# Complex reflection groups and K3 surfaces I

Cédric Bonnafé  IMAG, Université de Montpellier, CNRS, Montpellier, France
<EMAIL_ADDRESS>and  Alessandra Sarti  Université de Poitiers,
Laboratoire de Mathématiques et Applications, UMR 7348 du CNRS, TSA 61125, 11
bd Marie et Pierre Curie, 86073 Poitiers cedex 9, France<EMAIL_ADDRESS>poitiers.fr

  *     * scAbstract. scWe construct here many families of K3 surfaces that one can obtain as quotients of algebraic surfaces by some subgroups of the rank four complex reflection groups. We find in total 15 families with at worst $ADE$–singularities. In particular we classify all the K3 surfaces that can be obtained as quotients by the derived subgroup of the previous complex reflection groups. We prove our results by using the geometry of the weighted projective spaces where these surfaces are embedded and the theory of Springer and Lehrer-Springer on properties of complex reflection groups. This construction generalizes a previous construction by W. Barth and the second author.

scKeywords. K3 surfaces; complex reflection groups

sc2020 Mathematics Subject Classification. 14J28; 14J10; 20F55

sc[Français]

scGroupes de réflexions complexes et surfaces K3 I

scRésumé. scNous construisons ici de nombreuses familles de surfaces K3 que
l’on peut obtenir comme quotients de surfaces algébriques par certains sous-
groupes des groupes de réflexions complexes de rang 4. Nous trouvons au total
15 familles avec au plus des singularités $ADE$. En particulier, nous
classifions toutes les surfaces K3 pouvant être obtenues comme quotients par
le sous-groupe dérivé des groupes de réflexions complexes susmentionnés. Nous
démontrons nos résultats en utilisant la géométrie des espaces projectifs à
poids dans lesquels ces surfaces sont plongées et la théorie de Springer et
Lehrer-Springer sur les propriétés des groupes de réflexions complexes. Cette
construction généralise une construction précédente introduite par W. Barth et
la seconde auteure.

  *
cNovember 8, 2020Received by the Editors on June 17, 2020\.
Accepted on December 23, 2020.

IMAG, Université de Montpellier, CNRS, Montpellier, France

sce-mail<EMAIL_ADDRESS>
Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR 7348
du CNRS, TSA 61125, 11 bd Marie et Pierre Curie, 86073 Poitiers cedex 9,
France

sce-mail<EMAIL_ADDRESS>
scThe first author is partly supported by the ANR: Projects No
ANR-16-CE40-0010-01 (GeRepMod) and ANR-18-CE40-0024-02 (CATORE).

© by the author(s) This work is licensed under
http://creativecommons.org/licenses/by-sa/4.0/

###### Contents

  1. 1 Introduction
  2. 2 Notation, preliminaries
  3. 3 Reflection groups
  4. 4 Determining singularities
  5. 5 Invariant K3 surfaces
  6. 6 Singularities of ${\mathcal{Z}}(f)/\Gamma$
  7. Appendix A. Surfaces in weighted projective spaces
  8. Appendix B. Around ADE singularities

## 1\. Introduction

In this paper we describe a relation between complex reflection groups and K3
surfaces. A relation already appeared recently in the paper [BS21] by the
authors, where they use the reflection group denoted by $G_{29}$ in the
Shepard–Todd classification [ST54] to describe K3 surfaces with maximal finite
automorphism groups containing the Mathieu group $M_{20}$. The motivation for
our paper is an early paper of the second author [Sar01] and of W. Barth and
the second author [BS03] where they study first one parameter families of
surfaces of general type in the three dimensional complex projective space
containing four surfaces with a high number of nodes (i.e.
$A_{1}$–singularities). Then they study the quotients of these families by
some groups related to the platonic solids: tetrahedron, octahedron and
icosahedron and which they call bipolyhedral groups. These turn out to be
subgroups of some complex reflection groups and they show that the quotients
are K3 surfaces with $ADE$–singularities. In this paper we show that these
examples are only a few examples of K3 surfaces that one can produce by using
complex reflection groups. Moreover the theory of Springer and Lehrer–Springer
and some technical lemmas allow a deep understanding of the reason why the
quotients have trivial dualizing sheaf and admit only $ADE$–singularities.
This allows then to conclude that the minimal resolution are K3 surfaces. We
find in total 15 families of K3 surfaces.

More precisely we consider a complex reflection group $W$ acting on a four
dimensional complex vector space $V$. By Shephard–Todd/Chevalley/Serre Theorem
[Bro10, Theorem 4.1] there exist 4 algebraically independent polynomials which
are invariant under the action of $W$ and which generate algebraically the
ring of all $W$-invariant polynomials. We assume furthermore that $W$ is
generated by reflections of order $2$ and in Table 1 we give the list of the
degrees of the four invariant polynomials (observe that these degrees do not
depend on the polynomials) and of the codegrees corresponding to the degrees
of four invariant derivatives which generate the module of all $W$-invariant
derivatives. The aim of the paper is to study the quotient of the projective
zero set ${\mathcal{Z}}(f)$ of an homogeneous fundamental invariant $f$ of $W$
by some subgroup of $W$: the derived subgroup $W^{\prime}$ and the group
$W^{\scriptscriptstyle{\mathrm{SL}}}=\operatorname{{\mathrm{Ker}}}(\det)\cap
W$. A reason for this choice is that the simple structure of the invariant
ring of $W$ and the fact that $W$ is generated by reflections of order $2$
imply that ${\mathcal{Z}}(f)/\Gamma$ is a complete intersection in a weighted
projective space. More precisely the quotient surface
${\mathcal{Z}}(f)/W^{\scriptscriptstyle{\mathrm{SL}}}$ is a double cover of a
weighted projective plane whereas if $W^{\prime}$ is different from
$W^{\scriptscriptstyle{\mathrm{SL}}}$ then ${\mathcal{Z}}(f)/W^{\prime}$ is a
complete intersection in a four dimensional weighted projective space. We also
explain how to obtain explicit equations for ${\mathcal{Z}}(f)/W^{\prime}$ or
${\mathcal{Z}}(f)/W^{\scriptscriptstyle{\mathrm{SL}}}$.

It turns out that if the degree $d$ of $f$ is ”well chosen” and
${\mathcal{Z}}(f)$ has only ADE-singularities, then ${\mathcal{Z}}(f)/\Gamma$
is a K3 surface with $ADE$–singularities (“well chosen” means that the sum of
the degrees of the equations of the surface is equal to the sum of the weights
of the ambient weighted projective space: the set of pairs $(W,d)$ such that
$d$ is “well-chosen” is denoted by ${\mathcal{K}}_{3}$ and is described in
§5.2). To show that we have $ADE$–singularities on the double covers one has
to study carefully the singularities of the branching curve as well as the
singularities that one gets from the singularities of the weighted projective
plane. Our main Theorem 5.4 is the following:

######  Theorem 1.1.

Assume that $(W,d)\in{\mathcal{K}}_{3}$. Let be the subgroup $W^{\prime}$ or
$W^{\scriptscriptstyle{\mathrm{SL}}}$ of $W$ and let $f$ be a fundamental
invariant of $W$ of degree $d$ whose projective zero set ${\mathcal{Z}}(f)$
has only $ADE$–singularities.

Then ${\mathcal{Z}}(f)/\Gamma$ is a K3 surface with $ADE$–singularities.

In particular the theorem allows us to classify all the K3 surfaces that can
be obtained as quotient by $W^{\prime}$ or
$W^{\scriptscriptstyle{\mathrm{SL}}}$. Theorem 5.4 is a qualitative result,
that insures that one can build from invariants of some complex reflection
groups of rank $4$ several families of K3 surfaces with $ADE$–singularities.
However, it does not say anything about the types of the singularities and
important invariants of their minimal resolution (rank of the Picard number,
description of the transcendental lattice).

Theorem 5.4 (and its proof) improves previous works by Barth and the second
author [BS03] for two reasons:

  * -

By looking at all complex reflection groups of rank $4$, it enlarges
considerably the class of examples of K3 surfaces that can be constructed as
above. It shows moreover that the discovery of families of K3 surfaces in
[BS03] is not just ”an accident” but it is strictly related to polynomial
invariants of complex reflection groups and their action on these.

  * -

The main difficulty is to prove that ${\mathcal{Z}}(f)/\Gamma$ has only ADE
singularities. Our proof involves some general facts about singularities (see
Appendix Appendix B. Around ADE singularities) and more complex reflection
group theory (as the theory of Springer and Lehrer–Springer on eigenspaces of
elements of complex reflection groups): as a consequence, our proof avoids as
much as possible (but not completely) a case-by-case analysis, and so may also
be viewed not only as a generalization but also as an enlightenment of results
from [BS03].

An important feature of the K3 surfaces constructed in Theorem 5.4 is that
most of them have big Picard number, and generally as big as possible compare
to the number of moduli of the family they belong to. In particular, we can
build in this way several (more than thirty) K3 surfaces with Picard number
$20$, often called singular K3 surfaces, whose moduli space is a
$0$-dimensional subspace of the $20$-dimensional moduli space of K3 surfaces.
This will be explained in the sequel to this paper [BS-II], [BS-III], where we
aim to complete the qualitative results of this first part by quantitative
results whenever $W$ is assumed to be primitive (i.e. $W=G_{28}$, $G_{29}$,
$G_{30}$ or $G_{31}$). We will for instance compute the transcendental lattice
of the singular K3 surfaces, and describe explicit elliptic fibrations in most
cases. Note that the case where $(W,d)=(G_{28},6)$ or $(G_{30},12)$ and
$\Gamma=W^{\prime}$ was already treated by Barth and the second author [BS03]:
these examples will be revisited thanks to our new techniques, and more
geometrical informations will be given. Note also that, by taking Galois
invariant models for complex reflection groups as in [MM10], it turns out that
all our families of K3 surfaces are defined over ${\mathbb{Q}}$ and, in
particular, all the singular K3 surfaces we obtain are defined over
${\mathbb{Q}}$: this fact is only checked case-by-case [BS-II], [BS-III] but
would deserve a general explanation.

The paper is organized as follows: Section 2 contains basic facts on the
action of groups of matrices on homogeneous polynomials and Section 3 recalls
facts on reflection groups, in particular we find equations for the quotient
surfaces and we recall basic facts from Springer and Lehrer-Springer theory,
that we use in the next sections (and in the sequel to this paper [BS-II],
[BS-III]) to describe the singularities that we have on the quotient surfaces.
In Section 4 we give several useful facts to describe the singularities of the
quotient surfaces in particular in the case that these are a double cover of a
weighted projective plane. Note that Sections 2-4 are written in a greater
generality (reflection groups acting on vector spaces of any finite dimension)
as they might be of general interest. In Section 5 we describe how to obtain
K3 surfaces. In Table 2 we recall the degrees of the equations and the
weighted projective spaces where the (singular) surfaces are embedded. We give
in this section the main part of the proof of our main Theorem 5.4. We finish
the proof in Section 6 where we show that the quotient K3 surfaces have at
worst $ADE$–singularities.

Acknowledgements. We wish to thank Sylvain Brochard for useful discussions
about the results of Appendix Appendix B. Around ADE singularities. We thank
also Enrica Floris and Étienne Mann for several interesting discussions.

## 2\. Notation, preliminaries

If $d\geq 1$, we denote by ${\boldsymbol{\mu}}_{d}$ the group of $d$-th roots
of unity in ${\mathbb{C}}^{\times}$ and we fix a primitive $d$-th root of
unity $\zeta_{d}$ (we will use the standard notation $i=\zeta_{4}$). If
$l_{1},\dots,l_{r}$ are positive integers, then
${\mathbb{P}}(l_{1},\dots,l_{r})$ denotes the corresponding weighted
projective space.

We fix an $n$-dimensional ${\mathbb{C}}$-vector space $V$ and we denote by
${\mathbb{P}}(V)$ its associated projective space. If $v\in
V\setminus\\{0\\}$, we denote by $[v]\in{\mathbb{P}}(V)$ the line it defines
(i.e. $[v]={\mathbb{C}}v$). The group
${\mathbf{G}}{\mathbf{L}}_{\mathbb{C}}(V)$ acts on the algebra
${\mathbb{C}}[V]$ of polynomial functions on $V$ as follows: if
$g\in{\mathbf{G}}{\mathbf{L}}_{\mathbb{C}}(V)$ and $f\in{\mathbb{C}}[V]$, we
write

$\kern 0.5pt\vphantom{f}^{g}\kern-0.5ptf(v)=f(g^{-1}\cdot v).$

If $g\in{\mathbf{G}}{\mathbf{L}}_{\mathbb{C}}(V)$ and
$\zeta\in{\mathbb{C}}^{\times}$, we denote by $V(g,\zeta)$ the
$\zeta$-eigenspace of $g$. If $v\in V(g,\zeta)$ and $f\in{\mathbb{C}}[V]^{g}$,
then

(2.1) $\kern
0.5pt\vphantom{(\partial_{v}f)}^{g}\kern-0.5pt(\partial_{v}f)=\zeta\partial_{v}f.$

If $G$ is a subgroup of ${\mathbf{G}}{\mathbf{L}}_{\mathbb{C}}(V)$, we write
$PG$ for its image in ${\mathbf{P}}{\mathbf{G}}{\mathbf{L}}_{\mathbb{C}}(V)$.
Recall that a subgroup $G$ of ${\mathbf{G}}{\mathbf{L}}_{\mathbb{C}}(V)$ is
called primitive if there does not exist a decomposition
$V=V_{1}\oplus\cdots\oplus V_{r}$ with $V_{i}\neq 0$ and $r\geq 2$ such that
$G$ permutes the $V_{i}$’s. If $S$ is a subset of $V$, we denote by $G_{S}$
(resp. $G(S)$) the setwise (resp. pointwise) stabilizer of $S$ (so that $G(S)$
is a normal subgroup of $G_{S}$ and $G_{S}/G(S)$ acts faithfully on $S$). Note
that $G_{S}=G_{{\mathbb{C}}S}$ and $G(S)=G({\mathbb{C}}S)$, where
${\mathbb{C}}S$ denotes the linear span of $S$. The derived subgroup of $G$
will be denoted by $G^{\prime}$, and we set
$G^{\scriptscriptstyle{\mathrm{SL}}}=G\cap{\mathbf{S}}{\mathbf{L}}_{\mathbb{C}}(V)$.
Note that $G^{\prime}\subset G^{\scriptscriptstyle{\mathrm{SL}}}$ and that the
inclusion might be strict. We state here a totally trivial result which will
be used extensively and freely along this series of papers:

######  Lemma 2.2.

Let $g\in{\mathbf{G}}{\mathbf{L}}_{\mathbb{C}}(V)$, let $\zeta$ be a root of
unity of order $d$, let $v\in V$ be such that $g(v)=\zeta v$ and let
$f\in{\mathbb{C}}[V]^{g}$ be homogeneous of degree $e$ not divisible by $d$.
Then $f(v)=0$.

