2.1 Reflection groups and the orbit mapping

Throughout the text, $W$ stands for a (complex) reflection group acting on a ${\mathbb C}$-vector space $\mathcal V$. We write $p={\operatorname{dim}} \mathcal V$. The elements of the group may be assumed to be unitary with respect to a fixed hermitian form. This hermitian form is used to define orthogonality throughout the text.

Let $K\leqslant \mathcal V$ be the maximal subspace of $\mathcal V$ where $W$ acts trivially. The rank of $W$ is $\operatorname{rk} W={\operatorname{dim}} K^\bot$.

The orbit of a subset $S\subseteq \mathcal V$ (or better, the union of the orbits of the points in $S$) is denoted by $WS\subseteq \mathcal V$. We adopt the convention that the action of $\sigma\in W$ on a function $H$ in $\mathcal O_{\mathcal V}$ is $(\sigma H)(u)=H(\sigma^{-1}u).$ Hence, a zeroset $S=V(J)$ is transformed by $\sigma$ into the set $\sigma S=\{\sigma u\mid u\in S\}=V(\sigma J)$. The stabilizer of $S$ is

\begin{equation*}W^S=\{\sigma\in W\mid \sigma S=S\}.\end{equation*}

and the pointwise stabilizer of $S$ is

\begin{equation*}W_S=\{\sigma\in W\mid \sigma u=u,\text{for all }u\in S\}.\end{equation*}

$W^S$ is the maximal subgroup of $W$ acting on $S$. The pointwise stabilizer $W_S$ is a normal subgroup of $W^S$ and the quotient $W^S/W_S$ acts faithfully on $S$.

The union of the reflecting hyperplanes of all reflections in $W$ is called the hyperplane arrangement and is written as $\mathscr A\subset\mathcal V$. The reflecting hyperplanes induce a partition of $\mathcal V$ into subsets, called facets, consisting of those points contained in exactly the same hyperplanes. The set of facets is denoted by $\mathscr C$ and called the complex of $W$. Clearly, facets $C\in \mathscr C$ are open subsets of linear subspaces, hence have the same tangent space everywhere, which may be identified with the closure $\overline C$. Similarly, we often write $C^\bot$ instead of $T_yC^\bot$, for any $y\in C$. The facet $C_0$ containing the origin of $\mathcal V$ is precisely $K$, and it is the only closed facet in the complex $\mathscr C$.

The celebrated Shephard–Todd–Chevalley Theorem characterizes reflection groups as the only finite subgroups of the group of unitary transformations of $\mathcal V$ for which the quotient mapping $\mathcal V\to \mathcal V/W$ can be realized as a polynomial mapping

\begin{equation*}\omega\colon \mathcal V\to {\mathbb C}^p\end{equation*}

whose coordinate functions $\omega_1,\dots,\omega_p$ are homogeneous polynomials (they are a set of generators of the ring of $W$-invariant polynomials). The degrees $d_1,\dots,d_p$ of the coordinate functions $\omega_i$ satisfy the following identities:

• • $\prod_i d_i=\vert W\vert$,

• • $\sum_i (d_i-1)=\#$ reflections in $ W$.

Moreover, it is known that the degrees $d_1,\dots,d_p$ are uniquely determined by $W$ up to ordering. Also, the rank of $W$ is equal to the number of degrees $d_i$ that are greater than one.

Geometrically speaking, the map $\omega$ identifies an orbit to a point, that is, for any $u\in\mathcal V$, we have $\omega^{-1}(\omega(u))=Wu$.

Reflection groups act faithfully, hence $\omega$ is generically $|W|$-to-one, and it is well known that the ramification locus of $\omega$ is the hyperplane arrangement of W. More precisely, given a point $u\in \mathcal V$, contained in a facet $C\in \mathscr C$, we have $\ker \operatorname{d}\!\omega_u=C^\bot$.

By the universal property of quotient mappings, the mapping $\omega$ is well defined up to $\mathcal L$-equivalence (that is, up to changes of coordinates in the target). As it will become clear, replacing $\omega$ by an $\mathcal L$-equivalent map does not change the $\mathcal A$-class of the associated reflection mappings. Since we study reflection mappings up to $\mathcal A$-equivalence, we may regard $\omega$ as being unique, and abusively call it the orbit map of $W$.

Example 2.2. (Products of cyclic groups)

The product ${\mathbb Z}/{d_1}\times\dots\times {\mathbb Z}/{d_p}$ is a reflection group acting on ${\mathbb C}^p$ by

\begin{equation*}(a_1,\dots, a_p)\cdot (u_1,\dots,u_p)=(\xi_1^{a_1}u_1,\dots,\xi_p^{a_p}u_p)\text{, with }\xi_j= e^{\frac{2\pi i}{d_j}}.\end{equation*}

An element $(a_1,\dots, a_p)$ is a reflection if and only if exactly one $a_i$ is nonzero. The orbit map is defined by

\begin{equation*}\omega(u_1,\dots, u_p)=(u_1^{d_1},\dots, u_p^{d_p}).\end{equation*}

Example 2.3. (Dihedral groups, $D_{2n}$)

These are the groups of symmetries of a regular polygon of $n$ sides. We will use the group $D_8$, consisting of the identity, four reflections

\begin{equation*} \sigma_1=\left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right),\quad \sigma_2=\left(\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}\right),\quad \sigma_3=\left(\begin{array}{cc}-1 & 0 \\ 0 & 1\end{array}\right),\quad \sigma_4=\left(\begin{array}{cc}0 & -1 \\ -1 & 0\end{array}\right) \end{equation*}

and three rotations

\begin{equation*} \rho_1=\left(\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right),\quad \rho_2=\left(\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right),\quad \rho_3=\left(\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right). \end{equation*}

The orbit map of $D_8$ is the mapping $\omega\colon {\mathbb C}^2\to{\mathbb C}^2$, given by

\begin{equation*}(u,v)\mapsto(u^2+v^2,u^2v^2).\end{equation*}

Example 2.4. (The group $\mathfrak S_4$ of symmetries of the tetrahedron)

Consider a regular tetrahedron centred at the origin of ${\mathbb R}^3$, for example, the one with vertices

\begin{equation*}V_1=\left(1, 0, \frac{-1}{\sqrt{2}}\right)\quad V_2=\left(-1, 0, \frac{-1}{\sqrt{2}}\right),\quad V_3=\left( 0,1, \frac{1}{\sqrt{2}}\right),\quad V_4=\left(0,-1, \frac{1}{\sqrt{2}}\right).\end{equation*}

The group of unitary automorphisms of the tetrahedron is a reflection group, isomorphic as an abstract group to the permutation group $\mathfrak S_4$ of its four vertices. This group is generated by the permutations $(i\;i+1)$, which in matrix form are

\begin{align*}&(1\;2)=\left(\begin{array}{ccc}-1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right),\; (2\;3)=\frac{1}{2}\left(\begin{array}{ccc}1 & -1 & -\sqrt{2} \\-1 & 1 & -\sqrt{2} \\-\sqrt{2} & -\sqrt{2} & 0\end{array}\right),\\ & (3\;4)=\left(\begin{array}{ccc}1 & 0 & 0 \\0 & -1 & 0 \\0 & 0 & 1\end{array}\right). \end{align*}

The group $\mathfrak S_4$ contains $6$ reflections, which are precisely the permutations $(i\;j)$. The orbit map is

\begin{equation*}\omega(u,v,w)=\big(u^2+v^2+w^2,(u+v)(u-v)w,(2u^2-w^2)(2v^2-w^2)\big).\end{equation*}

2.2. Reflection mappings

Take an embedding $h\colon \mathcal X\hookrightarrow \mathcal V$ of an $n$-dimensional complex manifold $\mathcal X$ into $\mathcal V$. The image of $h$ is an $n$-dimensional complex submanifold

\begin{equation*}\mathcal Y=h(\mathcal X)\subseteq \mathcal V.\end{equation*}

A reflection mapping is a map obtained as the composition of the orbit map $\omega$ and the embedding $h$, that is,

\begin{equation*}f=\omega\circ h\colon \mathcal X\to {\mathbb C}^p.\end{equation*}

To make the involved reflection group explicit, we sometimes say that $f$ is a $W$-reflection mapping.

Later, in Section 6, we discuss how to know that certain map-germs are not reflection map-germs. But, since usually we study map-germs up to $\mathcal A$-equivalence, there is some ambiguity in what it means to be a reflection mapping, and it is convenient to fix our language early on. Temporarily, call the reflection map-germs $f=\omega\circ h$ defined above honest reflection map-germs. We can now make our abuse of notation official:

CAUTION 2.5.

We say that a map-germ $f\colon (X,0)\to ({\mathbb C}^p,0)$ is a $W$-reflection map-germ if it is $\mathcal A$-equivalent to an honest $W$-reflection map-germ, that is, if there exist two biholomorphisms $(X,0)\to (X',0)$ and $({\mathbb C}^p,0)\to ({\mathbb C}^p,0)$, an embedding $h\colon (X',0)\hookrightarrow (\mathcal V,0)$, an action of $W$ as a reflection group on $\mathcal V$ and an orbit mapping $\omega \colon \mathcal V\to{\mathbb C}^p$ for such action, giving rise to a commuting diagram

The same abuse of language applies to global mappings and, if we say that $f$ is a reflection mapping, we just mean that $f$ is a $W$-reflection mapping for some reflection group $W$.

A further source of ambiguity comes from the fact that, when we say that $W$ is a reflection group, we are not specifying the vector space $\mathcal V$ on which $W$ acts and, even for fixed $\mathcal V$, we only know how $W$ acts on $\mathcal V$ up to conjugation by a rigid automorphism of $\mathcal V$. Furthermore, recall that, even if $\mathcal V$ and the action are fixed, the orbit mapping $\omega$ is only determined up to $\mathcal L$ equivalence. The following proposition justifies that one should not worry very much about this.

