=-({a_{44}}/{a_{21}})y^3+\beta _sy^{10}+O(11)$, $s=1,2$, with $\beta _1-\beta _2\ne 0$ if and only if $CndH_4\ne 0$. Then, ${\mathcal {D}}_p\cdot {\mathcal {D}}_{2p}=10$.

The values of $T$ can be computed using the models ${\bf P}^k_l$. We have $\lambda _j=x+\vartheta _{3j}y^2+\vartheta _{(3l-1)j}y^{3l-2}$ and $\lambda _{j,j'}=(\vartheta _{3j}-\vartheta _{3j'})y+(\vartheta _{3l-1,j}-\vartheta _{3l-1,j'})y^{3l-3}.$ By lemma 3.6, $T_{j,j'}=\operatorname {dim}_\mathbb {C} ({\mathcal {O}_2}/{\langle x,y\rangle })=1$ when $3\nmid k$ or when $k=3p$ but $\{j,j'\}\neq \{p,2p\}$. For $k=3p$, we show that $\vartheta _{3l-1,p}\neq \vartheta _{3l-1,2p}$ so $T_{p,2p}=3l-3$.

Remark 4.15 The singularity ${\bf P}^4_2\colon (x,y)\mapsto (x,y^4,xy+y^3+y^5)$ is topologically equivalent to the singularity $T_4\colon (x,y)\mapsto (x,y^4,xy+y^3)$ in [Reference Mond22]. Observe that according to proposition 4.14, if $k$ is not divisible by $3$, then the $y^{3l-1}$ term can be removed from the expression of ${\bf P}^k_l$, without changing the topological class of the germ.

Proposition 4.16 Suppose that $a_{11}=a_{22}=a_{33}=0$ and $a_{21}a_{44}\neq 0$. Any $k$-folding map-germ $F_k$ in case ${\rm {(a)}}$ or satisfying the additional conditions in ${\rm {(b)}}$, ${\rm {(c)}}$ or ${\rm {(d)}}$ is finitely $\mathcal {A}$-determined and is topologically equivalent to one of the following map-germs:

\[ {\bf Q}^k_3\colon(x,y)\mapsto(x,y^k,xy+y^4+y^{5}+y^{6}) \]

\[ {\bf Q}^k_4\colon(x,y)\mapsto(x,y^k,xy+y^4+y^{6}+y^{8}) \]

\[ \widetilde{{\bf Q}}^k_4\colon(x,y)\mapsto(x,y^k,xy+y^4+y^{5}+y^{7}) \]

The invariants are as in table 5. We have ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=3$ except when $j$ and $j'$ are in the sets $J$ or $J'$ below.

1. (a) If $3\nmid k$ and $4\nmid k$, then there are no exceptional branches and the germs in this stratum are topologically equivalent to ${\bf Q}_3^k$.

2. (b) If $k=3p$ and $4\nmid k$, then $J=\{p,2p\}$.

   1. (b1) If $CndH_2\ne 0$, then ${\mathcal {D}}_p\cdot {\mathcal {D}}_{2p}=4$ and $F_k$ is topologically equivalent to ${\bf Q}_3^k$.

   2. (b2) If $CndH_2=0$ and $CndH_3\ne 0$, then ${\mathcal {D}}_p\cdot {\mathcal {D}}_{2p}=7$ and $F_k$ is topologically equivalent to ${\bf Q}_4^k$.

3. (c) If $k=4p$, $3\nmid k$, then $J=\{p,2p,3p\}$.

   1. (c1) If $CndQm_5\ne 0$, then ${\mathcal {D}}_{j}\cdot {\mathcal {D}}_{j'}=5$, for all $j,j'\in J, j\ne j'$, and $F_k$ is topologically equivalent to ${\bf Q}_3^k$.

   2. (c2) If $CndQm_5=0$ and $CndQm_6\ne 0$, then ${\mathcal {D}}_{j}\cdot {\mathcal {D}}_{j'}=6$, for all $j,j'\in J, j\ne j'$, and $F_k$ is topologically equivalent to $\widetilde {\bf Q}_4^k$.