###### Proof.

As $f\in{\mathbb{C}}[V]^{g}$, we have $f(g(v))=f(v)$. But $f(g(v))=f(\zeta
v)=\zeta^{e}f(v)$ because $f$ is homogeneous of degree $e$. So the result
follows from the fact that $\zeta^{e}\neq 1$. ∎

If $X$ is a complex algebraic variety and $x\in X$, we denote by
${\mathrm{T}}_{x}(X)$ the tangent space of $X$ at $x$. If
$f\in{\mathbb{C}}[V]$ is homogeneous, we will denote by ${\mathcal{Z}}(f)$ the
projective (possibly non-reduced) hypersurface in
${\mathbb{P}}(V)\simeq{\mathbb{P}}^{n-1}$ defined by $f$. Its singular locus
will be denoted by ${\mathcal{Z}}_{\mathrm{sing}}(f)$.

######  Lemma 2.3.

Let $G$ be a finite subgroup of ${\mathbf{G}}{\mathbf{L}}_{\mathbb{C}}(V)$,
let $f\in{\mathbb{C}}[V]^{G}$ be homogeneous and let $v\in
V^{G}\setminus\\{0\\}$ be such that $f(v)=0$. Then $G$ acts trivially on
${\mathrm{T}}_{[v]}({\mathbb{P}}(V))/{\mathrm{T}}_{[v]}({\mathcal{Z}}(f))$.

###### Proof.

Since $G$ is finite, there exists a $G$-stable subspace $E$ of $V$ such that
$V=E\oplus{\mathbb{C}}v$. Let $\alpha\in V^{*}$ be such that $\alpha(v)=1$ and
$\alpha(E)=0$. The affine chart $U_{\alpha}$ of ${\mathbb{P}}(V)$ defined by
$\alpha\neq 0$ is identified with $v+E$ and, after translation, is identified
with $E$: through this identification, ${\mathcal{Z}}(f)\cap U_{\alpha}$ is
the affine hypersurface defined by the polynomial $F\in{\mathbb{C}}[E]$, where
$F(e)=f(v+e)$. Since $v$ is $G$-invariant, $F$ is also $G$-invariant. Let us
denote by $F_{i}$ its $i$-th homogeneous component: it is $G$-invariant. Then
$F_{0}=0$ because $f(v)=0$. Moreover, ${\mathrm{T}}_{[v]}({\mathbb{P}}(V))=E$
and
${\mathrm{T}}_{[v]}({\mathcal{Z}}(f))=\operatorname{{\mathrm{Ker}}}(F_{1})$
(and these identifications are $G$-equivariant since $v\in V^{G}$).

But ${\mathbb{C}}F_{1}$ is the dual space to
$E/\operatorname{{\mathrm{Ker}}}(F_{1})$: since $G$ acts trivially on
${\mathbb{C}}F_{1}$, this shows that $G$ acts trivially on
$E/\operatorname{{\mathrm{Ker}}}(F_{1})={\mathrm{T}}_{[v]}({\mathbb{P}}(V))/{\mathrm{T}}_{[v]}({\mathcal{Z}}(f))$.
∎

The next result is just an easy generalization of [Sar01, §6].

######  Corollary 2.4.

Let $G$ be a finite subgroup of ${\mathbf{G}}{\mathbf{L}}_{\mathbb{C}}(V)$
such that $\dim V^{G}=1$ and let $f\in{\mathbb{C}}[V]^{G}$ be non-zero and
homogeneous. We assume that $f$ vanishes at $V^{G}$, viewed as a point of
${\mathbb{P}}(V)$. Then $V^{G}$ is a singular point of ${\mathcal{Z}}(f)$.

###### Proof.

Let $v\in V^{G}\setminus\\{0\\}$. We keep the notation of the proof of the
previous Lemma 2.3 ($E$, $\alpha$, $F$, $F_{i}$). Since $V^{G}={\mathbb{C}}v$,
we have $E^{G}=0$ and so, by semisimplicity, we have that
$(E/\operatorname{{\mathrm{Ker}}}(F_{1}))^{G}=0$. But $G$ acts trivially on
$E/\operatorname{{\mathrm{Ker}}}(F_{1})$ by Lemma 2.3. Therefore,
$E/\operatorname{{\mathrm{Ker}}}(F_{1})=0$ so
${\mathrm{T}}_{[v]}({\mathcal{Z}}(f))=\operatorname{{\mathrm{Ker}}}(F_{1})=E$
and so ${\mathcal{Z}}(f)$ is singular at $[v]=V^{G}$. ∎

######  Remark 2.5.

The previous lemma might be used to explain the construction of several
singular curves and surfaces constructed by the two authors [Sar01], [Bon19].
Let us explain how to proceed.

Let $G$ be a finite subgroup of ${\mathbf{G}}{\mathbf{L}}_{\mathbb{C}}(V)$,
and let $H_{1},\dots,H_{r}$ be a set of representatives of conjugacy classes
of maximal subgroups of $G$ among the subgroups $H$ satisfying
$\dim(V^{H})=1$. Let $N_{k}=N_{G}(H_{k})$, let $v_{k}\in
V^{H_{k}}\setminus\\{0\\}$ and let k denote the $G$-orbit of $[v_{k}]$ in
${\mathbb{P}}(V)$. Then

(2.6) $|{}_{k}|=\frac{|G|}{|N_{k}|}.$

For this, it is sufficient to prove that $N_{k}=G_{[v_{k}]}$. First, $N_{k}$
stabilizes $V^{H_{k}}=[v_{k}]$, which proves that $N_{k}\subset G_{[v_{k}]}$.
Conversely, $G_{[v_{k}]}$ normalizes $G_{v_{k}}$. But $H_{k}\subset G_{v_{k}}$
by construction and, by the maximality of $H_{k}$, this implies that
$H_{k}=G_{v_{k}}$.

Now, we fix two linearly independent homogeneous polynomials $f_{1}$ and
$f_{2}$ of the same degree such that $f_{1}(v_{k})\neq 0$ for all $k$. We also
set $\lambda_{k}=f_{2}(v_{k})/f_{1}(v_{k})$. Then it follows from Corollary
2.4 that

(2.7) ${\mathcal{Z}}(f_{2}-\lambda_{k}f_{1})$ contains k in its singular
locus.

It also shows that, if $G$ is defined over a subfield $K$ of ${\mathbb{C}}$,
then the points of k (which are singular points of
${\mathcal{Z}}(f_{2}-\lambda_{k}f_{1})$) have coordinates in $K$.

######  Corollary 2.8.

Let $G$ be a finite subgroup of ${\mathbf{G}}{\mathbf{L}}_{\mathbb{C}}(V)$
such that $\dim V^{G}=2$, let $f\in{\mathbb{C}}[V]^{G}$ be homogeneous and
non-zero and let $v\in V^{G}\setminus\\{0\\}$ be such that $f(v)=0$. Let $L$
be the line ${\mathbb{P}}(V^{G})$ and assume that $[v]$ is a smooth point of
${\mathcal{Z}}(f)$. Then the intersection of $L$ with ${\mathcal{Z}}(f)$ is
transverse at $[v]$.

###### Proof.

We keep again the notation of the proof of Lemma 2.3 ($E$, $\alpha$, $F$,
$F_{i}$). Since $\dim V^{G}=2$, this forces $\dim E^{G}=1$. Since $[v]$ is
smooth, this means that $F_{1}\neq 0$. It then follows from Lemma 2.3 that
$E=E^{G}\oplus\operatorname{{\mathrm{Ker}}}(F_{1})$. But
$E^{G}={\mathrm{T}}_{[v]}(L)$ and
$\operatorname{{\mathrm{Ker}}}(F_{1})={\mathrm{T}}_{[v]}({\mathcal{Z}}(f))$.
This shows the result. ∎

######  Corollary 2.9.

Let $G$ be a finite subgroup of ${\mathbf{G}}{\mathbf{L}}_{\mathbb{C}}(V)$
such that $\dim V^{G}=2$, and let $f\in{\mathbb{C}}[V]^{G}$ be homogeneous and
non-zero. Let $L$ be the line ${\mathbb{P}}(V^{G})$ and assume that
$L\subset{\mathcal{Z}}(f)$. Then $L\subset{\mathcal{Z}}_{\mathrm{sing}}(f)$.

###### Proof.

Let $v\in V^{G}\setminus\\{0\\}$ and assume that $[v]$ is a smooth point of
${\mathcal{Z}}(f)$. Then the intersection of $L$ with ${\mathcal{Z}}(f)$ is
not transverse at $[v]$ because $L\subset{\mathcal{Z}}(f)$: this contradicts
Corollary 2.8. ∎

## 3\. Reflection groups

We fix a finite subgroup $W$ of ${\mathbf{G}}{\mathbf{L}}_{\mathbb{C}}(V)$ and
we set

$\operatorname{Ref}(W)=\\{s\in W\leavevmode\nobreak\ |\leavevmode\nobreak\
\dim(V^{s})=n-1\\}.$

Hypothesis. We assume throughout this paper that
$W=\langle\operatorname{Ref}(W)\rangle.$ In other words, $W$ is a complex
reflection group. The number $\operatorname{codim}\nolimits(V^{W})$ is called
the rank of $W$.$A_{\displaystyle{A_{A}}}$

Standard arguments allow to reduce most questions about reflection groups to
questions about irreducible reflection groups. These last ones have been
classified by Shephard-Todd and we refer to Shephard-Todd numbering [ST54] for
such groups: there is an infinite family $G(de,e,r)$ with $d$, $e$, $r\geq 0$
(they are of rank $r$ if $(d,e)\neq(1,1)$ and of rank $r-1$ otherwise) and
$34$ exceptional ones numbered from $G_{4}$ to $G_{37}$ (they are exactly the
primitive complex reflection groups). If $W$ can be realized over the field of
real numbers, then it is a Coxeter group and we will also use the notation
${\mathrm{W}}(X_{i})$ where $X_{i}$ is the type of some Coxeter graph. For
instance, the group $G_{30}$ in Shephard-Todd numbering is the Coxeter group
${\mathrm{W}}(H_{4})$.

### 3.1. Invariants

By Shephard-Todd/Chevalley/Serre Theorem [Bro10, Theorem 4.1], there exist $n$
algebraically independent homogeneous elements $f_{1}$, $f_{2},\dots,f_{n}$ of
${\mathbb{C}}[V]^{W}$ such that

${\mathbb{C}}[V]^{W}={\mathbb{C}}[f_{1},f_{2},\dots,f_{n}].$

Let $d_{i}=\deg(f_{i})$. A family $(f_{1},f_{2},\dots,f_{n})$ satisfying the
above property is called a family of fundamental invariants of $W$. Observe
that by a result of Marin-Michel [MM10], these polynomials can be defined over
the rational numbers (for more details, see [BS-II]). A homogeneous element
$f\in{\mathbb{C}}[V]^{W}$ is called a fundamental invariant if it belongs to a
family of fundamental invariants. Whereas such a family is not uniquely
defined even up to permutation, the list $(d_{1},d_{2},\dots,d_{n})$ is well-
defined up to permutation and is called the list of degrees of $W$: it will be
denoted by ${\mathrm{Deg}}(W)$.

Notation. From now on, and until the end of this paper, we fix a family
${\mathbf{f}}=(f_{1},f_{2},\dots,f_{n})$ of fundamental invariants and we set
$d_{i}=\deg(f_{i})$.$A_{\displaystyle{A_{A}}}$

The following equalities are well-known [Bro10, Theorem 4.1]:

(3.1) $|W|=d_{1}d_{2}\cdots
d_{n}\qquad\text{and}\qquad|\operatorname{Ref}(W)|=\sumop\displaylimits_{i=1}^{n}(d_{i}-1).$

Also, as $W$ acts irreducibly on $V$, its center $|{\mathrm{Z}}(W)|$ consists
of homotheties, so it is cyclic. Moreover by [Bro10, Proposition 4.6],

(3.2) $|{\mathrm{Z}}(W)|={\mathrm{Gcd}}(d_{1},d_{2},\dots,d_{n}).$

The ${\mathbb{C}}[V]$-module $\operatorname{Der}({\mathbb{C}}[V])$ of
derivatives of the algebra ${\mathbb{C}}[V]$ is naturally graded in such a way
that $\partial_{v}$ has degree $-1$ for all $v\in V$. By Solomon Theorem
[Bro10, Theorem 4.44 and §4.5.4], the graded ${\mathbb{C}}[V]^{W}$-module
$\operatorname{Der}({\mathbb{C}}[V])^{W}$ of invariant derivatives is free of
rank $n$, hence it admits a homogeneous ${\mathbb{C}}[V]^{W}$-basis
$(D_{1},\dots,D_{n})$ whose respective degrees are denoted by
$d_{1}^{*}\dots,d_{n}^{*}$. Again, the family $(D_{1},\dots,D_{n})$ is not
uniquely defined even up to permutation, but the list
$(d_{1}^{*},d_{2}^{*},\dots,d_{n}^{*})$ is well-defined up to permutation and
is called the list of codegrees of $W$: it will be denoted by
${\mathrm{Codeg}}(W)$.

We conclude this subsection by a general easy result which follows immediately
from the fact that ${\mathbb{C}}[V]^{W}$ is a graded polynomial algebra whose
weights are given by ${\mathrm{Deg}}(W)$.

######  Proposition 3.3.

The map

$\begin{array}[]{rccc}{\pi_{\mathbf{f}}}:&{{\mathbb{P}}(V)}&\longrightarrow&{{\mathbb{P}}(d_{1},d_{2},\dots,d_{n})}\\\
&{[v]}&\longmapsto&{[f_{1}(v):f_{2}(v):\cdots:f_{n}(v)]}\end{array}$

is well-defined and induces an isomorphism

${\mathbb{P}}(V)/W\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathbb{P}}(d_{1},d_{2},\dots,d_{n}).$

Moreover, $\pi_{\mathbf{f}}$ induces by restriction an isomorphism

${\mathcal{Z}}(f_{1})/W\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathbb{P}}(d_{2},\dots,d_{n}).$

### 3.2. Reflecting hyperplanes

If $s\in\operatorname{Ref}(W)$, then the hyperplane $V^{s}$ is called the
reflecting hyperplane of $s$ (or a reflecting hyperplane of $W$). We denote by
${\mathcal{A}}$ the set of reflecting hyperplanes of $W$. If $X$ is a subset
of $V$, then, by Steinberg-Serre Theorem [Bro10, Theorem 4.7], $W(X)$ is
generated by reflections and so is generated by the reflections whose
reflecting hyperplane contains $X$: such a subgroup is called a parabolic
subgroup of $W$.