Proposition 2.6. Let $f\colon ({\mathbb C}^n,0)\to ({\mathbb C}^{p},0)$ be a $W$-reflection map-germ. Fix an action of $W$ on a $p$-dimensional vector space $\mathcal V$ and an orbit mapping $\omega\colon \mathcal V\to {\mathbb C}^p$ for such action. Then, there exists a germ of embedding $h\colon ({\mathbb C}^n,0)\hookrightarrow (\mathcal V,0)$ such that,

\begin{equation*}f\sim_{\mathcal L}\omega\circ h.\end{equation*}

Proof. Consider the diagram

obtained as follows: The top rectangle is given by the $\mathcal A$-equivalence of $f$ with an honest reflection map-germ $\omega'\circ h'$. The vector spaces $\mathcal V'$ and $\mathcal V$ have both dimension $p$, and $\mathcal V'$ comes equipped with a Hermitian form that makes the action of $W$ on $\mathcal V'$ unitary. Then, there is a rigid isomorphism $\mathcal V'\to \mathcal V$ that transforms the action of $W$ into our preferred one. Finally, the remaining vertical arrow $({\mathbb C}^p,0)\to ({\mathbb C}^p,0)$ corresponds to the $\mathcal L$-equivalence between $\omega'$ and the orbit mapping obtained by the composition $\mathcal V'\to \mathcal V\to {\mathbb C}^p$. The result follows by letting $h$ be the composite $({\mathbb C}^n,0)\to ({\mathbb C}^n,0)\hookrightarrow (\mathcal V',0)\to (\mathcal V,0)$,

Clearly, one may safely replace the embedding $h\colon \mathcal X\to \mathcal Y\subseteq \mathcal V$ by the inclusion $\mathcal Y\hookrightarrow \mathcal V$. This way, up to $\mathcal A$-equivalence, every reflection mapping takes the form

\begin{equation*}f=\omega\vert_{\mathcal Y}\colon \mathcal Y\to {\mathbb C}^p.\end{equation*}

Whether we are thinking of a reflection map-germ as a composite $\omega\circ h\colon({\mathbb C}^n,0)\to({\mathbb C}^p,0)$ or as a restriction $\omega\vert_\mathcal Y\colon (\mathcal Y,0)\to ({\mathbb C}^p,0)$ will be clear from the context.

To finish fixing our notation, locally at any point, $\mathcal Y$ is defined in $\mathcal V$ by a collection of regular equations, which we write as

\begin{equation*}L=(L_1,\dots,L_{p-n})=0.\end{equation*}

(technically, it would be better to express our results in terms of the ideal sheaf $I(\mathcal Y)$ of holomorphic functions vanishing on $\mathcal Y$. For the sake of clarity, we have chosen to ignore this issue, which can be fixed by a standard gluing process).

Example 2.7. (Reflected graphs)

Take a reflection group $W$ acting on $\mathcal V$, and a mapping $H\colon \mathcal V\to {\mathbb C}^p$. We may regard $W$ as a reflection group acting on $\mathcal V\times {\mathbb C}^p$, by extending the action trivially on ${\mathbb C}^p$, and take $\mathcal Y\subseteq \mathcal V\times {\mathbb C}^p$ to be the graph of $H$ (or equivalently, take the graph embedding $h\colon \mathcal V\to \mathcal V\times {\mathbb C}^p$). The resulting reflection mapping $\mathcal V\to {\mathbb C}^n\times {\mathbb C}^p$ is called a reflected graph and has the form

\begin{equation*}x\mapsto \big(\omega(x),H(x)\big).\end{equation*}

A typical and much studied example of reflected graphs are folding maps

\begin{equation*}(x,y)\mapsto (x,y^2, H(x,y)),\quad x\in {\mathbb C}^{n-1},y\in {\mathbb C}.\end{equation*}

Observe that, after a target change of coordinates, $H(x,y)$ can be taken to be of the form $yP(x,y^2)$. Similarly, double folds are $\mathbb Z/2\times \mathbb Z/2$-reflected graphs of the form $(x,y)\mapsto (x^2,y^2,xP_1(x^2,y^2)+yP_2(x^2,y^2)+xyP_3(x^2,y^2))$ and $k$-folding mappings are ${\mathbb Z}/k$-reflected graphs $(x,y)\mapsto(x,y^k,H(x,y))$ (See the Introduction for references for folding maps, double folds and $k$-folding mappings).

An interesting example is the $\mathbb Z/k\times \mathbb Z/k$-reflection graph $(x,y)\mapsto (x^k,y^k, xy)$, defined by the graph of the function $H(x,y)=xy$. It parametrizes the $A_{k-1}$-singularity $\{Z^k=XY\}$, but it does so in a generically $k$-to-one way, as explained in Example 2.12 (observe that $A_{k-1}$ being normal prevents it from being parametrized in a generically one-to-one way).

Also, taking the group $D_8$ (Example 2.3) and the functions $H_1(x,y)=x+2y$ and $H_2(x,y)=x^2+xy-y^2+x^3+x^2y-2xy^2-y^3$, we obtain the reflection graphs

\begin{equation*}f^{D_8}_1\colon(x,y)\mapsto (x^2+y^2,x^2y^2,x+2y),\end{equation*}

\begin{equation*}f^{D_8 }_2\colon (x,y)\mapsto (x^2+y^2,x^2y^2,2x^2+3xy-y^2+2x^3+8x^2y-2xy^2-2y^3).\end{equation*}

The last two examples are depicted in Figure 1.

Figure 1.

Construction of the reflected graphs $f_1^{D_8}$ and $f_2^{D_8}$.

Example 2.8. (Reflection mappings with unbounded multiplicity)

The map-germs

\begin{equation*}f^{(d_1,d_2,d_3)}\colon(x,y)\mapsto (x^{d_1}, y^{d_2},(x+y)^{d_3}),\end{equation*}

\begin{equation*}f^{(d_1,\dots,d_5)}\colon(x,y,z)\mapsto (x^{d_1}, y^{d_2},z^{d_3},(x+y+z)^{d_4},(x-y+2z)^{d_5}),\end{equation*}

with $d_i$ pairwise coprime positive integers, belong to a family of map-germs $({\mathbb C}^n,0)\to({\mathbb C}^{2n-1},0)$, introduced in [Reference Peñafort Sanchis22, Theorems 9.5 and 9.6]), which, to this day, are the only known family of $\mathcal A$-finite map-germs whose coordinate functions have unbounded order. These are the germs studied by Brasselet, Ruas, Silva, and Thuy mentioned in the Introduction.

2.3 Unfoldings and $W$-unfoldings

In the theory of singularities of mappings, the notion of deformation of a space-germ is replaced by that of unfolding of a map-germ. An unfolding of a map-germ $f\colon (\mathcal Y,S)\to (\mathcal Z,z)$ is a map-germ

\begin{equation*}F\colon (\mathcal Y,S)\times ({\mathbb C}^r,0)\to (\mathcal Z,z)\times ({\mathbb C}^r,0)\end{equation*}

of the form $F(x,t)=(f_t(x),t)$, satisfying $f_0=f$. Fixed a small representative of $F$ and a value $\epsilon$ of the parameter $t$, the map $f_\epsilon$ is called a perturbation of $f$.

A reflection mapping may be perturbed into a non-reflection mapping, for example, by perturbing $\omega$ in a way unrelated to the action of the group $W$. At the same time, perturbing $\mathcal Y$ while keeping $\omega$ intact gives a family of reflection mappings containing the original reflection mapping defined by $\mathcal Y$. If we want to study unfoldings of reflection mappings without leaving the reflection mapping setting, these are the deformations we want to consider. In the following lines, we formalize this construction.

Let $f\colon \mathcal Y\to {\mathbb C}^p$ be a reflection mapping, and consider a complex submanifold $\widetilde{\mathcal Y}\subseteq \mathcal V\times \Delta$, such that $\widetilde{\mathcal Y}\to \Delta$ defines a trivial deformation of $\mathcal Y$ over an open subset $\Delta\subseteq {\mathbb C}^r$ containing the origin. We may extend the action of $W$ to $\mathcal V\times {\mathbb C}^r$, trivially on the ${\mathbb C}^r$ coordinates, so that the corresponding orbit mapping is $\widetilde\omega=\omega\times {\operatorname{id}}_{{\mathbb C}^r}$.

Definition 2.9. In the above setting, the reflection mapping

\begin{equation*}F=\widetilde\omega_{\vert_{\widetilde{\mathcal Y}}}\colon \widetilde{\mathcal Y}\to {\mathbb C}^p\times \Delta\end{equation*}

is called a $W$-unfolding of $f$. If $f\colon (\mathcal Y,S)\to (\mathcal Z,z)$ is a germ of reflection mapping, a $W$-unfolding of $f$ is the germ at $S\times\{0\}$ of a $W$-unfolding of a representative.

Observe that the triviality condition on the deformation $\widetilde{\mathcal Y}\to \Delta$ ensures that $(\widetilde{\mathcal Y},S\times \{0\})$ and $(\mathcal Y,S)\times (\Delta,0)$ are isomorphic. In particular, every $W$-unfolding of a germ $f$ is, up to $\mathcal A$-equivalence, an unfolding of $f$.