4. (d) If $k=12p$, the exceptional contact between double-point branches occurs when the indices are in $J=\{4p,8p\}$ or $J'=\{3p,6p,9p\}$. There are three strata:

   1. (d1) If $CndH_2\ne 0$ and $CndQm_5\ne 0$, then ${\mathcal {D}}_{j}\cdot {\mathcal {D}}_{j'}=4$ (resp. ${\mathcal {D}}_{j}\cdot {\mathcal {D}}_{j'}=5$) for all distinct pairs with $j,j'$ in $J$ (resp. $J'$), and $F_k$ is topologically equivalent to ${\bf Q}_3^k$.

   2. (d2) If $CndH_2=0, CndH_3\ne 0$ and $CndQm_5\ne 0$, then ${\mathcal {D}}_{j}\cdot {\mathcal {D}}_{j'}=7$ (resp. ${\mathcal {D}}_{j}\cdot {\mathcal {D}}_{j'}=5$) for all distinct pairs with $j,j'$ in $J$ (resp. $J'$), and $F_k$ is topologically equivalent to ${\bf Q}_4^k$.

   3. (d3) If $CndQm_5=0, CndQm_6\ne 0,$ and $CndH_2\ne 0$, then ${\mathcal {D}}_{j}\cdot {\mathcal {D}}_{j'}=4$ (resp. ${\mathcal {D}}_{j}\cdot {\mathcal {D}}_{j'}=6$) for all distinct pairs with $j,j'$ in $J$ (resp. $J'$), and $F_k$ is topologically equivalent to $\widetilde {\bf Q}_4^k$.

Proof. The branches $\mathcal {D}_j$ can be parametrized by $t\mapsto (\gamma _j(t),t)$ with

\begin{align*} \gamma_j(t)& ={-}\frac{1}{a_{21}}\vartheta_{4j}a_{44}t^3 -\frac{1}{a_{21}^2}(\vartheta_{2j}\vartheta_{4j}a_{32}a_{44}-\vartheta_{5j}a_{21}a_{55})t^4\\ & \quad -\frac{1}{a_{21}^3} (\vartheta_{2j}^2 \vartheta_{4j} a_{32}^2 a_{44}-\vartheta_{3j} \vartheta_{4j} a_{43} a_{44} a_{21}\nonumber\\ & \quad -\vartheta_{2j} \vartheta_{5j} a_{21}a_{32}a_{55}+\vartheta_{6j} a_{66} a_{21}^2) t^5+O(6). \end{align*}

(a) If $3\nmid k$, $4\nmid k$, then $\vartheta _{4j}\ne \vartheta _{4j'}$ for all $j,j'$ with $j\ne j'$ (lemma 3.6). Therefore, ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=3$ for all distinct pairs.

(b) If $k=3p$ and $4\nmid p$, then $\vartheta _{4j}\ne \vartheta _{4j'}$ for all distinct pairs with $j$ or $j'$ not in $J=\{p,2p\}$. For such pairs, ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=3$.

Using the fact that $\vartheta _{6p}=\vartheta _{6(2p)}=\vartheta _{3p}=\vartheta _{3(2p)}=0$, $\vartheta _{4p}=\vartheta _{4(2p)}=1$ and $\vartheta _{5p}=\vartheta _{2p}\neq \vartheta _{5(2p)}=\vartheta _{2(2p)}$, the parametrization of the exceptional branch $\mathcal {D}_p$ becomes

\[ \gamma_p(t)={-}\frac{a_{44}}{a_{21}}t^3 -\frac{\vartheta_{2p}(a_{32}a_{44}-a_{21}a_{55})}{a_{21}^2}t^4-\frac{\vartheta_{2p}^2a_{32} (a_{32}a_{44}-a_{21}a_{55})}{a_{21}^3} t^5+O(6). \]

A parametrization of $\mathcal {D}_{2p}$ is obtained by replacing $\vartheta _{2p}$ by $\vartheta _{2(2p)}$ in the expression of $\gamma _p$. Therefore, ${\mathcal {D}}_p\cdot {\mathcal {D}}_{2p}=4$ if and only if $a_{32}a_{44}-a_{21}a_{55}\ne 0$, i.e. $CndH_2\ne 0$.