If $H\in{\mathcal{A}}$, then the group $W(H)$ is cyclic (indeed, by
semisimplicity, it acts faithfully on $V/H$ which has dimension $1$) and we
denote its order by $e_{H}$. Note that $W_{H}\setminus\\{1\\}$ is the set of
reflections of $W$ whose reflecting hyperplane is $H$, so

(3.4)
$|\operatorname{Ref}(W)|=\sumop\displaylimits_{H\in{\mathcal{A}}}(e_{H}-1).$

We denote by $\alpha_{H}$ an element of $V^{*}$ such that
$H=\operatorname{{\mathrm{Ker}}}(\alpha_{H})$. In particular, if all the
reflections have order $2$, then $|\operatorname{Ref}(W)|=|{\mathcal{A}}|$.
Finally, note the following equality [Bro10, Remark 4.48]

(3.5) $|{\mathcal{A}}|=\sumop\displaylimits_{i=1}^{n}(d_{i}^{*}+1).$

If is a $W$-orbit in ${\mathcal{A}}$, then we denote by $e$ the common value
of the $e_{H}$’s for $H\in\Omega$. We then set

$J=\prodop\displaylimits_{H\in\Omega}\alpha_{H}.$

Then there exists a unique polynomial $P_{{\mathbf{f}},\Omega}$ in variables
$x_{1},\dots,x_{n}$ of respective weights $d_{1},\dots,d_{n}$ such that

(3.6) $J^{e}=P_{{\mathbf{f}},\Omega}(f_{1},\dots,f_{n}).$

Note that $P_{{\mathbf{f}},\Omega}$ is homogeneous of degree $e|\Omega|$. Then
(see [Sta77, Theorem 2.3 and Corollary 4.3] or [LT09, Theorem 9.19 and
Corollary 9.21])

(3.7)
${\mathbb{C}}[V]^{W^{\prime}}={\mathbb{C}}[f_{1},\dots,f_{n},(J)_{\Omega\in{\mathcal{A}}/W}]$

and a presentation of ${\mathbb{C}}[V]^{W^{\prime}}$ is given by the relations
(3.6). Consequently:

######  Proposition 3.8.

Let ${}_{1},\dots,{}_{r}$ denote the $W$-orbits in ${\mathcal{A}}$. Then the
map

$\begin{array}[]{rccc}{\pi_{\mathbf{f}}^{\prime}}:&{{\mathbb{P}}(V)}&\longrightarrow&{{\mathbb{P}}(d_{1},d_{2},\dots,d_{n},|{}_{1}|,\dots,|{}_{r}|)}\\\
&{[v]}&\longmapsto&{[f_{1}(v):f_{2}(v):\cdots:f_{n}(v):J_{{}_{1}}(v):\cdots:J_{{}_{r}}(v)]}\end{array}$

is well-defined and induces an isomorphism

${\mathbb{P}}(V)/W^{\prime}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\\{[x_{1}:\cdots:x_{n}:j_{1}:\cdots:j_{r}]\in{\mathbb{P}}(d_{1},\dots,d_{n},|{}_{1}|,\dots,|{}_{r}|)\leavevmode\nobreak\
|\leavevmode\nobreak\ \hphantom{AAA}$
$\hphantom{AAAAAA}\forall\leavevmode\nobreak\ 1\leq k\leq
r,\leavevmode\nobreak\
j_{k}^{e_{{}_{k}}}=P_{{\mathbf{f}},{}_{k}}(x_{1},\dots,x_{n})\\}.$

Moreover, $\pi_{\mathbf{f}}^{\prime}$ induces by restriction an isomorphism

${\mathcal{Z}}(f_{1})/W^{\prime}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\\{[x_{2}:\cdots:x_{n}:j_{1}:\dots:j_{r}]\in{\mathbb{P}}(d_{2},\dots,d_{n},|{}_{1}|,\dots,|{}_{r}|)\leavevmode\nobreak\
|\leavevmode\nobreak\ \hphantom{AAA}$
$\hphantom{AAAAAA}\forall\leavevmode\nobreak\ 1\leq k\leq
r,\leavevmode\nobreak\
j_{k}^{e_{{}_{k}}}=P_{{\mathbf{f}},{}_{k}}(0,x_{2},\dots,x_{n})\\}.$

Note that $W/W^{\scriptscriptstyle{\mathrm{SL}}}$ is cyclic but
$W^{\scriptscriptstyle{\mathrm{SL}}}$ is not necessarily equal to the derived
subgroup $W^{\prime}$ of $W$. We have
$W^{\prime}=W^{\scriptscriptstyle{\mathrm{SL}}}$ if and only if
$|{\mathcal{A}}/W|=1$. Now, let

$J=\prodop\displaylimits_{H\in{\mathcal{A}}}\alpha_{H}=\prodop\displaylimits_{\Omega\in{\mathcal{A}}/W}J\in{\mathbb{C}}[V].$

It is well-defined up to a scalar and homogeneous of degree $|{\mathcal{A}}|$.
It is the generator of the ideal of the reduced subscheme of the ramification
locus of the morphism $V\to V/W$. Then by [Bro10, Remark 3.10 and Proposition
4.4]

(3.9) $\kern 0.5pt\vphantom{J}^{w}\kern-0.5ptJ=\det(w)^{-1}J$

for all $w\in W$. In particular
$J\in{\mathbb{C}}[V]^{W^{\scriptscriptstyle{\mathrm{SL}}}}$ and
$J^{|W/W^{\scriptscriptstyle{\mathrm{SL}}}|}\in{\mathbb{C}}[V]^{W}$. So there
exists a unique polynomial $P_{\mathbf{f}}\in{\mathbb{C}}[X_{1},\dots,X_{n}]$,
which is homogeneous of degree
$|W/W^{\scriptscriptstyle{\mathrm{SL}}}|\cdot|{\mathcal{A}}|$ if we assign to
$X_{i}$ the degree $d_{i}$, and such that

(3.10)
$J^{|W/W^{\scriptscriptstyle{\mathrm{SL}}}|}=P_{\mathbf{f}}(f_{1},\dots,f_{n}).$

######  Proposition 3.11.

Assume that the map $H\mapsto e_{H}$ is constant on ${\mathcal{A}}$ $($and let
$e$ denote this constant value, which coincides with
$|W/W^{\scriptscriptstyle{\mathrm{SL}}}|)$. Then
${\mathbb{C}}[V]^{W^{\scriptscriptstyle{\mathrm{SL}}}}={\mathbb{C}}[f_{1},f_{2},\dots,f_{n},J]$
and a presentation is given by the single equation  _( 3.10)_.

So the map

$\begin{array}[]{rccc}{\pi_{\mathbf{f}}^{\scriptscriptstyle{\mathrm{SL}}}}:&{{\mathbb{P}}(V)}&\longrightarrow&{{\mathbb{P}}(d_{1},d_{2},\dots,d_{n},|{\mathcal{A}}|)}\\\
&{[v]}&\longmapsto&{[f_{1}(v):f_{2}(v):\cdots:f_{n}(v):J(v)]}\end{array}$

is well-defined and induces an isomorphism

${\mathbb{P}}(V)/W^{\scriptscriptstyle{\mathrm{SL}}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\\{[x_{1}:\cdots:x_{n}:j]\in{\mathbb{P}}(d_{1},\dots,d_{n},|{\mathcal{A}}|)\leavevmode\nobreak\
|\leavevmode\nobreak\ j^{e}=P_{\mathbf{f}}(x_{1},\dots,x_{n})\\}.$

Moreover, $\pi_{\mathbf{f}}^{\scriptscriptstyle{\mathrm{SL}}}$ induces by
restriction an isomorphism

${\mathcal{Z}}(f_{1})/W^{\scriptscriptstyle{\mathrm{SL}}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\\{[x_{2}:\cdots:x_{n}:j]\in{\mathbb{P}}(d_{2},\dots,d_{n},|{\mathcal{A}}|)\leavevmode\nobreak\
|\leavevmode\nobreak\ j^{e}=P_{\mathbf{f}}(0,x_{2},\dots,x_{n})\\}.$

### 3.3. Eigenspaces, Springer theory

We now recall the basics of Springer and Lehrer-Springer theory: all the
results stated in this subsection can be found in [Spr74], [LS99a], [LS99b].
Note that some of the proofs have been simplified in [LM03]. Let us fix now a
natural number $e$. We set

$\mathclose{\,\mathit{d}}(e)=\\{1\leq k\leq n\leavevmode\nobreak\
|\leavevmode\nobreak\ e\leavevmode\nobreak\ \text{divides}\leavevmode\nobreak\
d_{k}\\},$ $\mathclose{\,\mathit{d}}^{*}(e)=\\{1\leq k\leq
n\leavevmode\nobreak\ |\leavevmode\nobreak\ e\leavevmode\nobreak\
\text{divides}\leavevmode\nobreak\ d_{k}^{*}\\},$
$\delta(e)=|\mathclose{\,\mathit{d}}(e)|\qquad\text{and}\qquad\delta^{*}(e)=|\mathclose{\,\mathit{d}}^{*}(e)|.$

With this notation, we have

(3.12) $\delta(e)=\max_{w\in W}\bigl{(}\dim V(w,\zeta_{e})\bigr{)}.$

In particular, $\zeta_{e}$ is an eigenvalue of some element of $W$ if and only
if $\delta(e)\neq 0$ that is, if and only if $e$ divides some degree of $W$.
In this case, we fix an element $w_{e}$ of $W$ such that

$\dim V(w_{e},\zeta_{e})=\delta(e).$

We set for simplification $V(e)=V(w_{e},\zeta_{e})$ and
$W(e)=W_{V(e)}/W(V(e))$: this subquotient of $W$ acts faithfully on $V(e)$.

If $f\in{\mathbb{C}}[V]$, we denote by $f^{[e]}$ its restriction to $V(e)$.
Note that if $i\notin\mathclose{\,\mathit{d}}(e)$, then $f_{i}^{[e]}=0$ by
Lemma 2.2. Let us recall here the results of Springer and Lehrer-Springer we
will need:

######  Theorem 3.13 (Springer, Lehrer-Springer).

Assume that $\delta(e)\neq 0$. Then:

  * ${\mathrm{(a)}}$

If $w\in W$, then there exists $x\in W$ such that $x(V(w,\zeta_{e}))\subset
V(e)$.

  * ${\mathrm{(b)}}$

$W(e)$ acts (faithfully) on $V(e)$ as a group generated by reflections.

  * ${\mathrm{(c)}}$

The family $(f_{k}^{[e]})_{k\in\mathclose{\,\mathit{d}}(e)}$ is a family of
fundamental invaraints of $W(e)$. In particular, the list of degrees of $W(e)$
consists of the degrees of $W$ which are divisible by $e$ (i.e.
${\mathrm{Deg}}(W(e))=(d_{k})_{k\in\mathclose{\,\mathit{d}}(e)}$).

  * ${\mathrm{(d)}}$

We have

$\bigcupop\displaylimits_{w\in W}V(w,\zeta_{e})=\bigcupop\displaylimits_{x\in
W}x(V(e))=\\{v\in V\leavevmode\nobreak\ |\leavevmode\nobreak\
\forall\leavevmode\nobreak\
k\in\\{1,2,\dots,n\\}\setminus\mathclose{\,\mathit{d}}(e),\leavevmode\nobreak\
f_{k}(v)=0\\}.$

  * ${\mathrm{(e)}}$

$\delta^{*}(e)\geq\delta(e)$ with equality if and only if
$W(V(w_{e},\zeta_{e}))=1$.

  * ${\mathrm{(f)}}$

If $\delta^{*}(e)=\delta(e)$, then $W(e)=W_{V(e)}=C_{W}(w_{e})$ and the family
of eigenvalues (with multiplicity) of $w_{e}$ is equal to
$(\zeta_{e}^{1-d_{k}})_{1\leq k\leq n}$. Moreover, if $w$ is such that $\dim
V(w,\zeta_{e})=\delta(e)$, then $w$ is conjugate to $w_{e}$.

######  Remark 3.14.

Let $k\in\\{1,2,\dots,n\\}$ be such that $\delta(d_{k})=\delta^{*}(d_{k})=1$.
Then $z_{k}=V(d_{k})$ is a line in $V$, so we can view it as an element of
${\mathbb{P}}(V)$. By Theorem 3.13(f), the stabilizer $W_{z_{k}}$ of $z_{k}$
acts faithfully on $V(d_{k})$, so it is cyclic and contains $w_{d_{k}}$. In
fact,

$W_{z_{k}}=\langle w_{d_{k}}\rangle.$

For proving this, let $e=|W_{z_{k}}|$. We just need to verify that $e=d_{k}$.
But $d_{k}$ divides $e$ and $\zeta_{e}$ is the eigenvalue of some elements of
$W$. So $e$ divides some $d_{j}$ by the remark following (3.12). Therefore,
$d_{k}$ divides $d_{j}$ and so $d_{k}=d_{j}$ because $\delta(d_{k})=1$. This
proves that $e=d_{k}$, as desired.

######  Corollary 3.15.

Assume that $\delta(e)=\delta^{*}(e)\neq 0$ and let
$k_{0}\in\\{1,2,\dots,n\\}$ be such that $d_{k_{0}}$ is divisible by $e$. Let
$v\in V(e)\setminus\\{0\\}$ and let $z=[v]$.

  * ${\mathrm{(a)}}$

The family of eigenvalues of $w_{e}$ for its action on the tangent space
${\mathrm{T}}_{z}({\mathbb{P}}(V))$ is equal to $(\zeta_{e}^{-d_{k}})_{k\neq
k_{0}}$.

  * ${\mathrm{(b)}}$

Let $f\in{\mathbb{C}}[V]^{W}$ be homogeneous of degree $d$ and assume that
$f(v)=0$. Then:

    * ${\mathrm{(b1)}}$

If $d\nequiv d_{k}\mod e$ for all $k\neq k_{0}$, then ${\mathcal{Z}}(f)$ is
singular at $z$.

    * ${\mathrm{(b2)}}$

Assume that ${\mathcal{Z}}(f)$ is smooth at $z$ and let $k_{1}\neq k_{0}$ be
such that $d\equiv d_{k_{1}}\mod e$ $($the existence of $k_{1}$ is guaranted
by $($_b1_ $))$. Then the family of eigenvalues of $w_{e}$ for its action on
the tangent space ${\mathrm{T}}_{z}({\mathcal{Z}}(f))$ is equal to
$(\zeta_{e}^{-d_{k}})_{k\neq k_{0},k_{1}}$.