Example 2.10. (A family of tetrahedral reflection mappings)

Let $\mathfrak S_4$ be the group of symmetries of a tetrahedron, as in Example 2.4. Consider the family of reflection mappings

\begin{equation*}f_t^{\mathfrak S_4}=\omega\vert_{\mathcal Y_t}\colon \mathcal Y_t\to {\mathbb C}^3\quad\text{with}\quad\mathcal Y_t=\{u=t(2v+w)\}.\end{equation*}

Equivalently, we may parametrize $\mathcal Y_t$ by $(x,y)\mapsto (t(2x+y),x,y)$ and think of $f_t^{\mathfrak S_4}$ as mappings ${\mathbb C}^2\to {\mathbb C}^3$ of the form

\begin{equation*}\big(x,y\big)\mapsto \Big(t^2(2x+ y)^2+x^2+y^2,\big(t^2(2x+ y)^2-x^2\big)y,\big(2t^2(2x+ y)^2-y^2\big)\big(2x^2-y^2\big)\Big)\end{equation*}

As it turns out, $\mathcal Y_0=\{u=0\}={\operatorname{Fix}}(1\; 2)$ is one of the reflecting hyperplanes of $\mathfrak S_4$. Since $\mathfrak S_4$ acts transitively on its reflecting hyperplanes, we have $W\mathcal Y_0=\mathscr A$, thus

\begin{equation*}{\operatorname{Im}} f_0^{\mathfrak S_4}=\omega(\mathcal Y_0)=\omega(\mathscr A),\end{equation*}

which means that the image of $f_0^{\mathfrak S_4}$ is precisely the discriminant of the orbit map $\omega$.

2.4 The degree of a reflection mapping

Before turning our attention to the image and double point spaces, we show how the degree of a reflection map is encoded by the stabilizers of $\mathcal Y$ in a very simple way. This is essential to our study of the image in the hypersurface case, but it applies to all dimensions.

In this work, when we talk about the degree of a mapping $f\colon \mathcal Y\to \mathcal Z$, we mean the number of preimages of a generic point in $f(\mathcal Y)$. For this to make sense, there must be an open dense subset of $f(\mathcal Y)$ on which this number of preimages is constant. This happens if $f(\mathcal Y)$ is irreducible and $f\colon\mathcal Y\to f(\mathcal Y)$ is proper. The degree of a finite map-germ is defined by taking a proper representative. For mappings where $f(\mathcal Y)$ fails to be irreducible, a different degree is associated to $f$ on each irreducible component of $f(\mathcal Y)$.

Example 2.12. Consider the mapping $f(x,y)=(x^k, y^k, xy)$ of Example 2.7. One sees that the subgroup of ${\mathbb Z}/k\times{\mathbb Z}/k$ preserving $\mathcal Y=\{u_3=u_1u_2\}$ is

\begin{equation*}({\mathbb Z}/k\times{\mathbb Z}/k)^\mathcal Y=\langle(1,-1)\rangle\cong {\mathbb Z}/k,\end{equation*}

while no non trivial element of ${\mathbb Z}/k\times{\mathbb Z}/k$ preserves $\mathcal Y$ pointwise. Therefore, the mapping $f$ is generically $k$-to-one. The mapping $f_0^{\mathfrak S_4}$ of Example 2.10 has

\begin{equation*}{\mathfrak{S}_4}^{\mathcal {Y}_0}=\langle (1\,2) ,(3\,4) \rangle\cong {\mathbb Z}/2\times{\mathbb Z}/2\quad \text{and}\quad {\mathfrak S_4}_{\mathcal {Y}_0}=\langle (1\,2) \rangle\cong {\mathbb Z}/2.\end{equation*}

Therefore, $f_0^{\mathfrak S_4}$ parametrizes the discriminant of $\omega$ in a two-to-one way.

In order to prove Theorems 3.2 and 5.2 below, we use a deformation lemma. Since the proof is slightly involved and the statement is quite clear, we postpone the proof until Appendix A.

2.5 Fitting ideals

It is well known that the image of a finite holomorphic mapping is an analytic set (indeed, it is enough for the mapping to be proper). However, given an unfolding $F=(f_t,t)$, the ideal of the image of $f_0$ need not be the same as the result of computing the ideal of the image of $F$ and replacing $t=0$. This is a problem for the study of singular mappings, where deformations are regarded as an essential part of the theory. Luckily enough, there is a solution consisting on declaring the image of a finite mapping $f\colon \mathcal Y\to \mathcal Z$ to be, rather than just a set, a complex space

\begin{equation*}{\operatorname{Im}} f=V(\mathcal F_0(f_*\mathcal O_{\mathcal Y})),\end{equation*}

where $\mathcal F_0(f_*\mathcal O_{\mathcal Y})$ stands for the $0$th Fitting ideal sheaf of the pushforward module. This sometimes gives ${\operatorname{Im}} f$ a non-reduced analytic structure, but this is the price we pay in order for the analytic structure to behave well under deformations. For map-germs, one uses the $0$th Fitting ideal, written $F_0(f_*\mathcal O_{\mathcal Y})$. For information about Fitting ideals, we refer to [Reference Mond and Pellikaan18]. We only include here the results we need.

Proposition 2.16. Let $f\colon X\to {\mathbb C}^{n+1}$ be a finite mapping defined on a reduced $n$-dimensional Cohen-Macaulay space, and assume the irreducible decomposition of $X$ to be $X_1,\dots,X_m$. Let $g_i$ be a generator of the ideal of $f(X_i)$ in ${\mathbb C}^{n+1}$, and let $d_i$ be the degree of $f$ restricted to $X_i$. Then $F_0(f_*\mathcal O_X)$ is a principal ideal, generated by

\begin{equation*}g= \prod_{i=1}^m g_i^{d_i}.\end{equation*}

The first result follows from [Reference Mond and Pellikaan18, Lemma 1.2], the second is [Reference Mond and Pellikaan18, Proposition 3.2]. The next one is a slight modification, tailored to our needs:

Proposition 2.17. Let $f\colon X\to {\mathbb C}^{n+1}$ be a finite map-germ defined on an $n$-dimensional Cohen-Macaulay space. Let $X_1,\dots,X_r$ be $n$-dimensional Cohen-Macaulay complex spaces, forming a set-theoretical decomposition $X=\cup_{i=1}^rX_i$, where $X_i$ and $X_j$ have no common components if $i\neq j$. Assume each $X_i$ to be isomorphic to $X$ on an open dense subset of $X_i$. Then,

\begin{equation*}F_0(f_*\mathcal O_X)=\prod_{i=1}^r F_0((f\vert_{X_i})_* \mathcal O_{X_i}).\end{equation*}

Proof. Let $\tilde X$ be the disjoint union of the spaces $X_i$ and take the obvious mapping $\tilde f\colon \tilde X\to {\mathbb C}^{n+1}$. Since being Cohen-Macaulay is a local property satisfied by each of the $X_i$, the space $\tilde X$ is Cohen-Macaulay. Since

\begin{equation*}\tilde f_*\mathcal O_{\tilde X}=\bigoplus_{i=1}^r (f\vert_{X_i})_* \mathcal O_{X_i},\end{equation*}

it follows from the construction of Fitting ideals that $F_0(\tilde f_*\mathcal O_{\tilde X})=\Pi_{i=1}^r F_0((f\vert_{X_i})_* \mathcal O_{X_i})$. Since $f$ and $\tilde f$ are the same map on an open dense subset of their targets, it follows that the stalks of $F_0(\tilde f_*\mathcal O_{\tilde X})$ and $F_0(f_*\mathcal O_{X})$ are the same on that open dense subset. But $F_0(\tilde f_*\mathcal O_{\tilde X})$ and $F_0(f_*\mathcal O_{X})$ are principal ideals, hence they must agree.

2.6 Computing the image via elimination of variables

Computing explicit expressions of sections of orbit mappings can be hard and, even if we manage to find them, the formula $\Pi_{\sigma\in W}\,\sigma L\circ s$ becomes too tedious to compute by hand when we take bigger reflection groups. If we try to implement the expression in some computer software, we face the fact that the coordinate functions of sections are non-polynomial, which is a problem when using commutative algebra software such as Singular. Here we introduce two alternative ways of computing the ideal $F_0(f_*\mathcal O_\mathcal Y)$ defining the image, without resorting to sections of $\omega$. As a trade-off, this methods use elimination of variables, which is less explicit, but it is a task computers are happy to do, or at least to try to.

In the case at hand, elimination of variables means what follows: Consider the ring ${\mathbb C}[u,X]$ of polynomials on the variables $u_1,\dots,u_p$ and $X_1,\dots X_p$ (the same works for, say, the ring of germs of holomorphic functions at the origin). Contained in ${\mathbb C}[u,X]$, there is the ideal

\begin{equation*}J=\langle\prod_{\sigma\in W}\sigma L(u)\rangle+\langle X_1-\omega_1(u),\dots,X_p-\omega_p(u)\rangle,\end{equation*}

and the subring ${\mathbb C}[X]$ of polynomials in $X_1,\dots X_p$. Then, $J\cap {\mathbb C}[X]$ is an ideal in ${\mathbb C}[X]$, said to be obtained by eliminating the $u$ variables in $J$. Alternatively, consider the ring homomorphism

\begin{equation*}\omega^*\colon {\mathbb C}[X]\to {\mathbb C}[u],\end{equation*}

given by $H\mapsto H\circ \omega$. Then, the elimination can be expressed as

\begin{equation*}J\cap {\mathbb C}[X]=(\omega^*)^{-1}\langle\prod_{\sigma\in W}\sigma L(u)\rangle.\end{equation*}

Proposition 3.9. For any reflection mapping $f\colon \mathcal Y^n\to{\mathbb C}^{n+1}$, ${\operatorname{Im}} f$ is the zero locus of

\begin{equation*}F_0(f_*\mathcal O_\mathcal Y)=(\omega^*)^{-1}\langle \prod_{\sigma\in W}\sigma L \rangle=\left((\omega^*)^{-1}\langle L \rangle\right)^{|W^\mathcal Y:W_\mathcal Y|}.\end{equation*}

Proof. The first equality was shown in the proof of Theorem 3.2. For the second equality, let

\begin{equation*}K=\langle L(u)\rangle+\langle X_1-\omega_1(u),\dots,X_p-\omega_p(u)\rangle,\end{equation*}

Since $V(K)\cong V(L)=\mathcal Y$, it follows that $K$ is radical, hence $K\cap {\mathbb C}[X]$ is radical. This means that $K\cap {\mathbb C}[X]$ is the (radical) ideal of the image of $f$. At the same time, we know by Proposition 2.11 that the degree of $f$ is $|W^\mathcal Y:W_\mathcal Y|$. Putting both things together, the equality $F_0(f_*\mathcal O_\mathcal Y)=(K\cap {\mathbb C}[X])^{|W^\mathcal Y:W_\mathcal Y|}$ is a direct application of Proposition 2.16.