When $CndH_2= 0$, we have

\[ \gamma^1_{p}={-}\frac{a_{44}}{a_{21}}t^3-\frac{a_{31} a_{44}^2 - a_{21} a_{44}a_{54} + a_{21}^2 a_{77} }{a_{21}^3}t^6+\\ \frac{\vartheta^2_{p}CndH_3}{a^4_{21}}t^7+O(8). \]

For $\gamma _{2p}$, we replace $\vartheta ^2_{p}$ by $\vartheta ^2_{2p}$ in $\gamma _{p}$. It follows that ${\mathcal {D}}_p\cdot {\mathcal {D}}_{2p}=7$ if and only if $CndH_3\ne 0$.

(c) If $k=4p$ and $3\nmid p$, we have ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=3$ for $j$ or $j'$ not in $\{p,2p,3p\}$.

The parametrization of $\mathcal {D}_p$ becomes $\gamma _{p}(t)={a_{55}}/{a_{21}}t^4+ {\vartheta ^2_{p}(a_{32}a_{55} - a_{21}a_{66})}/{a_{21}^2} t^5+O(6),$ and similarly for $\gamma _{2p}$ and $\gamma _{3p}$ replacing $p$ by $2p$ and $3p$ respectively. Consequently, the branches $\mathcal {D}_p$, $\mathcal {D}_{2p}$, $\mathcal {D}_{3p}$ have pairwise order of contact $5$ if and only if $a_{32}a_{55} - a_{21}a_{66}\ne 0$, i.e. $CndQm_5\ne 0.$

When $CndQm_5=0$, $\gamma _{p}=-{a_{55}}/{a_{21}}t^4- \vartheta _{2p}\vartheta _{3p}({a_{21} a_{77} - a_{43} a_{55}})/ {a_{21}^2}t^6+O(7)$, with similar adjustments as above for $\gamma _{2p}$ and $\gamma _{3p}$. Therefore, the three exceptional branches have pairwise order of contact $6$ if and only if $a_{21} a_{77}-a_{43} a_{55}\ne 0$, i.e. $CndQm_6\ne 0$.

(d) This follows by lemma 3.6 and (b) and (c) above.

The contact between the branches determines $\mu (\mathcal {D})$, the topological types and their associated strata. It remains to compute $T$ for each normal form.

For ${\bf Q}^k_3$ we have $\lambda _j=x+\vartheta _{4j}y^3+\vartheta _{5j}y^{5}+\vartheta _{6j}y^5,$ and $\lambda _{j,j'}=(\vartheta _{4j}-\vartheta _{4j'})y^2+(\vartheta _{5j}-\vartheta _{5j'})y^3+(\vartheta _{6j}-\vartheta _{6j'})y^4.$ Using the properties of $\vartheta _{sj}$ in lemma 3.6, we have $T_{j,j'}=2$ unless $k=3p$ and $j,j'\in \{p,2p\}$, or $k=4p$ and $j,j'\in \{p,2p,3p\}$. In the first case we get $T_{j,j'}=3$ and in the second $T_{j,j'}=4$. The invariant for the germ ${\bf Q}^k_4$ differs from ${\bf Q}^k_3$ only when $k$ is divisible by $3$. For $k=3p$, we have $T_{p,2p}=6$ and $T_{j,j'}=4$ if $k=4q$ and $j,j'\in \{q,2q,3q\}$. All other indices $j$ and $j'$ give $T_{j,j'}=2$. Similarly, for the germ $\widetilde {\bf Q}^k_4$ and for $k=4p$, we get $T_{j,j'}=5$ if $j,j'\in \{p,2p,3p\}$. If $k=3q$, then $T_{q,2q}=3$. All other indices $j$ and $j'$ give $T_{j,j'}=2$.

Remark 4.17 When $k$ is divisible by $12$, the germs ${\bf Q}^k_4$ and $\widetilde {\bf Q}^k_4$ have the same invariants $C,T,\mu (D)$ and $r(D)$ but they are not topologically equivalent as their associated sets of contacts between double-points branches are distinct (see [Reference Zariski33]).