###### Proof.

By permuting if necessary the degrees, we may assume that $k_{0}=1$. Note that
$\zeta_{e}^{1-d_{1}}=\zeta_{e}$. Choose a basis $(v_{1},\dots,v_{n})$ of $V$
such that $v=v_{1}$ and $w(v_{k})=\zeta_{e}^{1-d_{k}}v_{k}$ for all
$k\in\\{1,2,\dots,n\\}$ (see Theorem 3.13(e)).

(a) Identify ${\mathbb{P}}(V)$ with ${\mathbb{P}}^{n-1}({\mathbb{C}})$ through
the choice of this basis. Then the action of $w_{e}$ is transported to

$w_{e}\cdot[x_{1}:x_{2}:\cdots:x_{n}]=[\zeta_{e}x_{1}:\zeta_{e}^{1-d_{2}}x_{2}:\cdots:\zeta_{e}^{1-d_{n}}x_{n}]=[x_{1}:\zeta_{e}^{-d_{2}}x_{2}:\cdots:\zeta_{e}^{-d_{n}}x_{n}].$

Since $z=[1:0:\cdots:0]$, this shows (a).

(b) Let us work in the affine chart “$x_{1}=1$”, identified with
${\mathbf{A}}^{n-1}({\mathbb{C}})$ through the coordinates
$(x_{2},\dots,x_{n})$. The equation of the tangent space
${\mathrm{T}}_{z}({\mathcal{Z}}(f))$ is given in this chart by

$\sumop\displaylimits_{k=2}^{n}(\partial_{v_{k}}f)(v)x_{k}=0.$

By (2.1),

$\kern
0.5pt\vphantom{(\partial_{v_{k}}f)}^{w}\kern-0.5pt(\partial_{v_{k}}f)(v)=\zeta_{e}^{1-d_{k}}(\partial_{v_{k}}f)(v).$

But

$\kern
0.5pt\vphantom{(\partial_{v_{k}}f)}^{w}\kern-0.5pt(\partial_{v_{k}}f)(v)=(\partial_{v_{k}}f)(w^{-1}(v))=(\partial_{v_{k}}f)(\zeta_{e}^{-1}v).$

As $\partial_{v_{k}}f$ is homogeneous of degree $d-1$, this implies that

$\zeta_{e}^{1-d_{k}}(\partial_{v_{k}}f)(v)=\zeta_{e}^{1-d}(\partial_{v_{k}}f)(v).$

Therefore, if $d\nequiv d_{k}\mod e$ for all $k\in\\{2,\dots,n\\}$, we get
$(\partial_{v_{k}}f)(v)=0$ for all $k\in\\{2,\dots,n\\}$, and so
${\mathcal{Z}}(f)$ is singular ar $[v]$. This shows (b1).

Now, if ${\mathcal{Z}}(f)$ is smooth at $z$, then there exists
$k_{1}\in\\{2,\dots,n\\}$ such that $\partial_{v_{k_{1}}}f(v)\neq 0$, and in
particular $d\equiv d_{k_{1}}\mod e$. Then there exists $k\in\\{2,\dots,n\\}$
such that $\partial_{v_{k}}f(v)\neq 0$. This shows that the action of $w_{e}$
on the one-dimensional space
${\mathrm{T}}_{z}({\mathbb{P}}(V))/{\mathrm{T}}_{[v]}({\mathcal{Z}}(f))$ is
given by multiplication by $\zeta_{e}^{-d_{k_{1}}}=\zeta_{e}^{-d}$. The proof
of (b2) is complete. ∎

## 4\. Determining singularities

An important step for analyzing the properties of the K3 surfaces constructed
in the next section is to determine the singularities of the variety
${\mathcal{Z}}(f)/\Gamma$ in the cases we are interested in (here, $f$ is a
fundamental invariant of $W$ and is a subgroup of $W$). We provide in this
section two different tools that will be used in the sequel to this paper [BS-
II], [BS-III], where particular examples will be studied.

### 4.1. Stabilizers

The singularity of ${\mathcal{Z}}(f)/\Gamma$ at the -orbit of
$z\in{\mathcal{Z}}(f)$ depends on the singularity of ${\mathcal{Z}}(f)$ at $z$
and the action of z on this (eventually trivial) singularity. We investigate
here some facts about the stabilizers $W_{z}$ and their action on the tangent
space ${\mathrm{T}}_{z}({\mathcal{Z}}(f))$.

Let $f$ denote a homogeneous invariant of $W$, let $d$ denote its degree and
let $v\in V\setminus\\{0\\}$ be such that $f(v)=0$. We set
$z=[v]\in{\mathcal{Z}}(f)\subset{\mathbb{P}}(V)$. We denote by
$\theta_{z}:W_{z}\longrightarrow{\mathbb{C}}^{\times}$ the linear character
defined by $w(v)=\theta_{z}(w)v$ for all $w\in W_{z}$. Then
$W_{v}=\operatorname{{\mathrm{Ker}}}(\theta_{z})$ and we denote by
$e_{z}=|{\mathrm{Im}}(\theta_{z})|$. So there exists $w\in W_{z}$ such that
$\theta_{z}(w)=\zeta_{e_{z}}$. In other words, $v\in V(w,\zeta_{e_{z}})$ and
so, by Theorem 3.13(a), we may, and we will, assume that $v\in
V(w_{e_{z}},\zeta_{e_{z}})=V(e_{z})$. This shows that

(4.1) $W_{z}=W_{v}\langle w_{e_{z}}\rangle.$

Recall from §3.2 that $W_{v}$ is a parabolic subgroup of $W$ and so is
generated by reflections. Note the following useful facts:

  * ${\mathrm{(a)}}$

Let $m=|{\mathrm{Z}}(W)|$ (recall from (3.2) that
$m={\mathrm{gcd}}({\mathrm{Deg}}(W))$). Since
${\boldsymbol{\mu}}_{m}={\mathrm{Z}}(W)\subset W_{z}$, $m$ divides $e_{z}$.

  * ${\mathrm{(b)}}$

If $\delta(e_{z})=\delta^{*}(e_{z})$, $f(v)=0$ and ${\mathcal{Z}}(f)$ is
smooth at $z$, then the eigenvalues of $w_{e_{z}}$ on the tangent space
${\mathrm{T}}_{z}({\mathcal{Z}}(f))$ are given by Corollary 3.15.

  * ${\mathrm{(c)}}$

Let $P$ be a parabolic subgroup of $W$ of rank $n-2$ and assume that
${\mathcal{Z}}(f)$ is smooth. Then $\dim V^{P}=2$ and so
$L={\mathbb{P}}(V^{P})$ is a line in ${\mathbb{P}}(V)$. Then $L$ intersects
${\mathcal{Z}}(f)$ transversally by Corollary 2.8, so
$|L\cap{\mathcal{Z}}(f)|=d$ because $f$ has degree $d$. Moreover,

(4.2) If $z\in L\cap{\mathcal{Z}}(f)$, then $W_{v}=P$.

Indeed, $P\subset W_{v}$ by construction and, if this inclusion is strict,
this means that $W_{v}$ has rank $n-1$ or $n$. But it cannot have rank $n$ for
otherwise $W_{v}=W$ and $v=0$ (which is impossible). And it cannot have rank
$n-1$ because Corollary 2.4 would imply that ${\mathcal{Z}}(f)$ is singular at
$z$, contrarily to the hypothesis. This implies for instance that two smooth
points in $L\cap{\mathcal{Z}}(f)$ are in the same -orbit if and only if they
are in the same ${\mathrm{N}}(P)$-orbit.

Moreover, in this case, we have a $P$-equivariant isomorphism

(4.3) ${\mathrm{T}}_{[v]}({\mathcal{Z}}(f))\simeq V/V^{P}$

(see the proof of Corollary 2.8).

### 4.2. Singularities of double covers

If $n=4$, $\Gamma=W^{\scriptscriptstyle{\mathrm{SL}}}$ and $W$ is generated by
reflections of order $2$, then the surface ${\mathcal{Z}}(f)/\Gamma$ is the
double cover of a weighted projective plane. Most of (but not all) the
singularities of ${\mathcal{Z}}(f)/\Gamma$ may be then analyzed through the
singularities of the branch locus of this cover.

So we fix a double cover $\pi:Y\to X$ between two irreducible algebraic
surfaces and we assume that $Y$ is normal and $X$ is smooth. By the purity of
the branch locus, the branch locus $R$ of $\pi$ is empty or pure of
codimension $1$ (i.e. pure of dimension $1$). The next well-known fact (see
for instance [BPVdV84, Part III, §7]) will help us in our explicit
computations:

######  Proposition 4.4.

Let $y\in Y$ be such that $x=\pi(y)$ belongs to $R$. We assume that $x$ is an
ADE curve singularity of the branch locus $R$. Then $y$ is an ADE surface
singularity of the same type.

## 5\. Invariant K3 surfaces

Hypothesis. In this section, and only in this section, we assume that $n=4$
and that $W$ is irreducible and generated by reflections of order
$2$.$A_{\displaystyle{A_{A}}}$

### 5.1. Classification

We provide in Table 1 the list of irreducible complex reflection groups $W$ of
rank $4$ which are generated by reflections of order $2$ together with the
following informations: the order of $W$, the order of $W/{\mathrm{Z}}(W)$
(which is the group that acts faithfully on ${\mathbb{P}}(V)$), the order of
$W^{\prime}$ and the lists of degrees and codegrees. We also recall their
notation in Shephard-Todd classification [ST54] as well as their Coxeter name
whenever they are real.

${{{\begin{array}[]{!{\vline width 2pt} c !{\vline width 2pt} c|c|c|c!{\vline
width 2pt}}\cr\hrule
height=2.0pt}W&|W|&|W/{\mathrm{Z}}(W)|&|W^{\prime}|&\begin{array}[]{c}{\mathrm{Deg}}(W)\\\
{\mathrm{Codeg}}(W)\end{array}\\\ \cr\hrule
height=2.0pt}G(1,1,5)={\mathrm{W}}(A_{4})\simeq{\mathfrak
S}_{5}&120&120&60&\begin{array}[]{c}2,3,4,5\\\ 0,1,2,3\end{array}\\\ \hline\cr
G(e,e,4),e\geq
2&24e^{3}&24e^{3}/{\mathrm{gcd}}(e,4)&12e^{3}&\begin{array}[]{c}e,2e,3e,4\\\
0,e,2e,3e-4\end{array}\\\ \hline\cr G(2e,e,4),e\geq
1&384e^{3}&192e^{3}/{\mathrm{gcd}}(e,4)&96e^{3}&\begin{array}[]{c}2e,4e,6e,8\\\
0,2e,4e,6e\end{array}\\\ \hline\cr
G_{28}={\mathrm{W}}(F_{4})&1\,152&576&288&\begin{array}[]{c}2,6,8,12\\\
0,4,6,10\end{array}\\\ \hline\cr
G_{29}&7\,680&1\,920&3\,840&\begin{array}[]{c}4,8,12,20\\\
0,8,12,16\end{array}\\\ \hline\cr
G_{30}={\mathrm{W}}(H_{4})&14\,400&7\,200&7\,200&\begin{array}[]{c}2,12,20,30\\\
0,10,18,28\end{array}\\\ \hline\cr
G_{31}&46\,080&11\,520&23\,040&\begin{array}[]{c}8,12,20,24\\\
0,12,16,28\end{array}\\\ \cr\hrule height=2.0pt}\end{array}$ Table 1.
Irreducible complex reflection groups of rank $4$ generated by reflections of
order $2$

Recall that $G(2,1,4)={\mathrm{W}}(B_{4})$ and $G(2,2,4)={\mathrm{W}}(D_{4})$.
Note that the hypothesis on the order of the reflections implies that

(5.2) $|W^{\scriptscriptstyle{\mathrm{SL}}}|=\frac{|W|}{2}.$

In particular, $W^{\prime}\neq W^{\scriptscriptstyle{\mathrm{SL}}}$ if and
only if $W=G_{28}$ or $W=G(2e,e,4)$ for some $e\geq 1$. Also, note the
following diagram of non-trivial inclusions between those of the complex
reflection groups which are contained in a primitive one (here,
$H\stackrel{{\scriptstyle\vartriangleleft}}{{\lhook\joinrel\longrightarrow}}G$
means that $H$ is a normal subgroup of $G$).

(5.3) $\begin{array}[]{c}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
22.20903pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 46.20903pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
22.20905pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
65.41806pt\raise-40.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\scriptstyle{\vartriangleleft}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 127.10487pt\raise-40.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
65.41806pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
127.10487pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{G_{28}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
146.56737pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
183.04515pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
183.04515pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{G_{31}}$}}}}}}}{\hbox{\kern-3.0pt\raise-80.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
46.20903pt\raise-80.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{G(4,4,4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
90.6271pt\raise-80.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
99.62709pt\raise-80.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\scriptstyle{\vartriangleleft}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 114.62709pt\raise-80.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
114.62709pt\raise-80.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{G(4,2,4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
151.51971pt\raise-69.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
186.0577pt\raise-44.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}\ignorespaces\end{array}$

### 5.2. K3 surfaces

Equations of surfaces of the form ${\mathcal{Z}}(f)/W^{\prime}$ or
${\mathcal{Z}}(f)/W^{\scriptscriptstyle{\mathrm{SL}}}$ (where $f$ is a
fundamental invariant of degree $d$) in a weighted projective space are
provided by Propositions 3.8 and 3.11. Whenever some arithmetic conditions on
$d$ and the degrees of $W$ are satisfied, it can then be proven thanks to
results of Appendix Appendix A. Surfaces in weighted projective spaces (and
particularly Corollary A.3) that the canonical sheaf of
${\mathcal{Z}}(f)/W^{\prime}$ or
${\mathcal{Z}}(f)/W^{\scriptscriptstyle{\mathrm{SL}}}$ is trivial (provided
that ${\mathcal{Z}}(f)$ is normal, so that the quotient is also normal and the
canonical sheaf is well-defined): it turns out that, in most cases, the
quotient ${\mathcal{Z}}(f)/W^{\prime}$ or
${\mathcal{Z}}(f)/W^{\scriptscriptstyle{\mathrm{SL}}}$ has only ADE
singularities and positive Euler characteristic so that their minimal
resolution are K3 surfaces. A particular feature of these examples is that
their minimal resolution have always a big Picard number, as big as possible
compare to the number of moduli of the family. Note that some of these
examples were already studied by Barth and the second author [BS03]: we
revisit these cases and simplify some arguments using more theory about
complex reflection groups.

We denote by ${\mathcal{K}}_{3}$ the set of pairs $(W,d)$ where $W$ is an
irreducible complex reflection group of rank $4$ and $d$ is a positive integer
satisfying one of the following conditions:

  * •

$W=G(1,1,5)$, $G(4,2,4)$ or $G_{29}$, and $d=4$.