In practice, our experience shows that computing $(\omega^*)^{-1}\langle \Pi_{\sigma\in W}\sigma L \rangle$ is faster than computing $(\omega^*)^{-1}\langle L \rangle$, probably due to the fact that $\Pi_{\sigma\in W}\sigma L$ is a $W$-invariant function.

Example 3.10. Consider the family of $\mathfrak{S_4}$-reflection mappings of Example 2.10. Sections of the orbit mapping of $\mathfrak{S_4}$ have horrendous expressions, so computing ${\operatorname{Im}} f$ by means of the expression $\Pi_{\sigma\in W}\,\sigma L\circ s$ of Theorem 3.2 is not convenient. In contrast, Singular computes $(\omega^*)^{-1}\langle \Pi_{\sigma\in W}\sigma L \rangle$ in no time. The equation of the image of the unfolding $F=(f_t,t)$ is too long to write here but, for $t=0$ and $t=1$, we obtain the following equations:

\begin{equation*}{\operatorname{Im}} f^{\mathfrak{S_4}}_0=V(g),\quad g=(2 x^3 y^2+ x^4 z-27 y^4-18 x y^2 z-2 x^2 z^2+z^3)^2, \end{equation*}

\begin{align*} {\operatorname{Im}} f^{\mathfrak{S_4}}_1&=V(g),\quad g=614656x^{12}-174822592x^9y^2-16020256x^{10}z\\ &\quad+10356692964x^6y^4+800288220x^7y^2z+153738321x^8z^2\\ &\quad-198333009364x^3y^6-33901243950x^4y^4z-662345364x^5y^2 z^2\\ &\quad-685828516 x^6 z^3+1202174306137 y^8 +372758486548 x y^6 z\\ &\quad+7876328208 x^2 y^4 z^2-1163406956 x^3 y^2 z^3+1546928326 x^4 z^4 \\ &\quad +40000919994 y^4 z^3+1284226020 x y^2 z^4-1713759300 x^2 z^5\\ &\quad+741200625 z^6. \end{align*}

The hypersurces ${\operatorname{Im}} f^{\mathfrak{S_4}}_0$ and ${\operatorname{Im}} f^{\mathfrak{S_4}}_1$ are depicted in Figure 2. Recall that ${\operatorname{Im}} f^{\mathfrak{S_4}}_0$ is the discriminant of $\omega$, with non-reduced structure, and that the exponent $2$ in the equation reflects the fact that $f^{\mathfrak{S_4}}_0$ has degree two (see Example 2.12).

Figure 2.

The images of $f_0^{\mathfrak S_4}$ and $f_1^{\mathfrak S_4}$.

4.1. Decomposition of the double point space $K_2(f)$

Here we study, in the case of reflection mappings, the double point space $K^2(f)$ studied by Ronga [Reference Ronga24] and Kleiman [Reference Kleiman10]. We skip many details in the description of this double point space for general mappings, for which we refer to [Reference Nuño-Ballesteros and Peñafort Sanchis20, Reference Ronga24]. Let $\mathcal Y$ be an $n$-dimensional complex manifold and assume that it admits a global coordinate system (This is for the sake of simplicity, a standard gluing process shows the results in this section to be valid in general). We write the blowup of the product of two copies of $\mathcal Y$ along the diagonal as

\begin{equation*}B_2(\mathcal Y)=\operatorname{Bl}_{\Delta\mathcal Y}(\mathcal Y\times \mathcal Y)=\{(u,u',v)\in \mathcal Y\times \mathcal Y\times \mathbb P^{n-1}\mid v\wedge(u'-u)=0\},\end{equation*}

where $v\wedge(u'-u)=0$ is a shortcut to indicate the vanishing of all $2\times 2$ minors of the matrix

\begin{equation*} \left(\begin{array}{ccc} u'_1-u_1 & \dots & u'_n-u'_n \\ v_1 & \dots & v_n \end{array}\right). \end{equation*}

We write the exceptional divisor as $E=\Delta\mathcal Y\times \mathbb P^{n-1}$.

Now let $f\colon \mathcal Y\to \mathcal Z$ be a holomorphic mapping between manifolds (not necessarily a reflection mapping) of dimensions $n$ and $p$, both admitting a global coordinate system. Think of $u'-u$ and $f(u')-f(u)$ as a vectors with entries in $\mathcal O_{\mathcal Y\times \mathcal Y}$, of sizes $n$ and $p$, respectively. By Hilbert Nullstellensatz, there exist a $p\times n$ matrix $\alpha_f$, with entries in $\mathcal O_{\mathcal Y\times \mathcal Y}$, such that

\begin{equation*}f(u')-f(u)=\alpha_f\cdot (u'-u)\end{equation*}

Then, Kleiman’s double point space is

\begin{equation*}K_2(f)=\{(u,u',v)\in B_2(\mathcal Y)\mid \alpha_f(u,u')\cdot v=0\}.\end{equation*}

The proof that $K_2(f)$ does not depend on $\alpha$ is in [Reference Mond16, Proposition 3.1]. Away from the exceptional divisor, $K_2(f)$ is just the fibred product

\begin{equation*}K_2(f)\setminus E \cong (\mathcal Y\times_{\mathcal Z}\mathcal Y)\setminus \Delta\mathcal Y,\end{equation*}

via the blowup map. On the exceptional divisor, $K_2(f)$ keeps track of the kernel of the differential of $f$, as follows:

\begin{equation*}K_2(f)\cap E=\{(u,u,v)\in B_2(\mathcal Y)\mid v\in \ker \operatorname{d}\!f_u\}.\end{equation*}

The double point spaces $K_2$ enjoy the following key functorial property, very much related to the construction of reflection mappings. First, any embedding of complex manifolds, $\mathcal Y\hookrightarrow \mathcal V$, induces an embedding $B^2(\mathcal Y)\hookrightarrow B^2(\mathcal V)$.

Proposition 4.2. ([Reference Nuño-Ballesteros and Peñafort Sanchis20, Theorem 2.13])

Given a mapping $F\colon \mathcal V\to \mathcal Z$ between complex manifolds, the double point space of the restriction $f=F_{\vert_\mathcal Y}\colon \mathcal Y\to \mathcal Z$ is

\begin{equation*}K_2(f)=K_2(F)\cap B_2(\mathcal Y).\end{equation*}

In particular, the double points of reflection mappings are slices of the double points of orbit mappings. Conveniently enough, we understand $K_2(\omega)$ quite well. Fixed $\sigma \in W\setminus\{{\operatorname{id}}\}$, let $r={\operatorname{dim}} {\operatorname{Fix}} \sigma$ and choose linear mappings $\ell_\sigma\colon \mathcal V\to {\mathbb C}^{p-r}$ and $\ell_\sigma^\bot\colon \mathcal V\to {\mathbb C}^{r}$, so that

\begin{equation*}{\operatorname{Fix}} \sigma=V(\ell_\sigma)\qquad\text{and}\qquad({\operatorname{Fix}} \sigma)^\bot=V(\ell_\sigma^\bot).\end{equation*}

The following description of $K^2(\omega)$ is found (with different notation) in [Reference Peñafort Sanchis22, Theorem 7.6]:

Proposition 4.3. The double point space $K_2(\omega)$ is a $p$-dimensional reduced locally complete intersection, with irreducible decomposition

\begin{equation*}K_2(\omega)=\bigcup_{\sigma\in W\setminus\{1\}}K_2^\sigma,\end{equation*}

where $K_2^\sigma$ is the blow-up of $\{(u,\sigma u)\mid u\in \mathcal V\}$ along $\Delta {\operatorname{Fix}} \sigma$, embedded in $B_2(\mathcal V)$ as

\begin{equation*}K_2^\sigma=\{(u,\sigma u,v)\in B^2(\mathcal V)\mid \ell_\sigma^\bot(v)=0\}.\end{equation*}

In view of Proposition 4.2, the space $K_2$ of reflection mappings must inherit a $W\setminus\{1\}$-indexed decomposition:

As we just mentioned, this gives the set-theoretical decomposition

\begin{equation*}K_2(f)=\bigcup_{\sigma\in W\setminus\{1\}}K_2^\sigma(f).\end{equation*}

Observe that neither $K_2(f)$ nor $K_2^\sigma(f)$ need to be reduced, so there is no natural way to upgrade the previous decomposition into a union of complex spaces, because (to the best of the authors’ knowledge) the meaning of a union in the category of complex spaces is unclear. In any case, since the order of $W$ is finite, $K_2(\omega)$ and $K_2^\sigma$ are locally equal topological spaces at points $z\in K_2^\sigma$ not contained in any other $K_2^\tau,\tau \neq \sigma$. Moreover, since both spaces are reduced, $K_2(\omega)$ and $K_2^\sigma$ are locally isomorphic complex spaces at such points. This means that the comparison of analytic structures in the above equality is only problematic at points where different branches $K_2^\sigma(f)$ and $K_2^\tau(f)$ meet. To be precise, we have observed what follows:

Now we want to refine our description of $K_2^\sigma(f)$. Again, we may think of $L$ and $\sigma L$ as vectors with entries in $\mathcal O_{\mathcal V}$. Then, there is a $(p-n)\times (p-r)$ matrix $\alpha_L^\sigma$, with entries in $\mathcal O_{\mathcal V}$, such that

\begin{equation*}\sigma^{-1}L-L=\alpha_L^\sigma \cdot \ell_\sigma.\end{equation*}

Also, consider the involution $\iota\colon \mathcal V\times \mathcal V\to \mathcal V\times \mathcal V$, given by $(u,u')\mapsto (u',u)$. This mapping lifts to an involution $\tilde \iota \colon B^2(\mathcal V)\to B^2(\mathcal V)$ taking $K_2(f)$ onto itself.