Proposition 4.18 Suppose that $a_{11}=a_{22}=a_{33}=a_{44}=0$ and $a_{21}a_{55}\neq 0$. Any $k$-folding map-germ $F_k$ in case ${\rm {(a)}}$ or satisfying the additional conditions in ${\rm {(b)}}$ or ${\rm {(c)}}$ is finitely $\mathcal {A}$-determined and is topologically equivalent to

\[ {\bf R}^k_4\colon(x,y)\mapsto(x,y^k,xy+y^5+y^6+y^7). \]

The invariants associated to the germs in the stratum are as in table 5. We have ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=4$ except for the exceptional pairs of branches below.

1. (a) If $4\nmid k$ and $5\nmid k$, there are no additional conditions and no exceptional branches.

2. (b) If $k=4p$, $5\nmid k$ and $CndQm_5\ne 0$, then ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=5$ for all distinct pairs with $j,j'$ in $J=\{p,2p,3p\}$.

3. (c) If $k=5p$, $4\nmid k$ and $CndRm_5\ne 0$, ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=5$ for all distinct pairs with $j,j'$ in $J=\{p,2p,3p,4p\}$.

4. (d) If $k=20p$, $CndQm_5\ne 0$ and $CndRm_5\ne 0$, then ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=5$ for all distinct pairs with $j,j'$ in $J=\{5p,10p,15p\}$ or in $J'=\{4p,8p,12p,16p\}$.

Proof. The branches $\mathcal {D}_j$ can be parametrized by $t\mapsto (\gamma _j(t),t)$ with

\begin{align*} \gamma_j(t)& ={-}\frac{1}{a_{21}}\vartheta_{5j}a_{55}t^4+ \frac{1}{a_{21}^2}(\vartheta_{2j}\vartheta_{5j}a_{32}a_{55} - \vartheta_{6j}a_{21}a_{66})t^5\\ & \quad +\frac{1}{a_{21}^3} (a_{21} (\vartheta_{2j}\vartheta_{6j}a_{32} a_{66} - \vartheta_{7j}a_{21} a_{77})\nonumber\\ & \quad - ((\vartheta_{2j})^2a_{32}^2 - \vartheta_{3j}a_{21} a_{43})\vartheta_{5j} a_{55})t^6+O(7). \end{align*}

(a) If $4\nmid k$ and $5\nmid k$, then $\vartheta _{5j}\ne \vartheta _{5j'}$ for all $j,j'$ with $j\ne j'$ (by lemma 3.6). Therefore, ${\mathcal {D}}_j\cdot {\mathcal {D}}_{j'}=4$ for all distinct pairs.

(b) If $k=4p$ and $5\nmid k$, then $\vartheta _{5p}=1$, $\vartheta _{6p}=\vartheta _{2p}$, and we can write $\gamma _p(t)=-a_{55}/a_{21}t^4+\vartheta _{2p}CndQm_5/a_{21}^2t^5+O(6)$. We get similarly $\gamma _{2p}(t)$ and $\gamma _{3p}(t)$ by substituting $\vartheta _{2p}$ by, respectively, $\vartheta _{2(2p)}$ and $\vartheta _{2(3p)}$ in $\gamma _p$. The result follows as $\vartheta _{2s}\ne \vartheta _{2s'}$ for $s,s'\in \{p,2p,3p\}$, $s\ne s'$.

(c) If $k=5p$ and $4\nmid k$, then $\vartheta _{5p}=0$, $\vartheta _{6p}=1$, $\vartheta _{7p}=\vartheta _{2p}$, so $\gamma _p(t)=-a_{66}/a_{21}t^5+\vartheta _{2p}CndRm_5/a_{21}^2t^6+O(7)$. The expressions $\gamma _{sp}$, $s=2,3,4,$ are obtaining by substituting $\vartheta _{2p}$ by $\vartheta _{2(sp)}$ in $\gamma _p$.

(d) The case $k=20p$ follows in a similarly way to that of proposition 4.16(d).

The calculations for $T$ are similar to those in the proof of proposition 4.16. We obtain $T_{j,j'}=4$ if $k=4p$ and $j,j'\in \{p,2p,3p\}$, and $T_{j,j'}=5$ if $k=5q$ and $j,j'\in \{q,2q,3q,4q\}$. All remaining indices $j$ and $j'$ give $T_{j,j'}=3$.

Remark 4.19 The singularities ${\bf Q}^4_3$ and ${\bf R}^4_4$ have the same invariants, but the contact of their double-point branches shows that they are not topologically equivalent.