  * •

$W=G(2e,2e,4)$ with $e$ odd, and $d\in\\{4e,6e\\}$.

  * •

$W=G(4e,4e,4)$, and $d=4e$.

  * •

$W=G(2,1,4)$, and $d\in\\{4,6\\}$.

  * •

$W=G_{28}$, and $d\in\\{6,8\\}$.

  * •

$W=G_{30}$, and $d=12$.

  * •

$W=G_{31}$, and $d=20$.

######  Theorem 5.4.

Assume that $(W,d)\in{\mathcal{K}}_{3}$. Let be the subgroup $W^{\prime}$ or
$W^{\scriptscriptstyle{\mathrm{SL}}}$ of $W$ and let $f$ be a fundamental
invariant of $W$ of degree $d$ such that ${\mathcal{Z}}(f)$ has only ADE
singularities.

Then ${\mathcal{Z}}(f)/\Gamma$ is a K3 surface with ADE singularities.

The proof of Theorem 5.4 will be given in §5.5 and Section 6. As an immediate
consequence, we get:

######  Corollary 5.5.

Under the hypotheses of Theorem 5.4, the minimal resolution of
${\mathcal{Z}}(f)/\Gamma$ is a smooth projective K3 surface.

### 5.3. Numerical informations

Before proving this Theorem 5.4, let us make some remarks. By Propositions 3.8
and 3.11, the variety ${\mathcal{Z}}(f)/\Gamma$ is a weighted complete
intersection (see [Dol82, §3.2] for the definition) in a weighted projective
space (it is defined by one or two equations). If
$\Gamma=W^{\scriptscriptstyle{\mathrm{SL}}}$, then ${\mathcal{Z}}(f)/\Gamma$
is a weighted hypersurface in a weighted projective space of dimension $3$
(see Proposition 3.11). If $\Gamma=W^{\prime}$, then ${\mathcal{Z}}(f)/\Gamma$
is a codimension $2$ weighted complete intersection in a weighted projective
space of dimension $4$ (see Proposition 3.8). We give in Table 2 the list of
the weights of the ambient projective space as well as the list of the degrees
of the equations in all the different cases (we also give the description of
${\mathcal{Z}}(f)/W$ as a weighted projective space).

By looking at this Table 2, the reader might think that we have build
infinitely many families of K3 surfaces, by letting the integer $e$ vary in
the fourth and fifth group considered. However, as it will be explained in
§5.4 (see the isomorphisms (5.12), (5.13) and (5.15)), the general group with
parameter $e$ and the particular group for $e=1$ give exactly the same
families of surfaces.

Also, it turns out that $G(2,2,4)^{\prime}=G(2,1,4)^{\prime}$, and since
invariants of $G(2,1,4)={\mathrm{W}}(B_{4})$ are invariants for
$G(2,2,4)={\mathrm{W}}(D_{4})$, this shows that two of the families of K3
surfaces constructed with $G(2,1,4)$ are contained in families built from
$G(2,2,4)$: note however that, for these particular examples, having the two
points of view give different embeddings in weighted projective spaces.

As a consequence, we have build 15 families of K3 surfaces (note that the
families corresponding to the groups $G(4,2,4)$ and $G_{29}$ are
$0$-dimensional, as there is, up to scalar, a single quartic polynomial
invariant by each of these groups). If we exclude the “easy” case of the
quotient of a quartic by a finite subgroup of
${\mathbf{S}}{\mathbf{L}}_{4}({\mathbb{C}})$ (see §6.1), it remains $8$ non-
zero dimensional families of K3 surfaces whose construction is non-trivial.

${{{{{{{{{{{\begin{array}[]{!{\vline width 2pt} c !{\vline width 2pt}
c|c|c|c|c!{\vline width 2pt}}\cr\hrule
height=2.0pt}W&d&\Gamma\hfil&\text{Ambient
space}&\begin{array}[]{c}\text{degree(s) of}\\\
\text{equation(s)}\end{array}&{\mathcal{Z}}(f)/W\\\ \cr\hrule
height=2.0pt}\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathrm{W}}(A_{4})\simeq{\mathfrak
S}_{5}&4&W^{\prime}=W^{\scriptscriptstyle{\mathrm{SL}}}&{\mathbb{P}}(2,3,5,10)&20&{\mathbb{P}}(2,3,5)\\\
\cr\hrule height=0.8pt}\hbox{\multirowsetup$\begin{array}[]{c}G(2,1,4)\\\
={\mathrm{W}}(B_{4})\end{array}$}&\hbox{\multirowsetup
4}&W^{\prime}&\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathbb{P}}(1,3,4,6,2)&12,4&\hbox{\multirowsetup${\mathbb{P}}(1,3,4)$}\\\
\cr\cline{3-5}\cr\cr&&W^{\scriptscriptstyle{\mathrm{SL}}}&\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathbb{P}}(1,3,4,8)&16&\\\
\cr\cline{2-6}\cr\cr&\hbox{\multirowsetup
6}&W^{\prime}&\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathbb{P}}(1,1,2,3,1)&6,2&\hbox{\multirowsetup${\mathbb{P}}(1,1,2)$}\\\
\cr\cline{3-5}\cr\cr&&W^{\scriptscriptstyle{\mathrm{SL}}}&\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathbb{P}}(1,1,2,4)&8&\\\
\cr\hrule
height=0.8pt}\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}\hbox{\multirowsetup$G(4,2,4)$}&\hbox{\multirowsetup$4$}&W^{\prime}&{\mathbb{P}}(2,3,2,1,6)&2,12&\hbox{\multirowsetup${\mathbb{P}}(1,3,1)$}\\\
\cr\cline{3-5}\cr\cr&&W^{\scriptscriptstyle{\mathrm{SL}}}&\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathbb{P}}(2,3,2,7)&14&\\\
\cr\hrule height=0.8pt}\hbox{\multirowsetup$\begin{array}[]{c}G(2e,2e,4)\\\
\text{$e$
odd}\end{array}$}&4e&W^{\prime}=W^{\scriptscriptstyle{\mathrm{SL}}}&\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathbb{P}}(1,3,2,6)&12&{\mathbb{P}}(1,3,2)\\\
\cr\cline{2-6}\cr\cr&6e&W^{\prime}=W^{\scriptscriptstyle{\mathrm{SL}}}&\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathbb{P}}(1,1,1,3)&6&{\mathbb{P}}(1,1,1)\\\
\cr\hrule
height=0.8pt}G(4e,4e,4)&4e&W^{\prime}=W^{\scriptscriptstyle{\mathrm{SL}}}&\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathbb{P}}(2,3,1,6)&12&{\mathbb{P}}(2,3,1)\\\
\cr\hrule height=0.8pt}\hbox{\multirowsetup$\begin{array}[]{c}G_{28}\\\
={\mathrm{W}}(F_{4})\end{array}$}&\hbox{\multirowsetup
6}&W^{\prime}&\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathbb{P}}(1,2,3,3,3)&6,6&\hbox{\multirowsetup${\mathbb{P}}(1,2,3)$}\\\
\cr\cline{3-5}\cr\cr&&W^{\scriptscriptstyle{\mathrm{SL}}}&\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathbb{P}}(1,2,3,6)&12&\\\
\cr\cline{2-6}\cr\cr&\hbox{\multirowsetup
8}&W^{\prime}&\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathbb{P}}(1,1,2,2,2)&4,4&\hbox{\multirowsetup${\mathbb{P}}(1,1,2)$}\\\
\cr\cline{3-5}\cr\cr&&W^{\scriptscriptstyle{\mathrm{SL}}}&\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathbb{P}}(1,1,2,4)&8&\\\
\cr\hrule
height=0.8pt}G_{29}&4&W^{\prime}=W^{\scriptscriptstyle{\mathrm{SL}}}&\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathbb{P}}(2,3,5,10)&20&{\mathbb{P}}(2,3,5)\\\
\cr\hrule
height=0.8pt}G_{30}={\mathrm{W}}(H_{4})&12&W^{\prime}=W^{\scriptscriptstyle{\mathrm{SL}}}&\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathbb{P}}(1,2,3,6)&12&{\mathbb{P}}(1,2,3)\\\
\cr\hrule
height=0.8pt}G_{31}&20&W^{\prime}=W^{\scriptscriptstyle{\mathrm{SL}}}&\vphantom{\displaystyle{\frac{\displaystyle{A}}{\displaystyle{A}}}}{\mathbb{P}}(2,1,2,5)&10&{\mathbb{P}}(1,1,1)\\\
\cr\hrule height=2.0pt}\end{array}$ Table 2. Weights of ambient projective
spaces and degree(s) of equation(s) of ${\mathcal{Z}}(f)/\Gamma$

Hypothesis and notation. From now on, and until the end of this paper, we
assume that $(W,d)\in{\mathcal{K}}_{3}$, that is the subgroup $W^{\prime}$ or
$W^{\scriptscriptstyle{\mathrm{SL}}}$ of $W$, and that $f$ is a fundamental
invariant of $W$ of degree $d$ such that ${\mathcal{Z}}(f)$ has only ADE
singularities. We also fix a family ${\mathbf{f}}=(f,f_{1},f_{2},f_{3})$ of
fundamental invariants containing $f$ and we set
$d_{i}=\deg(f_{i})$.$A_{\displaystyle{A_{A}}}$

###### Proof of the results given in Table 2.

The proof follows from Table 1 and a case-by-case analysis. We will not give
details for all cases, we will only treat two cases (the reader can easily
check that all other cases can be treated similarly).

$\bullet$ Assume that $(W,d)=(G_{31},20)$ and that $\Gamma=W^{\prime}$
($=W^{\scriptscriptstyle{\mathrm{SL}}}$). Then $(d_{1},d_{2},d_{3})=(8,12,24)$
by Table 1 and $|{\mathcal{A}}|=60$ by (3.1) and (3.4). It then follows from
Proposition 3.11 that

${\mathcal{Z}}(f)/\Gamma=\\{[x_{1}:x_{2}:x_{3}:j]\in{\mathbb{P}}(8,12,24,60)\leavevmode\nobreak\
|\leavevmode\nobreak\ j^{2}=P_{\mathbf{f}}(0,x_{1},x_{2},x_{3})\\}.$

But ${\mathbb{P}}(8,12,24,60)={\mathbb{P}}(2,3,6,15)={\mathbb{P}}(2,1,2,5)$,
so that

${\mathcal{Z}}(f)/\Gamma=\\{[y_{1}:y_{2}:y_{3}:j]\in{\mathbb{P}}(2,1,2,5)\leavevmode\nobreak\
|\leavevmode\nobreak\ j^{2}=Q(y_{1},y_{2},y_{3})\\}$

for some polynomial $Q\in{\mathbb{C}}[Y_{1},Y_{2},Y_{3}]$. Hence
${\mathcal{Z}}(f)/\Gamma$ is defined by an equation of degree $10$ in
${\mathbb{P}}(2,1,2,5)$, as expected.

Finally, by Proposition 3.11,
${\mathcal{Z}}(f)/W\simeq{\mathbb{P}}(8,12,24)={\mathbb{P}}(2,3,6)\simeq{\mathbb{P}}(2,1,2)\simeq{\mathbb{P}}(1,1,1)$,
as expected.

$\bullet$ Assume that $(W,d)=(G_{28},6)$ and $\Gamma=W^{\prime}$. Then
$(d_{1},d_{2},d_{3})=(2,8,12)$ by Table 1. Note that $W^{\prime}\neq
W^{\scriptscriptstyle{\mathrm{SL}}}$ and that there are two $W$-orbits 1 and 2
of reflecting hyperplanes, which are both of cardinality $12$. It then follows
from Proposition 3.8, that

${\mathcal{Z}}(f)/\Gamma=\\{[x_{1}:x_{2}:x_{3}:j_{1}:j_{2}]\in{\mathbb{P}}(2,8,12,12,12)\leavevmode\nobreak\
|\leavevmode\nobreak\ j_{1}^{2}=P_{{\mathbf{f}},{}_{1}}(0,x_{1},x_{2},x_{3})$
$\hphantom{AAAAAAAAAAAAAAAA}\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\
j_{2}^{2}=P_{{\mathbf{f}},{}_{2}}(0,x_{1},x_{2},x_{3})\\}.$

But
${\mathbb{P}}(2,8,12,12,12)={\mathbb{P}}(1,4,6,6,6)={\mathbb{P}}(1,2,3,3,3)$,
so that

${\mathcal{Z}}(f)/\Gamma=\\{[y_{1}:y_{2}:y_{3}:j_{1}:j_{2}]\in{\mathbb{P}}(1,2,3,3,3)\leavevmode\nobreak\
|\leavevmode\nobreak\ j_{1}^{2}=Q_{1}(y_{1},y_{2},y_{3})\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ j_{2}^{2}=Q_{2}(y_{1},y_{3},y_{4})\\}$

for some polynomials $Q_{1}$ and $Q_{2}$ in ${\mathbb{C}}[Y_{1},Y_{2},Y_{3}]$.
So ${\mathcal{Z}}(f)/\Gamma$ is defined by two equations of degree $6$ in
${\mathbb{P}}(1,2,3,3,3)$, as expected.

Finally, by Proposition 3.8,
${\mathcal{Z}}(f)/W\simeq{\mathbb{P}}(2,8,12)={\mathbb{P}}(1,4,6)\simeq{\mathbb{P}}(1,2,3)$,
as expected. ∎

######  Remark 5.7.

The arithmetic of degrees and the classification of reflection groups imply
that it does not seem possible to find a complex reflection $W$ and a degree
$d$ of $W$ such that $W$ is not generated by reflections of order $2$ and
${\mathcal{Z}}(f)/\Gamma$ has a trivial canonical sheaf, except whenever
$d=4$. But this is in some sense the less exciting case, as it is shown by the
argument given in §6.1 below.

Also, note that if $e\notin\\{1,2,4\\}$, then $G(e,e,4)$ has a unique
invariant of degree $4$ that defines a quartic in
${\mathbb{P}}^{3}({\mathbb{C}})$, but this invariant is equal to $xyzt$, and
so ${\mathcal{Z}}(f)$ is not irreducible and does not fulfill the hypothesis
of Theorem 5.4. That is why this case does not appear in the list
${\mathcal{K}}_{3}$.

######  Remark 5.8.

If $(W,d)=(G_{28},6)$ or $(G_{30},12)$, and $\Gamma=W^{\prime}$, then the
above result was obtained by Barth and the second author [BS03]: the group was
denoted by $G_{d}$ in their paper (this must not be confused with Shephard-
Todd notation).