Proposition 4.6. With the above notations, the branch $K^\sigma_2(f)$ is the complex space

\begin{equation*}K_2^\sigma(f)=\{(u,\sigma u,v)\in K^\sigma_2 \mid L(u)=0, \alpha_L^\sigma(u)\cdot \ell_\sigma(v)=0\}\end{equation*}

and it has the following properties:

1. (1) The dimension of $K_2^\sigma(f)$ is at least $2n-p$. If $ K_2^\sigma(f)$ has dimension $2n-p$, then it is locally a complete intersection.

2. (2) The blowup mapping $B_2(\mathcal Y)\to \mathcal Y\times \mathcal Y$ takes $K_2^\sigma(f)\setminus E$ isomorphically to the space

   \begin{equation*}\{(u,\sigma u)\mid u\in (\mathcal Y\cap \sigma^{-1}\mathcal Y)\setminus{\operatorname{Fix}}\sigma\}.\end{equation*}

3. (3) As sets,

   \begin{equation*}K_2^\sigma(f)\cap E=\{(u,u,v)\mid u\in \mathcal Y\cap {\operatorname{Fix}} \sigma, v\in \mathbb P(T_uY\cap {\operatorname{Fix}} \sigma^\bot)\}.\end{equation*}

4. (4) The involution $\tilde \iota$ takes $K_2^\sigma(f)$ isomorphically to $K_2^{(\sigma^{-1})}(f)$.

Proof. The space in question is defined as $K_2^\sigma(f)=B^2(\mathcal Y)\cap K_2^\sigma$. Consider the $(p-n)\times p$ matrix $\alpha_L$, with entries in $\mathcal O_{\mathcal V\times \mathcal V}$, satisfying $L(u')-L(u)=\alpha_L(u,u')\cdot(u'-u)$. This allows to describe $B_2(\mathcal Y)$ as the closed submanifold of $B_2(\mathcal V)$

\begin{equation*}B_2(\mathcal Y)=\{(u,u',v)\in B_2(\mathcal V)\mid L(u)=0,\alpha_L(u,u')\cdot v=0\}.\end{equation*}

To see this, it suffices to check that the space on the right-hand side of the equality is the strict transform of $\mathcal Y\times \mathcal Y$ in $B_2(\mathcal V)$. This in turn is true because this space has dimension $2n$, making it a complete intersection by counting equations, and it is isomorphic to $\mathcal Y\times \mathcal Y$ away from $E$, while the equations prevent it from having irreducible components contained in $E$. Now, to finish the description of $K_2^\sigma(f)$ in the statement, it suffices to show that, for points $(u,\sigma u,v)\in K_2^\sigma$, the conditions $\alpha_L(u,\sigma u)\cdot v=0$ and $\alpha_L^\sigma(u)\cdot \ell_\sigma(v)=0$ are equivalent. For simplicity, we may identify $\mathcal V={\operatorname{Fix}} \sigma^\bot\oplus{\operatorname{Fix}} \sigma$, so that any vector is of the form $u=(u_1,u_2)$, with $\ell_\sigma(u)=u_1$ and $\ell_\sigma^\bot(u)=u_2$. Any point $(u,\sigma u,v)\in K_2^\sigma$ satisfies $\ell_\sigma^\bot(v)=0$, which means that the equation $\alpha_L(u,\sigma u)\cdot v=0$ has the form

\begin{equation*} \left(\begin{array}{@{}c|c@{}} A(u,\sigma u)& B(u,\sigma u) \end{array}\right) \left(\begin{array}{c} v_1 \\\hline 0 \end{array}\right)=0. \end{equation*}

Hence $A(u,\sigma u).l_\sigma=\sigma^{-1}L-L$ and we may take $\alpha_L^\sigma(u)=A(u,\sigma u)$.

Item (1) is an easy equation counting: $K_2^\sigma$ is a $p$-dimensional manifold, on which we impose $2(p-n)$ equations. To see item (2), observe that $(u,\sigma u)$ is contained in $E$ if and only if $u\in Fix \sigma$, and that away from $E$, the equations $\alpha_L^\sigma(u)\cdot \ell_\sigma(v)=0$ and $L(u')-L(u)=0$ are equivalent. Item (3) follows from the equalities $ B^2(\mathcal Y)\cap E=\{(u,u,v)\mid u\in \mathcal Y, v\in \mathbb PT_u\mathcal Y\}$ and $K_2^\sigma\cap E=\{(u,u,v)\in B^2(\mathcal V)\mid u\in {\operatorname{Fix}}\sigma,v\in {\operatorname{Fix}}\sigma^\bot\}$. For item (4), we already know that $\tilde \iota$ is an automorphism on $K_2(f)$ and, since it has the form $(u,\sigma u,v)\mapsto(\sigma u, u, v)$, it must take $K_2^\sigma(f)$ to $K_2^{(\sigma^{-1})}(f)$. This must happen isomorphically, since otherwise $\tilde \iota$ would fail to be an automorphism.

Example 4.7. Consider the map-germ

\begin{equation*}f^{(d_1,\dots,d_5)}\colon(x,y,z)\mapsto (x^{d_1}, y^{d_2},z^{d_3},(x+y+z)^{d_4},(x-y+2z)^{d_5}),\end{equation*}

from Example 2.8. The equations of $\mathcal Y$ are $L=(u_1+u_2+u_3-u_4,u_1-u_2+2u_3-u_5)=0$. To illustrate the calculation of $K_2^\sigma(f)$, take, for instance, the elements of the form $\sigma_i=(a_1,\dots,a_i,0,\dots,0)$, $i=1,\dots,5$, with $0 \lt a_i \lt d_i$ (see Example 2.2 for notation). The pointwise fixed spaces are given by the vanishing of $\ell_{\sigma_i}(u)=(u_1,\dots,u_i)$. From the equations $\sigma^{-1}L-L=\alpha_L^\sigma \cdot \ell_\sigma$ one concludes that $\alpha_L^{\sigma_i}$ is the matrix defined by the first $i$ columns of

\begin{equation*}\alpha_L^{\sigma_5}= \left(\begin{array}{ccccc}\xi_1^{a_1}-1&\xi_2^{a_2}-1&\xi_3^{a_3}-1&-1(\xi_4^{a_4}-1)&0 \\ \xi_1^{a_1}-1&-1(\xi_2^{a_2}-1)&2(\xi_3^{a_3}-1)&0&-1(\xi_5^{a_5}-1) \end{array}\right) \end{equation*}

Since, for $i=1,2$, $\alpha_L^{\sigma_i}$ is injective, the equation $\alpha_L^{\sigma_i} \cdot \ell_{\sigma_i}(v)=0$ forces $\ell_{\sigma_i}(v)=0$. This, added to the condition $\ell_{\sigma_i}^\bot(v)=0$ for points in $(u,\sigma u,v)\in K_2^{\sigma_i}$, forces $v=0$ and, since a point $v\in \mathbb P^{p-1}$ cannot have all its coordinates equal to zero, we conclude $K_2^{\sigma_i}(f)=\emptyset$. $K_2^{\sigma_3}(f)$ is a line lying on the exceptional divisor, and it is a complete intersection. $K_2^{\sigma_4}(f)$ is a line lying on the exceptional divisor. $K_2^{\sigma_5}(f)$ is the union of a line and a 2-dimensional component lying on the exceptional divisor. This branch is not a complete intersection (indeed, it is not even Cohen Macaulay).

4.2 Decomposition of the double point space $D^2(f)$

Now we study a different double point space. As a set, $D^2(f)\subseteq \mathcal V\times\mathcal V$ is just the projection of $K_2(f)$. As a complex space, this space was introduced in [Reference Mond16]. The results for which we give no proof can be found in [Reference Nuño-Ballesteros and Peñafort Sanchis19]. Let $f\colon \mathcal Y\to \mathcal Z$ be a holomorphic mapping between complex manifolds, as in the previous section. We define

\begin{equation*}D^2(f)=V((f\times f)^*(I_\Delta)+I_n(\alpha_f)),\end{equation*}

where $I_\Delta$ stands for ideal of the diagonal $\Delta \mathcal Z$ and $I_n(\alpha_f)$ is the ideal generated by the $n\times n$ minors of $\alpha_f$ (see [Reference Mond16, Proposition 3.1] for the independence of $\alpha_f$). As it is the case with $K_2(f)$, the space $D^2(f)$ has a nice analytic structure whenever it has the right dimension.

As we mentioned, set theoretically, $D^2(f)$ is the projection of $K_2(f)$, that is,

\begin{equation*}D^2(f)=\{(u,u')\in \mathcal Y\times \mathcal Y\mid u'\neq u,f(u')=f(u)\}\cup\{(u,u)\in \Delta X\mid \operatorname{rk}(\operatorname{d}\!f_u) \lt n\}.\end{equation*}

Going back to the reflection mapping setting, by letting $D_2^\sigma(f)$ be the image of $K^\sigma_2(f)$ in $\mathcal Y\times \mathcal Y$, we obtain a set-theoretical decomposition

\begin{equation*}D^2(f)=\bigcup_{\sigma\in W\setminus\{1\}}D_2^\sigma(f),\end{equation*}

and clearly $D_2^\sigma(f)=\{(u,\sigma u)\in\mathcal V\times\mathcal V\mid L(u)=L(\sigma u)=0, \ker \alpha_L^\sigma(u)\neq 0\}.$ We improve this description by giving $D^\sigma_2(f)$ an appropriate analytic structure by means of explicit equations.

Definition 4.9. Let $\sigma\in W\setminus\{1\}$ and $r={\operatorname{dim}} ({\operatorname{Fix}} \sigma)$. If $r\geqslant n$, then let

\begin{equation*}D_2^\sigma(f)=\{(u,\sigma u)\in\mathcal V\times\mathcal V\mid L(u)=L(\sigma u)=0, I_{p-r}(\alpha_L^\sigma(u))=0\},\end{equation*}

where $I_{p-r}(\alpha_L^\sigma)$ are the $(p-r)\times (p-r)$ minors of $\alpha_L^\sigma$. If $r \lt n$, then let

\begin{equation*}D_2^\sigma(f)=\{(u,\sigma u)\in\mathcal V\times\mathcal V\mid L(u)=L(\sigma u)=0\}.\end{equation*}

The proof that this definition is independent of the matrix $\alpha_L^\sigma$ is, mutatis mutandis, the one of [Reference Mond16, Proposition 3.1].

Proposition 4.10. For every $\sigma\in W\setminus\{1\}$, the complex space $D_2^\sigma(f)$ defined above is, as a set, the image of $K_2^\sigma(f)$ on $\mathcal Y\times \mathcal Y$, and it has the following properties:

1. (1) The dimension of $D_2^\sigma(f)$ is at least $2n-p$. If ${\operatorname{dim}} D_2^\sigma(f)=2n-p$, then $D_2^\sigma(f)$ is Cohen-Macaulay. If furthermore $\sigma$ is a reflection or ${\operatorname{dim}} {\operatorname{Fix}} \sigma \lt n$, then $D_2^\sigma(f)$ is locally a complete intersection.

2. (2) Away from $\Delta \mathcal Y$, we have an equality of complex spaces

   \begin{equation*}D_2^\sigma(f)\setminus \Delta \mathcal Y=\{(u,\sigma u)\mid u\in (\mathcal Y\cap \sigma^{-1}\mathcal Y)\setminus{\operatorname{Fix}}\sigma\}.\end{equation*}

3. (3) As sets,

   \begin{equation*}D_2^\sigma(f)\cap \Delta \mathcal Y=\{(u, u)\mid u\in \mathcal Y\cap {\operatorname{Fix}} \sigma, T_u\mathcal Y\cap {\operatorname{Fix}}\sigma^\bot\neq 0\}.\end{equation*}

4. (4) Let $(u,\sigma u)\in D_2^\sigma(f)$, assume that $u\notin {\operatorname{Fix}}\sigma$ and that $(u,\sigma u)\notin D_2^\tau(f)$, for all $\tau\neq \sigma$. Then, $D^2(f)$ and $D_2^\sigma(f)$ are locally isomorphic at $(u,\sigma u)$.

5. (5) The involution $ \iota$ takes $D_2^\sigma(f)$ isomorphically to $D_2^{(\sigma^{-1})}(f)$.

Proof. We know that the image of $K_2^\sigma(f)$ in $\mathcal Y\times \mathcal Y$ is the set of pairs $(u,\sigma u)\in\mathcal Y\times\mathcal Y$, such that $\ker \alpha_L^\sigma(u)\neq 0$. Letting $r={\operatorname{dim}} {\operatorname{Fix}} \sigma$, the matrix $\alpha_L^\sigma$ has size $(p-n)\times (p-r)$. Therefore, $\alpha_L^\sigma$ must have non-trivial kernel automatically if $r \lt n$. Whenever $r\geqslant n$, the conditions $\ker \alpha_L^\sigma(u)\neq 0$ and $I_{p-r}(\alpha_L^\sigma(u))=0$ are equivalent. This shows $D_2^\sigma(f)$ to be the image of $K_2^\sigma(f)$, as a set.

Now we prove item (1). If ${\operatorname{dim}} {\operatorname{Fix}} \sigma \lt n$ then $D_2^\sigma(f)$ is defined by the $2(p-n)$ equations $L(u)=L(\sigma u)=0$ on the $p$-dimensional manifold $\{(u,\sigma u)\mid u\in \mathcal V\}$, hence it is a complete intersection if it has dimension $2n-p$. If $\sigma$ is a reflection, then $\ell_\sigma$ consists of a single entry and, then, we may compute $\alpha_L^\sigma$ just by dividing the entries of $L$ by $\ell_\sigma$, that is,

\begin{equation*} \alpha_L^\sigma=\left(\frac{\sigma^{-1}L_1-L_1}{\ell_\sigma},\dots, \frac{\sigma^{-1}L_{p-n}-L_{p-n}}{\ell_\sigma}\right) \end{equation*}

Now $I_{p-r}(\alpha_L^\sigma)$ is generated by the entries of $\alpha_L^\sigma$. In this case, modulo the entries of $L$, the entries of $\sigma^{-1}L$ are generated by the entries of $I_{p-r}(\alpha_L^\sigma)$, and can be discarded as generators of the ideal of $D_2^\sigma$. Just as in the case of ${\operatorname{dim}} {\operatorname{Fix}} \sigma \lt n$, now $D_2^\sigma(f)$ is defined by $2(p-n)$ equations, and it is a complete intersection if it has dimension $2n-p$. If ${\operatorname{dim}} {\operatorname{Fix}} \sigma\geqslant n$, the fact that $D_2^\sigma(f)$ is a Cohen-Macaulay space whenever it has dimension $2n-p$ is a direct application of [Reference de Concini and Strickland5, Thm 2.7, Lemma 2.3].

Item (2) is obvious if $r \lt n$. In the case of $r\geqslant n$ we need to show that, locally on a point $(u,\sigma u)\in D_2^\sigma(f)\setminus \Delta\mathcal Y$, the $p-r$ minors of $\alpha_L^\sigma$ are in the ideal generated by the entries of $L$ and $\sigma^{-1}L$. Let $A$ be the square submatrix obtained by picking the rows $j_1,\dots,j_{p-r}$ of $\alpha_L^\sigma$. Let $b$ be the vector with entries $\sigma^{-1}L_{j_1}-L_{j_1},\dots,\sigma^{-1}L_{j_{p-r}}-L_{j_{p-r}}$. Since $u\notin{\operatorname{Fix}}\sigma$, there must be some $i$ such that $\ell_{i}(u)\neq 0$. Let $A'$ be the matrix obtained by substitution of the $i$-th column of $A$ by $b$. By Cramer’s Rule, we obtain $|A| = |A'|/\ell_i(u)$. The claim follows from the fact that $|A'|$ is clearly in the ideal generated by the entries of $L$ and $\sigma^{-1}L$.

Item (3) follows directly from Item (3) of Proposition 4.6.

Observe that item (2) can be restated as the fact that $D_2^\sigma(f)\setminus \Delta \mathcal Y$ and $K_2^\sigma(f)\setminus E$ are isomorphic. At the same time, $D^2(f)\setminus \Delta \mathcal Y$ and $K_2(f)\setminus E$ are isomorphic (this is true because both spaces are isomorphic to $(\mathcal Y\times_{\mathcal V}\mathcal Y)\setminus \Delta\mathcal Y$. This is easy to see from the equations of $D^2(f)$ and from the definition of $K_2(f)$ given in [Reference Nuño-Ballesteros and Peñafort Sanchis20]). From this point of view, item (4) is the same as item (2) from Proposition 4.6. Finally, item (5) is proven in the same way as item (4) from Proposition 4.6.

Example 4.11. Consider the map-germ

\begin{equation*}f^{(d_1,\dots,d_5)}\colon(x,y,z)\mapsto (x^{d_1}, y^{d_2},z^{d_3},(x+y+z)^{d_4},(x-y+2z)^{d_5}),\end{equation*}

from Example 4.7. With the notations that we used there, we obtain $D_2^{\sigma_1}(f)=D_2^{\sigma_2}(f)=\emptyset$ because, in both cases, $\alpha_{L}^{\sigma_i}$ has maximal rank. On the other hand, for $i=3,4,5$, ${\operatorname{dim}} Fix \sigma_i \lt 3$, therefore

\begin{equation*}D_2^{\sigma_i}(f)=\{(y,\sigma_i y)\mid L(y)=L(\sigma _iy)=0\}.\end{equation*}

These spaces must be lines, because their equations are linear and we know a priori that $f^{(d_1,\dots,d_5)}$ is $\mathcal A$-finite [Reference Peñafort Sanchis22, Lemma 9.10 and Proposition 9.8], which forces the double point space to be a reduced curve.

4.3 Decomposition of the double point space $D(f)$

If $f\colon \mathcal Y\to \mathcal Z$ is a finite holomorphic mapping between complex manifolds (not necessarily a reflection mapping), then the mapping $\pi\colon D^2(f)\to \mathcal Y$, given by $(y,y')\mapsto y$, is finite. The source double point space $D(f)$ is defined as the image of $\pi$, that is,

\begin{equation*}D(f)=V(\mathcal F_0(\pi_*\mathcal O_{D^2(f)})).\end{equation*}

As a set,

\begin{equation*}D(f)=\{u\in \mathcal Y\mid \vert f^{-1}(f(u))\vert \gt 1\}\cup\{u\in \mathcal Y\mid \operatorname{rk}(\operatorname{d}\!f_u) \lt {\operatorname{dim}} \mathcal Y\}.\end{equation*}

Now come back to the reflection mapping setting. Since we have the set-theoretical decomposition $D^2(f)=\cup_{\sigma}D_2^\sigma(f)$, then the set $D(f)=\pi(D^2(f))$ must be the union of the sets $\pi(D_2^\sigma(f))$.