### 5.4. About the families attached to ${\boldsymbol{G(2e,2e,4)}}$

Assume in this subsection, and only in this subsection, that $W=G(2e,2e,4)$
for some $e$. Recall that $G(2e,2e,4)$ is the group of monomial matrices in
${\mathbf{G}}{\mathbf{L}}_{4}({\mathbb{C}})$ with coefficients in
${\boldsymbol{\mu}}_{2e}$ and such that the product of the non-zero
coefficients is equal to $1$. Note that this implies that
$W^{\prime}=W^{\scriptscriptstyle{\mathrm{SL}}}$.

If $1\leq k\leq 4$, we denote by $\sigma_{k}$ the $j$-th elementary symmetric
function in the variables $x$, $y$, $z$, $t$, and let

$J_{1}=(x-y)(x-z)(x-t)(y-z)(y-t)(z-t).$

If $p\in{\mathbb{C}}[x,y,z,t]$ and $l\geq 1$ is an integer, we set
$p[l]=p(x^{l},y^{l},z^{l},t^{l})\in{\mathbb{C}}[x,y,z,t]$. For instance,
$\sigma_{1}[l]=\Sigma(x^{l})$. Then
$(\sigma_{1},\sigma_{1}[2],\sigma_{1}[3],\sigma_{4})$ is a family of
fundamental invariants of $G(1,1,4)\simeq{\mathfrak S}_{4}$. So there exists a
unique polynomial $P\in{\mathbb{C}}[x_{1},x_{2},x_{3},x_{4}]$ such that

(5.9) $J_{1}^{2}=P(\sigma_{1},\sigma_{1}[2],\sigma_{1}[3],\sigma_{4}).$

We do not need here the explicit form of $P$. If $a$, $b$, $c\in{\mathbb{C}}$,
we set

$F_{a,b,c}=a\sigma_{1}[4]+b\sigma_{4}+c\sigma_{1}[2]^{2}=a\Sigma(x^{4})+bxyzt+c(\Sigma(x^{2}))^{2},$
$G_{a,b,c}=\sigma_{1}[6]+a\sigma_{1}[4]\sigma_{1}[2]+b\sigma_{1}[2]^{3}+c\sigma_{1}[2]\sigma_{4}.$

With this notation, we may, and we will, choose as a family of fundamental
invariants of $W$ the family
$(\sigma_{1}[2e],\sigma_{1}[4e],\sigma_{1}[6e],\sigma_{4})$. Note that the
element $J\in{\mathbb{C}}[x,y,z,t]^{W^{\prime}}$ defined in §3.2 is equal to
$J_{1}[2e]$ (up to a scalar). Applying the endomorphism $p\mapsto p[2e]$ of
${\mathbb{C}}[x,y,z,t]$ to the formula (5.9) and using Proposition 3.11 gives

(5.10)
${\mathbb{P}}(V)/W^{\prime}\\!\simeq\\!\\{[x_{1}\\!:\\!x_{2}\\!:\\!x_{3}\\!:\\!x_{4}\\!:\\!j]\in{\mathbb{P}}(2e,4e,6e,4,12e)|j^{2}\\!=\\!P(x_{1},x_{2},x_{3},x_{4}^{2e})\\},$

because $\sigma_{4}[2e]=\sigma_{4}^{2e}$. Let us examine the situation
according to the parity of $e$.

#### 5.4.1. The case where $e$ is odd

Assume here that $e$ is odd. Then

${\mathbb{P}}(2e,4e,6e,4,12e)={\mathbb{P}}(e,2e,3e,2,6e)\simeq{\mathbb{P}}(1,2,3,2,6).$

So it follows from (5.10) that

(5.11)
${\mathbb{P}}(V)/W^{\prime}\simeq\\{[x_{1}:x_{2}:x_{3}:x_{4}:j]\in{\mathbb{P}}(1,2,3,2,6)\leavevmode\nobreak\
|\leavevmode\nobreak\ j^{2}=P(x_{1},x_{2},x_{3},x_{4}^{2})\\}.$

Now, a fundamental invariant of degree $6e$ of $W$ if of the form
$G_{a,b,c}[e]$. We deduce from (5.11) that

${\mathcal{Z}}(G_{a,b,c}[e])/W^{\prime}\simeq\\{[x_{1}:x_{2}:x_{4}:j]\in{\mathbb{P}}(1,2,2,6)\leavevmode\nobreak\
|\leavevmode\nobreak\
j^{2}=P(x_{1},x_{2},-ax_{1}x_{2}-bx_{1}^{3}-cx_{1}x_{4},x_{4}^{2})\\}$

and in particular

(5.12)
${\mathcal{Z}}(G_{a,b,c}[e])/G(2e,2e,4)^{\prime}={\mathcal{Z}}(G_{a,b,c})/G(2,2,4)^{\prime}.$

Similarly, a fundamental invariant of degree $4e$ of $W$ is of the form
$F_{a,b,c}[e]$ with $(a,b)\neq 0$ and

(5.13)
${\mathcal{Z}}(F_{a,b,c}[e])/G(2e,2e,4)^{\prime}={\mathcal{Z}}(F_{a,b,c})/G(2,2,4)^{\prime}.$

This shows that the varieties
${\mathcal{Z}}(G_{a,b,c}[e])/G(2e,2e,4)^{\prime}$ and
${\mathcal{Z}}(F_{a,b,c}[e])/G(2e,2e,4)^{\prime}$ do not depend on $e$.

#### 5.4.2. The case where $e$ is even

Assume here that $e=2e^{\prime}$ for some $e^{\prime}\geq 1$. Then

${\mathbb{P}}(2e,4e,6e,4,12e)={\mathbb{P}}(e^{\prime},2e^{\prime},3e^{\prime},1,6e^{\prime})={\mathbb{P}}(1,2,3,1,6).$

So it follows from (5.10) that

(5.14)
${\mathbb{P}}(V)/W^{\prime}\simeq\\{[x_{1}:x_{2}:x_{3}:x_{4}:j]\in{\mathbb{P}}(1,2,3,1,6)\leavevmode\nobreak\
|\leavevmode\nobreak\ j^{2}=P(x_{1},x_{2},x_{3},x_{4}^{4})\\}.$

Now, a fundamental invariant of $G(4e^{\prime},4e^{\prime},4)$ of degree
$4e^{\prime}$ is of the form $F_{a,b,c}[e^{\prime}]$ and a similar argument as
before shows that

(5.15)
${\mathcal{Z}}(F_{a,b,c}[e^{\prime}])/G(4e^{\prime},4e^{\prime},4)^{\prime}={\mathcal{Z}}(F_{a,b,c})/G(4,4,4)^{\prime}.$

Again, the variety
${\mathcal{Z}}(F_{a,b,c}[e^{\prime}])/G(4e^{\prime},4e^{\prime},4)^{\prime}$
does not depend on $e^{\prime}$.

#### 5.4.3. Complements

Note for future reference (see §6.3) the following fact:

######  Lemma 5.16.

If ${\mathcal{Z}}(F_{a,b,c})$ is irreducible, then it is smooth or has only
$A_{1}$ singularities.

###### Proof.

Assume that ${\mathcal{Z}}(F_{a,b,c})$ is irreducible and singular. Let us
first assume that $a=0$. Then we may assume that $b=1$ and the irreducibility
of ${\mathcal{Z}}(F_{0,1,c})$ forces $c\neq 0$. An easy computation then shows
that the only singular points of ${\mathcal{Z}}(F_{0,1,c})$ are the ones
belonging to the $G(2,2,4)$-orbit of $p=[0:0:i:1]$. But the homogeneous
component of degree $2$ of $F_{0,1,c}(x,y,i+z,1)$ is $ixy-4z^{2}$, which is a
non-degenerate quadratic form in $x$, $y$, $z$. So $p$ is an $A_{1}$
singularity of ${\mathcal{Z}}(F)$, as expected.

Le us now assume that $a\neq 0$, and even that $a=1$. Assume that we have
found $(b_{0},c_{0})\in{\mathbb{C}}^{2}$ such that
${\mathcal{Z}}(F_{1,b_{0},c_{0}})$ admits a singular point
$q=[x_{0}:y_{0}:z_{0}:t_{0}]$ which is not an $A_{1}$ singularity. Then, by
permuting the coordinates if necessary, we may assume that $t_{0}\neq 0$. So
let us work in the affine chart $t\neq 0$ and set
$F_{b,c}^{\circ}=F_{1,b,c}(x,y,z,1)$. Let $H_{b,c}$ denote the Hessian matrix
of $F_{b,c}^{\circ}$. Then $(x_{0},y_{0},z_{0},b_{0},c_{0})$ belongs to the
variety

${\mathcal{X}}=\\{(x,y,z,b,c)\in{\mathbb{A}}^{5}({\mathbb{C}})\leavevmode\nobreak\
|\leavevmode\nobreak\ F^{\circ}_{b,c}(x,y,z)=\frac{\partial
F^{\circ}_{b,c}}{\partial x}(x,y,z)=$ $\frac{\partial
F^{\circ}_{b,c}}{\partial y}(x,y,z)=\frac{\partial F^{\circ}_{b,c}}{\partial
z}(x,y,z)=\det(H_{b,c}(x,y,z))=0\\}.$

Now let $\pi:{\mathbb{A}}^{5}({\mathbb{C}})\to{\mathbb{A}}^{2}({\mathbb{C}})$,
$(x,y,z,\beta,\gamma)\mapsto(\beta,\gamma)$. A Magma [BCP97] computation:

>
>     > A5<x,y,z,b,c>:=AffineSpace(Rationals(),5);
>     > A2<b0,c0>:=AffineSpace(Rationals(),2);
>     > pi:=map<A5 -> A2 | [b,c]>;
>     > Fbco:=(x^4+y^4+z^4+1)+b*x*y*z+c*(x^2+y^2+z^2+1)^2;
>     > Hbc:=Matrix(CoordinateRing(A5),3,3,
>     >   [[Derivative(Derivative(Fbco,k),l) :
>     >     k in [1..3]] : l in [1..3]]);
>     > X:=Scheme(A5,[Fbco] cat [Derivative(Fbco,k) : k in [1,2,3]]
>     >   cat [Determinant(Hbc)]);
>     > MinimalBasis(ReducedSubscheme(pi(X)));
>     [
>         c0 + 1/2,
>         b0^2 - 16
>     ]
>

shows that $\pi({\mathcal{X}})=\\{(4,-1/2),(-4,-1/2)\\}$. This implies that
$(b_{0},c_{0})\in\\{(4,-1/2),(-4,-1/2)\\}$. But $F_{4,-1/2}$ and $F_{-4,-1/2}$
are not irreducible (they are divisible by $x-y-z+t$ and $x+y+z-t$
respectively): this contradicts the hypothesis. ∎

######  Remark 5.17.

Observe that the previous family ${\mathcal{Z}}(F_{a,b,c}$) with $(a,b)=(1,0)$
was studied in [Sar04], where it is shown that the family contains exactly
four singular surfaces with $4,8,12,16$ $A_{1}$–singularities. In particular
the surface with $16$ $A_{1}$–singularities is a Kummer K3 surface.

### 5.5. Proof of Theorem 5.4

The rest of this paper is devoted to the proof of this Theorem 5.4. Note that
it proceeds by a case-by-case analysis, but this case-by-case analysis is
widely simplified by the general facts about complex reflection groups
recalled in the previous sections.

###### Proof of Theorem 5.4.

Assume that the hypotheses of Theorem 5.4 are satisfied. For proving that
${\mathcal{Z}}(f)/\Gamma$ is a K3 surface with ADE singularities, we need to
show the following facts:

  * ${\mathrm{(S)}}$

The surface ${\mathcal{Z}}(f)/\Gamma$ has only ADE singularities.

  * ${\mathrm{(E)}}$

The Euler characteristic of ${\mathcal{Z}}(f)/\Gamma$ is positive.

  * ${\mathrm{(C)}}$

The canonical divisor of ${\mathcal{Z}}(f)/\Gamma$ is trivial.

Indeed, if ${\tilde{X}}$ denotes the minimal resolution of
${\mathcal{Z}}(f)/\Gamma$ and if (S), (E) and (C) are proved, then
${\tilde{X}}$ has a trivial canonical sheaf by (S) and (C), so by the
classification of smooth algebraic surfaces ${\tilde{X}}$ is a K3 surface or
an abelian surface. But, by (S), the Euler characteristic of ${\tilde{X}}$ is
greater than or equal to the one of ${\mathcal{Z}}(f)/\Gamma$, so is also
positive by (E). Since the Euler characteristic of an abelian surface is $0$,
we deduce that ${\tilde{X}}$ is a smooth K3 surface.

The technical step is to prove (S), namely that ${\mathcal{Z}}(f)/\Gamma$ has
only ADE singularities. This will be postponed to the next Section 6. So
assume here that (S) is proved.

Let us now prove the statement (E), namely that the Euler characteristic of
${\mathcal{Z}}(f)/\Gamma$ is positive. Since ${\mathcal{Z}}(f)$ has only
isolated singularities by (S), it follows from [Dim92, Theorem 4.3] that
${\mathrm{H}}^{1}({\mathcal{Z}}(f),{\mathbb{C}})=0$. Since ${\mathcal{Z}}(f)$
has only ADE singularities, it is rationally smooth [KL79, Definition A1]. As
it is also projective, one can apply Poincaré duality and so
${\mathrm{H}}^{3}({\mathcal{Z}}(f),{\mathbb{C}})$ is the dual of
${\mathrm{H}}^{1}({\mathcal{Z}}(f),{\mathbb{C}})$, hence is equal to $0$. So
${\mathcal{Z}}(f)$ has no odd cohomology and since
${\mathrm{H}}^{j}({\mathcal{Z}}(f)/\Gamma,{\mathbb{C}})={\mathrm{H}}^{j}({\mathcal{Z}}(f),{\mathbb{C}})$,
this shows that ${\mathcal{Z}}(f)/\Gamma$ has no odd cohomology. So its Euler
characteristic is positive.

Now it remains to prove (C), namely that the canonical sheaf of
$X={\mathcal{Z}}(f)/\Gamma$ is trivial. For this, we use Corollary A.3, so we
need to prove that $X$ satisfies the hypotheses (H1), (H2), (H3), (H4) and
(H5) of Appendix Appendix A. Surfaces in weighted projective spaces.
Statements (H1), (H2) and (H4) are easily checked thanks to Table 2 while (H5)
follows from (S). So it remains to prove (H3), namely that $X$ is a well-
formed weighted complete intersection. There are two cases:

  * $\bullet$

If $\Gamma=W^{\scriptscriptstyle{\mathrm{SL}}}$, then $X$ is a weighted
hypersurface of degree $e$ in some ${\mathbb{P}}(l_{0},l_{1},l_{2},l_{3})$,
and, according to [IF00, §6.10], $X$ is well-formed if, for all $0\leq a<b\leq
3$, $\gcd(l_{a},l_{b})$ divides $e$. This is easily checked with Table 2.

  * $\bullet$

If $\Gamma=W^{\prime}\neq W^{\scriptscriptstyle{\mathrm{SL}}}$, then $X$ is a
weighted complete intersection defined by two equations of degree $e_{1}$ and
$e_{2}$ in some ${\mathbb{P}}(l_{0},l_{1},l_{2},l_{3},l_{4})$, and, according
to [IF00, §6.11], $X$ is well-formed if the following two properties are
satisfied:

    * –

For all $0\leq a<b\leq 4$, $\gcd(l_{a},l_{b})$ divides $e_{1}$ or $e_{2}$.

    * –

For all $0\leq a<b<c\leq 4$, $\gcd(l_{a},l_{b},l_{c})$ divides $e_{1}$ and
$e_{2}$.

Again, this is easily checked with Table 2.

The proof of Theorem 5.4 is complete, up to the proof of (S). ∎

## 6\. Singularities of ${\mathcal{Z}}(f)/\Gamma$

The aim of this section is to complete the proof of Theorem 5.4, by proving
that, under its hypotheses, the surface ${\mathcal{Z}}(f)/\Gamma$ has only ADE
singularities. The first subsection is devoted to the (easy) case where $d=4$
while the third subsection deals with the case where $(W,d)=(G(2e,2e,4),4e)$
for some odd $e$ by reduction to the $d=4$ case. The other cases are treated
in the second and fourth section.

### 6.1. The case ${\boldsymbol{d=4}}$

Assume in this subsection, and only in this subsection, that $d=4$. This case
is somewhat particular and requires its own treatment. It is also well known
in the literature, but we recall the discussion for convenience of the reader.
First, note that the hypothesis implies that ${\mathcal{Z}}(f)$ is already a
K3 surface (with eventually ADE singularities) and we denote by $\omega$ a
non-degenerate global holomorphic $2$-form on the smooth locus (it is well-
defined up to a scalar). By hypothesis,
$\Gamma\subset{\mathbf{S}}{\mathbf{L}}_{4}({\mathbb{C}})$, so preserves
$\omega$. So ${\mathcal{Z}}(f)/\Gamma$ inherits a non-degenerate global
holomorphic $2$-form $\omega$ on its smooth locus.

Now, let $p:X\to{\mathcal{Z}}(f)$ denote a minimal resolution of
${\mathcal{Z}}(f)$, and let $\omega_{X}$ denote the unique non-degenerate
$2$-form on $X$ extending $\omega$. Then $X$ is a K3 surface which inherits an
action of which stabilizes $\omega$: so this is a symplectic action and so
$X/\Gamma$ is a K3 surface with ADE singularities [Nik76, §5]. Let $q:Y\to
X/\Gamma$ denote a minimal resolution. We have a commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\displaystyle{p}}$$\textstyle{{\mathcal{Z}}(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\displaystyle{q}}$$\textstyle{X/\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\displaystyle{p}}$$\textstyle{{\mathcal{Z}}(f)/\Gamma,}$

where $p$ is induced by $p$. This shows that $p\circ
q:Y\to{\mathcal{Z}}(f)/\Gamma$ is a symplectic resolution, so
${\mathcal{Z}}(f)/\Gamma$ is a K3 surface with ADE singularities (and $Y$ is
its minimal resolution). This completes the proof of Theorem 5.4 whenever
$d=4$.

### 6.2. The case where ${\boldsymbol{{\mathcal{Z}}(f)/W}}$ is smooth

Assume in this subsection, and only in this subsection, that
${\mathcal{Z}}(f)/W$ is smooth. By examining Table 2, this occurs only if
$(W,d)=(G(2e,2e,4),6e)$ or $(G_{31},20)$. In both cases,
$\Gamma=W^{\prime}=W^{\scriptscriptstyle{\mathrm{SL}}}$ is of index $2$ in
$W$, and so the result follows from Corollary B.7.

### 6.3. The case where ${\boldsymbol{W=G(2e,2e,4)}}$

Assume in this subsection, and only in this subsection, that $W=G(2e,2e,4)$.
The case where $e$ is odd and $d=6e$ is treated in the previous subsection
6.2. If $e$ is odd and $d=4e$, then it follows from the isomorphism (5.13) and
Lemma 5.16 that we may assume that $e=1$. Then $d=4$ and this case is treated
in §6.1. If $e=2e^{\prime}$ is even and $d=4e^{\prime}$, then it follows from
the isomorphism (5.13) and Lemma 5.16 that we may assume that $e^{\prime}=1$.
Then $d=4$ and this case is treated in §6.1.

### 6.4. Remaining cases

According to the cases treated in §6.1, §6.2 and §6.3, we may now work under
the following hypothesis:

Hypothesis. From now on, and until the end of this section, we assume that
$d\neq 4$ and $W\in\\{G(2,1,4),G_{28},G_{30}\\}$.$A_{\displaystyle{A_{A}}}$

#### 6.4.1. The case where $\Gamma=W^{\scriptscriptstyle{\mathrm{SL}}}$

Assume in this subsection, and only in this subsection, that
$\Gamma=W^{\scriptscriptstyle{\mathrm{SL}}}$. In this case, is a subgroup of
index $2$ of $W$. Recall from Propositions 3.3 and 3.11 that

(6.1) ${\mathcal{Z}}(f)/W\simeq{\mathbb{P}}(d_{1},d_{2},d_{3})$

and

(6.2)
${\mathcal{Z}}(f)/W^{\scriptscriptstyle{\mathrm{SL}}}\simeq\\{[x_{1}:x_{2}:x_{3}:j]\in{\mathbb{P}}(d_{1},d_{2},d_{3},|{\mathcal{A}}|)\leavevmode\nobreak\
|\leavevmode\nobreak\ j^{2}=P_{\mathbf{f}}(0,x_{1},x_{2},x_{3})\\}.$

We denote by
$\rho:{\mathcal{Z}}(f)/W^{\scriptscriptstyle{\mathrm{SL}}}\longrightarrow{\mathcal{Z}}(f)/W$
the canonical map, let ${\mathcal{U}}$ denote the smooth locus of
${\mathbb{P}}(d_{1},d_{2},d_{3})$ and let ${\mathcal{S}}$ denote the set of
singular points of ${\mathbb{P}}(d_{1},d_{2},d_{3})$.

Now, ${\mathcal{U}}=\pi_{\mathbf{f}}^{-1}({\mathcal{U}})/W$ is smooth so
$\rho^{-1}({\mathcal{U}})=\pi_{\mathbf{f}}^{-1}({\mathcal{U}})/\Gamma$
contains only ADE singularities by Corollary B.7 (because
$\Gamma=W^{\scriptscriptstyle{\mathrm{SL}}}$ has index $2$ in $W$). Hence, it
remains to show that the points in $\rho^{-1}({\mathcal{S}})$ are smooth or
ADE singularities. Let $p_{1}=[1:0:0]$, $p_{2}=[0:1:0]$ and $p_{3}=[0:0:1]$ in
${\mathbb{P}}(d_{1},d_{2},d_{3})$. Then
${\mathcal{S}}\subset\\{p_{1},p_{2},p_{3}\\}$. The following fact is checked
by a case-by-case analysis:

######  Lemma 6.3.

Assume that $d\neq 4$ and $W\in\\{G(2,1,4),G_{28},G_{30}\\}$. If
$p_{k}\in{\mathbb{P}}(d_{1},d_{2},d_{3})$ is singular, then:

  * ${\mathrm{(a)}}$

$\delta(d_{k})=\delta^{*}(d_{k})=1$.

  * ${\mathrm{(b)}}$

$\det(w_{d_{k}})=1$.

  * ${\mathrm{(c)}}$

$p_{k}$ is an $A_{j}$ singularity of ${\mathbb{P}}(d_{1},d_{2},d_{3})$ for
some $j\geq 1$.

The proof will be given below. Let us first explain why this lemma might help
to check that the points in $\rho^{-1}({\mathcal{S}})$ are smooth or ADE
singularities. So let $p_{k}\in{\mathcal{S}}$ and let
${}_{k}=\pi_{\mathbf{f}}^{-1}(p_{k})$. Then

${}_{k}=\\{p\in{\mathbb{P}}(V)\leavevmode\nobreak\ |\leavevmode\nobreak\
\forall\leavevmode\nobreak\ 1\leq j\neq k\leq 3,\leavevmode\nobreak\
f_{j}(p)=f_{k}(p)=0\\}.$

By Lemma 6.3, $\dim V(d_{k})=1$, so we might view $V(d_{k})\in{\mathbb{P}}(V)$
as a point $z_{k}\in{\mathcal{Z}}(f)$. We denote by
$z_{k}^{\scriptscriptstyle{\mathrm{SL}}}$ the image of $z_{k}$ in
${\mathcal{Z}}(f)/\Gamma$. By Theorem 3.13(d), we have that k is the $W$-orbit
of $z_{k}$. But the stabilizer of $z_{k}$ in $W$ is $\langle w_{d_{k}}\rangle$
by Remark 3.14, so it is contained in by Lemma 6.3. So the map $\rho$ is étale
at $z_{k}^{\scriptscriptstyle{\mathrm{SL}}}$, and so the singularity of
${\mathcal{Z}}(f)/\Gamma$ at $z_{k}^{\scriptscriptstyle{\mathrm{SL}}}$ is
equivalent to the singularity of ${\mathbb{P}}(d_{1},d_{2},d_{3})$ at $p_{k}$,
hence is an $A_{j}$ singularity by Lemma 6.3. This completes the proof of
Theorem 5.4 whenever $\Gamma=W^{\scriptscriptstyle{\mathrm{SL}}}$ and
$(W,d)\neq(G(2e,2e,4),4e)$, provided that Lemma 6.3 is proved. This is done
just below:

###### Proof of Lemma 6.3.

Let us examine the different cases:

  * $\bullet$

Type $G(2,1,4)$. Assume here that $W=G(2,1,4)$. Then $d=6$ and
$(d_{1},d_{2},d_{3})=(2,4,8)$. But
${\mathbb{P}}(2,4,8)\simeq{\mathbb{P}}(1,2,4)\simeq{\mathbb{P}}(1,1,2)$, so
${\mathcal{S}}=\\{p_{3}\\}$ and $p_{3}$ is an $A_{1}$ singularity. Moreover,
$d_{3}=8$ and it follows from Table 1 that $\delta(8)=\delta^{*}(8)=1$. Also,
Theorem 3.13(f) implies that the eigenvalues of $w_{8}$ are
$(\zeta_{8}^{-5},\zeta_{8}^{-1},\zeta_{8}^{-3},\zeta_{8})$, so
$\det(w_{8})=1$.

  * $\bullet$

Type $G_{28}$. Assume here that $W=G_{28}$. If $d=6$, then
$(d_{1},d_{2},d_{3})=(2,8,12)$: but in that case
${\mathbb{P}}(2,8,12)={\mathbb{P}}(1,4,6)={\mathbb{P}}(1,2,3)$ so
${\mathcal{S}}=\\{p_{2},p_{3}\\}$ and so $d_{k}\in\\{8,12\\}$ (and note that
$p_{2}$ is an $A_{1}$ singularity, while $p_{3}$ is an $A_{2}$ singularity).
If $d=8$, then $(d_{1},d_{2},d_{3})=(2,6,12)$ and we have
${\mathbb{P}}(2,6,12)={\mathbb{P}}(1,3,6)={\mathbb{P}}(1,1,2)$ so
${\mathcal{S}}=\\{p_{3}\\}$ and so $d_{k}=12$ (and note that $p_{3}$ is an
$A_{1}$ singularity). It follows fom Table 1 that
$\delta(8)=\delta^{*}(8)=1=\delta(12)=\delta^{*}(12)$. Also, Theorem 3.13(f)
implies that $\det(w_{8})=\zeta_{8}^{4-d-d_{1}-d_{2}-d_{3}}=\zeta_{8}^{-24}=1$
and $\det(w_{12})=\zeta_{12}^{4-d-d_{1}-d_{2}-d_{3}}=\zeta_{12}^{-24}=1$.

  * $\bullet$

Type $G_{30}$. Assume here that $W=G_{30}$. Then $d=12$ and
$(d_{1},d_{2},d_{3})=(2,20,30)$. But we have then
${\mathbb{P}}(2,20,30)={\mathbb{P}}(1,10,15)={\mathbb{P}}(1,2,3)$ so
${\mathcal{S}}=\\{p_{2},p_{3}\\}$, so $d_{k}\in\\{20,30\\}$. Note also that
$p_{2}$ is an $A_{1}$-singularity while $p_{3}$ is an $A_{2}$-singularity. It
follows from Table 1 that $\delta(20)=\delta^{*}(20)=1$
$=\delta(30)=\delta^{*}(30)$. Also, Theorem 3.13(f) implies that
$\det(w_{20})=\zeta_{20}^{4-d-d_{1}-d_{2}-d_{3}}=\zeta_{20}^{-60}=1$ and
$\det(w_{30})=\zeta_{30}^{4-d-d_{1}-d_{2}-d_{3}}=\zeta_{30}^{-60}=1$.

The proof of Lemma 6.3 is complete. ∎

#### 6.4.2. The case where $\Gamma=W^{\prime}\neq
W^{\scriptscriptstyle{\mathrm{SL}}}$

This case can only occur if $W=G(2,1,4)$ or $G_{28}$.

  * $\bullet$

Type $G(2,1,4)$. Assume here that $W=G(2,1,4)$. Then $d=6$ and $f$ is also an
invariant of degree $6$ of $G(2,2,4)$. Since
$W^{\prime}=G(2,2,4)^{\prime}=G(2,2,4)^{\scriptscriptstyle{\mathrm{SL}}}$, the
result follows from the previous subsection.