We obtain the set-theoretical decomposition

\begin{equation*}D(f)=\bigcup_{\sigma\in W\setminus\{1\}}D_\sigma(f).\end{equation*}

Since the equations $u'=\sigma u$ are among the equations defining $D_2^\sigma(f)$, the spaces $\pi(D_2^\sigma(f))$ and $D_2^\sigma(f)$ isomorphic. Hence the spaces $D_\sigma(f)$ may be described equivalently as follows: Let $r={\operatorname{dim}} {\operatorname{Fix}} \sigma$. If $r\geqslant n$, then

\begin{equation*}D_\sigma(f)=\{u\in\mathcal Y\cap\mathcal \sigma^{-1} \mathcal Y\mid I_{p-r}(\alpha_L^\sigma(u))=0\}.\end{equation*}

If $r \lt n$, then

\begin{equation*}D_\sigma(f)=\mathcal Y\cap\mathcal \sigma^{-1} \mathcal Y.\end{equation*}

As the spaces $D_2^\sigma(f)$ and $D_\sigma(f)$ are isomorphic, Proposition 4.10 can be recast as a result about $D_\sigma(f)$.

If there is no risk of confusion, it is common to write $D$ instead of $D(f)$. Similarly, we may sometimes write $D_\sigma$ for $D_\sigma(f)$. This notation appears, for example, in Example 5.3.

5.1 The double point curve $D(f)$ of a reflection mapping $\mathcal Y^2\to {\mathbb C}^3$

In these dimensions, the double point spaces are plane curves when they have the right dimension. Moreover, a germ $f\colon ({\mathbb C}^2,0)\to({\mathbb C}^3,0)$ is $\mathcal A$-finite if and only if $D(f)$ is a reduced plane curve [Reference Marar, Nuño Ballesteros and Peñafort Sanchis13]. For these mappings, one is interested in computing the Milnor number $\mu(D(f))$ and the delta invariant $\delta(D(f))$. This is much easier for reflection mappings than it is for arbitrary mappings, thanks to the following criterion, which follows easily by putting together Hironaka’s $\mu=2\delta-r+1$ formula, the additivity of $\delta$, and the decomposition $\lambda=\Pi_{\sigma\neq 1}\lambda_\sigma$ of Theorem 5.2.

Proposition 5.6. A reflection mapping $f\colon (\mathcal Y^2,y)\to(\mathbb C^3,0)$ is $\mathcal A$-finite if and only if all the Milnor numbers $\mu(D_\sigma(f))$ and all the intersection numbers $D_\sigma(f)\cdot D_\tau(f)$, with $\sigma\neq \tau$, are finite. In this case,

\begin{equation*}\delta(D(f))=\sum_{\sigma\in W\setminus\{1\}}\Big(\delta(D_\sigma(f))+\frac{1}{2}\sum_{\tau\in W\setminus\{1,\sigma\}}D_\sigma(f)\cdot D_\tau(f)\Big),\end{equation*}

and

\begin{equation*}\mu(D(f))=\sum_{\sigma\in W\setminus\{1\}}\Big(\mu(D_\sigma(f))+\sum_{\tau\in W\setminus\{1,\sigma\}}D_\sigma(f)\cdot D_\tau(f)\Big)+k-|W|+2,\end{equation*}

where $D_{\sigma_i}(f)\cdot D_{\sigma_j}(f)={\operatorname{dim}}_{\mathbb C}(\mathcal O_\mathcal{Y}/\left\langle \lambda_{\sigma_i},\lambda_{\sigma_j}\right\rangle)$ is the intersection number of the branches $D_{\sigma_i}(f)$ and $D_{\sigma_j}(f)$ and $k$ is the number of elements $\sigma$ for which $D_\sigma(f)=\emptyset$ (We follow the convention that $\delta(\emptyset)=\mu(\emptyset)=0$).

The data of the $\mu(D_\sigma(f))$, $\delta(D_\sigma(f))$, $D_{\sigma_i}(f)\cdot D_{\sigma_j}$ and whether a branch $D_\sigma(f)$ is empty can be stored easily in matrix form. Here is a convenient way to do it: First choose an ordering of the reflection group $W=\{\sigma_0,\sigma_1,\dots,\sigma_{|W|-1}\},$ where $\sigma_0=1$. Let $I$ be the $(|W|-1)\times(|W|-1)$ matrix with entries

\begin{equation*} I_{ij}=\left\{\begin{matrix} 0,& \mbox{if } i=j \mbox{and } D_{\sigma_i}(f)\neq\emptyset,\\ 1, &\mbox{if } i=j \mbox{and } D_{\sigma_i}(f)=\emptyset,\\ D_{\sigma_i}(f)\cdot D_{\sigma_j}(f),&\mbox{if }i\neq j\text{and }D_{\sigma_i}(f)\cdot D_{\sigma_j}(f) \lt \infty,\\ -1,&\mbox{if } i\neq j\text{and }D_{\sigma_i}(f)\cdot D_{\sigma_j}(f)=\infty, \end{matrix}\right. \end{equation*}

Let $M$ be the size $(|W|-1)$ vector with entries

\begin{equation*} M_i=\left\{\begin{matrix} 0,& \mbox{if } D_{\sigma_i}(f)=\emptyset,\\ \mu(D_{\sigma_i}(f))&\mbox{if } D_{\sigma_i}(f) \mbox{is a reduced curve}\\ -1, &\mbox{otherwise}\\ \end{matrix}\right. \end{equation*}

and define the vector $\Delta$ analogously, but replacing every $\mu(D_{\sigma_i}(f))$ with $\delta(D_{\sigma_i}(f))$. A $-1$ value anywhere indicates that $D(f)$ is non-reduced, hence that $f$ fails to be $\mathcal A$-finite. In the absence of any $-1$, the mapping $f$ is $\mathcal A$-finite and the formulae in 5.6 turn into

\begin{equation*} \mu(D(f))=\sum_{i=1}^{|W|-1}M_{i}+\sum_{i, j=1}^{|W|-1}I_{ij}-|W|+2, \end{equation*}

\begin{equation*}\delta(D(f))=\sum_{i=1}^{|W|-1}\Delta_i+\sum_{1\leqslant i \lt j\leqslant |W|-1}I_{ij},\end{equation*}

Furthermore, the number of branches of $D(f)$ is the sum of the number of branches of all the $D_\sigma(f)$, which is equal to

\begin{equation*} \sum_{i=1}^{|W|-1}(2\Delta_i-M_i-I_{ii})+|W|-1. \end{equation*}

Observe that the matrix $I$ is symmetric and that the vectors $M$ and $\Delta$ must contain the same value at the positions corresponding to $D_\sigma(f)$ and $D_{\sigma^{-1}}(f)$, since these are isomorphic spaces by virtue of Item (5) of Theorem 5.

Example 5.7. Consider the mapping $f_2^{D_8}$ of Example 2.7, whose double point branches were computed in Example 5.3. By ordering the group as $1,\sigma_1,\sigma_2,\sigma_3,\sigma_4,\rho_1,\rho_2,\rho_3$ (see Example 2.3), computing the Milnor numbers and the intersection number of all pairs of branches, one gets

\begin{equation*}M=(0,0,0,0,1,4,1),\quad \Delta=(0,0,0,0,1,3,1),\quad I=\begin{pmatrix} 0 & 1 & 1 & 1 & 2& 3 & 2\\ 1 & 0 & 1 & 1 & 2& 3 & 2\\ 1 & 1 & 0 & 1 & 2 & 3 & 2\\ 1 & 1 & 1 & 0 & 2 & 3 & 2\\ 2 & 2 & 2 & 2 & 0 & 6 & 6 \\ 3 & 3 & 3 & 3 & 6 & 0 & 6 \\ 2 & 2 & 2 & 2 & 6 & 6 & 0 \\ \end{pmatrix} \end{equation*}

From this, one computes $\mu(D(f_ 2^{D_8}))=6+104-8+2=104$ and $\delta(D(f_ 2^{D_8}))=5+52=57$.

This same method is efficient when applied to bigger groups. For example, for the reflection mapping given by the reflection group $\mathfrak S_4$ of Example 2.4 and the embedding $(x,y)\to((x+y)^2,x,y)$, the vectors and square matrix involved have size 23, but it is better to look at the information in this format than trying to understand the space $D(f)$ formed by all branches together. For this particular example, one gets $\mu(D(f))=399$ and $\delta(D(f))=190$.

Example 5.8. If $f$ is a reflection graph with homogeneous $H$, as in Remarks 3.8 and 5.5, then, by ordering $W$ as identity first, then reflections, and then non-reflections, one obtains

\begin{align*}M_i&=\left\{\begin{matrix} (d-2)^2, \, i=1,\dots,r,\\ (d-1)^2, \, i=r+1,\dots|W|-1 \end{matrix}\right.,\\ D_{\sigma}(f)\cdot D_{\tau}(f)&=\left\{\begin{matrix} (d-1)^2, \text{if } \sigma \text{and } \tau \text{are reflections},\\ d^2, \text{if } \sigma \text{and } \tau \text{are non-reflections},\\ d(d-1), \text{otherwise} \end{matrix}\right.\end{align*}

where $r$ is the number of reflections in $W$. Therefore,

\begin{equation*}\mu(D(f))=(1+d+r-d|W|)^2\end{equation*}

(observe that this also follows from Remark 5.5) and

\begin{equation*}\delta(D(f))=\frac{1}{2} (r (3 + r) - 2 d (1 + r) ( |W|-1) + d^2 ( |W|-1)^2).\end{equation*}

For instance, the map $f=f^{(d_1,d_2,d_3)}$ from Example 2.8 has $\mu(D(f))=(1 - d_1 - d_2 - d_3 + d_1 d_2 d_3)^2$, and $\delta(D(f))=1/2 (1 - d_2 - d_3 + d_1 ( d_2 d_3-1)) (2 - d_2 - d_3 + d_1 (d_2 d_3-1))$.

6.1 Obstructions from corank and $\mathcal A$-finiteness

An obvious restriction on the reflection groups that can potentially generate the $P_3$ map-germ is that they cannot have rank greater than two. This comes from looking at the corank of $f$. The corank of a finite map-germ $f\colon ({\mathbb C}^n,0)\to ({\mathbb C}^p,0)$ is

\begin{equation*}\operatorname{corank} f= n-\operatorname{rk} df_0.\end{equation*}

and we say that a $W$-reflection map-germ is essential if $\operatorname{corank} f=\operatorname{rk} W.$ The restriction on the rank of $W$ is the following: The rank of $W$ equals the dimension of $C_0^\bot$, for the facet $C_0$ passing through the origin. But, from the chain rule and the fact that $\ker d\omega_0=C_0^\bot$, it follows that

\begin{equation*}\operatorname{corank} f={\operatorname{dim}}(T_0\mathcal Y\cap C_0^\bot).\end{equation*}

Therefore, the dimension of $C_0^\bot$ must be at least one, but cannot be equal to three, because this would force $P_3$ to have corank $2$. The general statement is as follows:

Proposition 6.1. Assume that $f\colon ({\mathbb C}^n,0)\to({\mathbb C}^p,0)$ is a $W$-reflection map-germ of corank $k$. Then

\begin{equation*}k\leqslant\operatorname{rk}W\leqslant k+p-n.\end{equation*}

The mapping $P_3$ is not an arbitrary corank one mapping; it is $\mathcal A$-finite and this could be useful information to decide whether or not it is a reflection mapping. Even though they do not apply to map-germs $({\mathbb C}^2,0)\to({\mathbb C}^3,0)$ like $P_3$, this is a good place to point out some obstructions related to stability and $\mathcal A$-finiteness:

1. (1) There is no stable germ of reflection map of corank $\geqslant$ 2, and all stable essential reflection maps of corank one are fold maps [Reference Peñafort Sanchis22, Theorem 8.4].

2. (2) For $p \lt 2n-1$, there are no $\mathcal A$-finite reflection map-germs $f \colon ({\mathbb C}^n ,0) \to ({\mathbb C}^p ,0)$ with $\operatorname{corank} f \geqslant 2$, and all $\mathcal A$-finite germs of essential reflection maps of corank one are fold maps. [Reference Peñafort Sanchis22, Theorem 8.5].

Combining the above criteria with the corank bounds in Proposition 6.1, one sees that for dimensions $p \lt 2n-1$ the combination of $\mathcal A$-finiteness, high corank and being a reflection map-germ imposes serious restrictions on the possible groups $W$ to be used. Here is a particular case:

For dimensions $p=2n-1$ and $p=2n$ the conditions of having high corank, being $\mathcal A$-finite and being a reflection map-germ seem to be perfectly compatible, because we have an infinite supply of $\mathcal A$-finite reflection map-germs with arbitrary corank [Reference Peñafort Sanchis22, Theorems 9.5 and 9.6] (this may be the case for $p \gt 2n$ as well, but this case has not yet been studied).

Back to $P_3$, we know that only reflection groups of rank $\leqslant 2$ can potentially realize $P_3$ as a reflection map-germ. The list of such reflection groups is as follows [Reference Lehrer and Taylor11, Theorem 8.29]:

• • Cyclic groups, ${\mathbb Z}/d$.

• • Products of two cyclic groups, ${\mathbb Z}/d_1\times {\mathbb Z}/d_2$.

• • The imprimitive groups $G(m,p,2)$, with $m\geqslant 1$ and $p\vert m$ [Reference Lehrer and Taylor11, Chapter 2].

• • The exceptional Shephard–Todd groups of rank 2, $G_4,\dots,G_{22}$ [Reference Lehrer and Taylor11, Table D.1].

Our goal is to discard all of the above groups as candidates to produce the map-germ $P_3$. For this, we turn on the geometry of the double point spaces of $P_3$.

6.2 Obstructions from the number of double point branches

A standard calculation shows the double point space $D^2\subseteq {\mathbb C}^2\times {\mathbb C}^2$ of $P_3$ to be a singularity

\begin{equation*}D^2\cong D_4,\end{equation*}

that is, isomorphic to three lines intersecting pairwise transversely on a plane. Its irreducible decomposition may be written as

\begin{equation*}D^2=C_1^2\cup C_2^2\cup C_2^2.\end{equation*}

The double point curve $D\subseteq {\mathbb C}^2$ is reduced and it consists of the projections $C_i$ of the curves $C_i^2$. These turn out to be three regular curves intersecting each other tangentially at the origin, giving

\begin{equation*}D=C_1\cup C_2\cup C_3.\end{equation*}

If we assume $P_3$ to be a reflection map-germ, then $D$ must decompose as

\begin{equation*}C_1\cup C_2\cup C_3=\bigcup_{\sigma\in W\setminus \{1\}}D_\sigma,\end{equation*}

but be aware that the branches $D_\sigma$ may be empty or non-irreducible. Being a map-germ $({\mathbb C}^2,0)\to({\mathbb C}^3,0)$, they are computed as $D_\sigma=V(\lambda_\sigma)$, where

\begin{equation*}\lambda_\sigma= \begin{cases} \frac{\sigma^{-1} L-L}{\ell_\sigma},\quad \text{if}\ \sigma\ \text{is a reflection}\\ \sigma^{-1}L- L,\quad\text{otherwise} \end{cases} \end{equation*}

Now observe that this forces $D_\sigma\neq \emptyset$, for each $\sigma\neq 1$ which is not a reflection. This means that the group $W$ giving rise to $P_3$ must have at most three elements, which are neither the identity nor reflections. Since $W$ must have rank at most two, the degrees of $\omega$, up to ordering, must be

\begin{equation*}(1,d_1,d_2).\end{equation*}

Taking into account that $\vert W\vert=d_1d_2$ and that the number of reflections in $W$ is $d_1+d_2-2$, the restriction on the number of elements which are not reflections gives

\begin{equation*}d_1d_2-d_1-d_2+1\leqslant 3.\end{equation*}

For either $d_1=1$ or $d_2=1$, that is, for the cyclic groups ${\mathbb Z}/d$, this imposes no restrictions. Otherwise, since $d_1\geqslant 2$, we may rewrite the previous equation as

\begin{equation*}d_2\leqslant\frac{d_1+2}{d_1-1}.\end{equation*}

Since $d_2$ must be at least two as well, this imposes the condition $d_1\leqslant 4$. After reordering them so that $d_1\leqslant d_2$, the possible degrees are

\begin{equation*}(1,1,d),\ (1,2,2),\ (1,2,3),\ (1,2,4).\end{equation*}

None of the exceptional groups $G_4,\dots,G_{22}$ have admissible degrees [Reference Lehrer and Taylor11, Table D.3] and the groups $G(m,p,2)$ have degrees $(1,2m/p,m)$ [Reference Lehrer and Taylor11, Chapter 2, Section 8]. This leaves only $G(2,1,2)$, $G(2,2,2)$, $G(3,3,2)$ and $G(4,4,2)$. Taking into account the overlaps $G(2,2,2)\cong {\mathbb Z}/2\times {\mathbb Z}/2$ and $G(2,1,2)\cong G(4,4,2)$ [Reference Lehrer and Taylor11, Example 2.11] and the fact that $G(3,3,2)$ and $G(4,4,2)$ are the dihedral groups $D_6$ and $D_8$ of symmetries of a triangle and a square, respectively, we are reduced to cyclic groups, ${\mathbb Z}/d$, the products ${\mathbb Z}/2\times {\mathbb Z}/2$, ${\mathbb Z}/2\times {\mathbb Z}/3$ and ${\mathbb Z}/2\times {\mathbb Z}/4$ and the dihedral groups $D_6$ and $D_8$.

The obstruction based on the number of branches generalizes trivially as follows:

Proposition 6.3. Let $f\colon ({\mathbb C}^n,0)\to ({\mathbb C}^{n+1},0)$ be a generically one-to-one map-germ. Let $r(D)$ be the number of branches of the double point space $D(f)=V(\lambda)$, counted with multiplicity (that is, the number of factors in $\lambda$, counting repetitions). Assume $f$ to be a $W$-reflection map-germ and let $(d_1,\dots,d_{n+1})$ be the degrees of $W$. Then,

\begin{equation*}\prod_{i=1}^{n+1} d_i-\sum_{i=1}^{n+1}d_i\leqslant r(D)-n.\end{equation*}

Now we want to discard the cyclic groups ${\mathbb Z}/d$ with $d \gt 4$, because they would give either $D=\emptyset$ or a space $D$ with too many components. The key is that emptyness of $D_\sigma$ for a reflection $\sigma$ depends only on the reflecting hyperplane of $\sigma$ (see Item (3) of Proposition 4.6) Consequently, the condition $D_\sigma=\emptyset$ does not depend strictly on $\sigma$, but only on ${\operatorname{Fix}} \sigma$, and therefore reflections with the same reflecting hyperplane must be either all empty or all nonempty. Therefore a ${\mathbb Z}/d$-reflection map-germs with nonempty $D(f)$ must have at least $d-1$ branches (counted with multiplicity) in their spaces $D(f)$. It follows that $P_3$ can only be a ${\mathbb Z}/d$-reflection map-germ for $d\leqslant 4$. Therefore, we are reduced to the following list of groups:

• • The cyclic groups, ${\mathbb Z}/2$, ${\mathbb Z}/3$ and ${\mathbb Z}/4$.

• • The products ${\mathbb Z}/2\times {\mathbb Z}/2$, ${\mathbb Z}/2\times {\mathbb Z}/3$ and ${\mathbb Z}/2\times {\mathbb Z}/4$.

• • The dihedral groups $D_6$ and $D_8$.