  * $\bullet$

Type $G_{28}$. Then $d\in\\{6,8\\}$. Then $d_{1}=2$, $d_{3}=12$ and $d_{2}$ is
the unique element of $\\{6,8\\}\setminus\\{d\\}$. Then

${\mathcal{Z}}(f)/W^{\prime}=\\{[x_{1}:x_{2}:x_{3}:j_{1}:j_{2}]\in{\mathbb{P}}(2,d_{2},12,12,12)\leavevmode\nobreak\
|\leavevmode\nobreak\ $
$\hphantom{AAAAAAAA}j_{1}^{2}=P_{{\mathbf{f}},{}_{1}}(0,x_{1},x_{2},x_{3})\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\
j_{2}^{2}=P_{{\mathbf{f}},{}_{2}}(0,x_{1},x_{2},x_{3})\\}.$

The group $W^{\scriptscriptstyle{\mathrm{SL}}}/W^{\prime}$ has order $2$ (we
denote by $\sigma$ its non-trivial element) and it acts on
${\mathcal{Z}}(f)/W^{\prime}$ as follows:

$\sigma([x_{1}:x_{2}:x_{3}:j_{1}:j_{2}])=[x_{1}:x_{2}:x_{3}:-j_{1}:-j_{2}].$

So one can check that the ramification locus ${\mathcal{R}}$ of the morphism
$\theta:{\mathcal{Z}}(f)/W^{\prime}\longrightarrow{\mathcal{Z}}(f)/W^{\scriptscriptstyle{\mathrm{SL}}}$
is defined by $j_{1}=j_{2}=0$ in both cases. We only need to prove that
${\mathcal{R}}$ is finite: indeed, if it is finite, then $\theta$ is
unramified in codimension $1$ and
${\mathcal{Z}}(f)/W^{\scriptscriptstyle{\mathrm{SL}}}$ has only ADE
singularities as it was shown in §6.4.1, so ${\mathcal{Z}}(f)/W^{\prime}$ has
only ADE singularities by Lemma B.4.

Now,

$\pi_{\mathbf{f}}^{\prime-1}({\mathcal{R}})=\\{p\in{\mathbb{P}}(V)\leavevmode\nobreak\
|\leavevmode\nobreak\ j_{1}(p)=j_{2}(p)=f(p)=0\\}.$

We only need to prove that $\pi_{\mathbf{f}}^{\prime-1}({\mathcal{R}})$ is
finite. First, let

${\mathcal{H}}=\\{p\in{\mathbb{P}}(V)\leavevmode\nobreak\
|\leavevmode\nobreak\ j_{1}(p)=j_{2}(p)=0\\}.$

Then the irreducible components of ${\mathcal{H}}$ are lines of the form
${\mathbb{P}}(H_{1}\cap H_{2})$, where $H_{1}\in{}_{1}$ and $H_{2}\in{}_{2}$.
This means that we only need to prove that such a line cannot be entirely
contained in ${\mathcal{Z}}(f)$. So, let $H_{1}\in{}_{1}$ and $H_{2}\in{}_{2}$
and let $s_{k}$ denote the reflection of $W$ whose reflecting hyperplane is
$H_{k}$. Let $G=\langle s_{1},s_{2}\rangle$. Then $V^{G}=H_{1}\cap H_{2}$ so
$\dim V^{G}=2$. If ${\mathbb{P}}(V^{G})$ is entirely contained in
${\mathcal{Z}}(f)$, it then follows from Corollary 2.9 that it is contained in
${\mathcal{Z}}_{\mathrm{sing}}(f)$: but this contradicts the fact that
${\mathcal{Z}}(f)$ has only ADE singularities.

The proof of Theorem 5.4 is complete. ∎

## Appendix A. Surfaces in weighted projective spaces

Let $m\geq 3$ and let $l_{0}$, $l_{1},\dots,l_{m}$ be positive integers. We
denote by $x_{0}$, $x_{1},\dots,x_{m}$ the coordinates in the weighted
projective space ${\mathbb{P}}(l_{0},l_{1},\dots,l_{m})$ and we fix $m-2$
polynomials $F_{1},\dots,F_{m-2}$ in the variables $x_{0}$,
$x_{1},\dots,x_{m}$ which are homogeneous of degree $e_{1},\dots,e_{m-2}$
(where $x_{k}$ is given the degree $l_{k}$). We consider the variety

$X=\\{[x_{0}:x_{1}:\cdots:x_{m}]\in{\mathbb{P}}(l_{0},l_{1},\dots,l_{m})\leavevmode\nobreak\
|\leavevmode\nobreak\ \forall\leavevmode\nobreak\ 1\leq j\leq
m-2,\leavevmode\nobreak\ F_{j}(x_{0},x_{1},\dots,x_{m})=0\\}.$

Let ${\mathbb{P}}_{\mathrm{sm}}$ (resp. ${\mathbb{P}}_{\mathrm{sing}}$) denote
the smooth (resp. singular) locus of ${\mathbb{P}}(l_{0},l_{1},\dots,l_{m})$.
We assume throughout this section that the following hold:

  * ${\mathrm{(H1)}}$

The weighted projective space ${\mathbb{P}}(l_{0},l_{1},\dots,l_{m})$ is well-
formed, i.e.

$\gcd(l_{0},\dots,l_{j-1},l_{j+1},\dots,l_{m})=1$

for all $j\in\\{0,1,\dots,m\\}$.

  * ${\mathrm{(H2)}}$

The variety $X$ is a weighted complete intersection, i.e. $\dim(X)=2$.

  * ${\mathrm{(H3)}}$

The variety $X$ is well-formed, i.e.
$\operatorname{codim}\nolimits_{X}(X\cap{\mathbb{P}}_{\mathrm{sing}})\geq 2$.

  * ${\mathrm{(H4)}}$

$l_{0}+l_{1}+\cdots+l_{m}=e_{1}+\cdots+e_{m-2}$.

  * ${\mathrm{(H5)}}$

$X$ has only ADE singularities.

Note that we do not assume that $X$ is quasi-smooth (i.e. we do not assume
that the affine cone of $X$ in ${\mathbb{C}}^{m+1}$ is smooth outside the
origin [Dol82, §3.1.5]). The following result is certainly well-known but, due
to the lack of an appropriate reference (particularly in the non-quasi-smooth
case), we provide here an explicit proof:

######  Lemma A.1.

Under the hypotheses _(H1), (H2), (H3), (H4) and (H5)_ , the smooth locus of
the surface $X$ has a non-degenerate $2$-form.

###### Proof.

We set ${\mathbb{P}}={\mathbb{P}}(l_{0},l_{1},\dots,l_{m})$ for
simplification. Let $U$ denote the smooth locus of
$X\cap{\mathbb{P}}_{\mathrm{sm}}$. By (H3),
$X\cap{\mathbb{P}}_{\mathrm{sing}}$ has codimension $\geq 2$ in $X$ and so
$X\setminus U$ has codimension $\geq 2$ in $X$ by (H5). Again by (H5), it is
sufficient to prove that $U$ admits a non-degenerate $2$-form.

If $0\leq a\leq m$ and $1\leq j\leq m-2$, we denote ${\mathbb{P}}^{(a)}$ the
open subset of ${\mathbb{P}}$ defined by $x_{a}\neq 0$: we identify it with
${\mathbb{C}}^{m}/{\boldsymbol{\mu}}_{l_{a}}$, where the coordinates in
${\mathbb{C}}^{m}$ are denoted by $(x_{0},\dots,x_{a-1},x_{a+1},\dots,x_{m})$
and ${\boldsymbol{\mu}}_{l_{a}}$ acts through
$\zeta\cdot(x_{0},\dots,x_{a-1},x_{a+1},\dots,x_{m})=(\zeta^{l_{0}}x_{0},\dots,\zeta^{l_{a-1}}x_{a-1},\zeta^{l_{a+1}}x_{a+1},\dots,\zeta^{l_{m}}x_{m})$,
and we set

$F_{j}^{(a)}(x_{0},\dots,x_{a-1},x_{a+1},\dots,x_{m})=F_{j}(x_{0},\dots,x_{a-1},1,x_{a+1},\dots,x_{m}).$

The smooth locus of ${\mathbb{P}}^{(a)}$ will be denoted by
${\mathbb{P}}_{\mathrm{sm}}^{(a)}$ and, since ${\mathbb{P}}$ is well-formed by
(H1), the above action of ${\boldsymbol{\mu}}_{l_{a}}$ on ${\mathbb{C}}^{m}$
contains no reflection and so ${\mathbb{P}}_{\mathrm{sm}}^{(a)}$ is the
unramified locus of the morphism
${\mathbb{C}}^{m}\to{\mathbb{C}}^{m}/{{\boldsymbol{\mu}}_{l_{a}}}$.

Let $J^{(a)}=\bigl{(}\partial F_{j}^{(a)}/\partial x_{k}\bigr{)}_{1\leq j\leq
m-2,0\leq k\neq a\leq m}$ denote the Jacobian matrix of the family
$(F_{1}^{(a)},\dots,F_{m-2}^{(a)})$. If $b$, $c$ are two different elements of
$\\{0,1,\dots,m\\}$ which are different from $a$, we denote by $J_{b,c}^{(a)}$
the $(m-2)\times(m-2)$ minor of $J^{(a)}$ obtained by removing the two columns
numbered by $b$ and $c$. We set

${\mathbb{P}}_{{\mathrm{sm}},b,c}^{(a)}=\\{p\in{\mathbb{P}}_{\mathrm{sm}}^{(a)}\leavevmode\nobreak\
|\leavevmode\nobreak\ J_{b,c}^{(a)}(p)\neq 0\\}$ and
$U_{b,c}^{(a)}=X\cap{\mathbb{P}}_{{\mathrm{sm}},b,c}^{(a)}.$

By the above description of ${\mathbb{P}}_{\mathrm{sm}}^{(a)}$ and (H2), we
get

$U=\bigcupop\displaylimits_{\begin{subarray}{c}0\leq a,b,c\leq m\\\
|\\{a,b,c\\}|=3\end{subarray}}U_{b,c}^{(a)}.$

We now define a $2$-form $\omega_{b,c}^{(a)}$ on
${\mathbb{P}}_{{\mathrm{sm}},b,c}^{(a)}$ by

(A.2) $\omega_{b,c}^{(a)}=l_{a}\frac{dx_{b}\wedge dx_{c}}{J_{b,c}^{(a)}}.$

Let us first explain why this defines a $2$-form on
${\mathbb{P}}_{{\mathrm{sm}},b,c}^{(a)}$. This amounts to show that
$\omega_{b,c}^{(a)}$ is invariant under the action of
${\boldsymbol{\mu}}_{l_{a}}$ on the variables $(x_{k})_{0\leq k\neq a\leq m}$
given by $\xi\cdot x_{k}=\xi^{l_{k}}x_{k}$. But, if $M$ is a monomial in
$J_{b,c}^{(a)}$ of degree $e$ in the variables $(x_{k})_{0\leq k\neq a\leq
m}$, then

$None$ $e\equiv
e_{1}+\cdots+e_{m-2}-(l_{0}+l_{1}+\cdots+l_{m}-l_{a}-l_{b}-l_{c})\mod l_{a}$

because the variable $x_{a}$ is specialized to $1$. So
$\xi\in{\boldsymbol{\mu}}_{l_{a}}$ acts on $J_{b,c}^{(a)}$ by multiplication
by $\xi^{l_{b}+l_{c}}$ by (H4). So it acts trivially on $\omega_{b,c}^{(a)}$.

We now denote by $\omega_{U,b,c}^{(a)}$ the restriction of
$\omega_{b,c}^{(a)}$ to $U_{b,c}^{(a)}$. Note that
$U_{b,c}^{(a)}=U_{c,b}^{(a)}$ but that
$\omega_{U,b,c}^{(a)}=-\omega_{U,c,b}^{(a)}$, so we have to make some choice.
We denote by ${\mathcal{E}}$ the set of triples $(a,b,c)$ of elements of
$\\{0,1,\dots,m\\}$ such that $a<b<c$ or $c<a<b$ or $b<c<a$. Then again

$U=\bigcupop\displaylimits_{(a,b,c)\in{\mathcal{E}}}U_{b,c}^{(a)}$

and we want to show that the family of $2$-forms
$(\omega_{U,b,c}^{(a)})_{(a,b,c)\in{\mathcal{E}}}$ glue together to define a
$2$-form on $U$. The argument is standard and will be done in two steps.

First step: glueing inside an affine chart. We fix $a\in\\{0,1,\dots,m\\}$ and
we set $U^{(a)}=U\cap{\mathbb{P}}^{(a)}$. Let $b$, $c$, $b^{\prime}$,
$c^{\prime}\in\\{0,1,\dots,m\\}$ be such that $(a,b,c)$,
$(a,b^{\prime},c^{\prime})\in{\mathcal{E}}$. We need to prove that

$None$ $\left.\omega_{U,b,c}^{(a)}\right|_{U_{b,c}^{(a)}\cap
U_{b^{\prime},c^{\prime}}^{(a)}}=\left.\omega_{U,b^{\prime},c^{\prime}}^{(a)}\right|_{U_{b,c}^{(a)}\cap
U_{b^{\prime},c^{\prime}}^{(a)}}.$

Proving $(\sharp)$ is a computation in ${\mathbb{C}}^{m}$ and amounts to prove
that

$None$ $J_{b^{\prime},c^{\prime}}^{(a)}dx_{b}\wedge
dx_{c}=J_{b,c}^{(a)}dx_{b^{\prime}}\wedge dx_{c^{\prime}}$

on the variety ${\hat{X}}^{(a)}$ defined by
$F_{1}^{(a)}=\cdots=F_{m-2}^{(a)}=0$ inside ${\mathbb{C}}^{m}$. By applying a
power of the cyclic permutation $(0,1,\dots,m)$ to the coordinates, we may
(and we will) assume that $a=0$ (so that $0<b<c$ and
$0<b^{\prime}<c^{\prime}$). Since $F_{j}^{(0)}$ vanishes on ${\hat{X}}^{(0)}$,
its differential vanishes also on $X$, which implies that

$\forall 1\leq j\leq m-2,\leavevmode\nobreak\
\sumop\displaylimits_{k=1}^{m}\frac{\partial F_{j}^{(0)}}{\partial
x_{k}}dx_{k}=0\quad\text{on ${\hat{X}}^{(0)}$.}$

Then $(\sharp^{\prime})$ is an easy application of generalized Cramer’s rule
[GA95].

Second step: glueing affine charts. We denote by $\omega_{U}^{(a)}$ the
glueing of the $2$-forms $\omega_{U,b,c}^{(a)}$, where $b$, $c$ are such that
$(a,b,c)\in{\mathcal{E}}$. Let $a$, $a^{\prime}\in\\{0,1,\dots,m\\}$. We need
to prove that

$None$ $\omega^{(a)}|_{U^{(a)}\cap
U^{(a^{\prime})}}=\omega^{(a^{\prime})}|_{U^{(a)}\cap U^{(a^{\prime})}}.$

For simplifying the notation, we will assume that $(a,a^{\prime})=(0,1)$, the
general case being treated similarly. We will denote by $(x_{k})_{0\leq k\neq
a\leq m}$ the coordinates on ${\mathbb{P}}^{(a)}$ and $(x_{k}^{\prime})_{0\leq
k\neq a^{\prime}\leq m}$ the coordinates on ${\mathbb{P}}^{(a^{\prime})}$.
Also for simplifying the notation, we will assume that $U_{1,2}^{(0)}\cap
U_{(2,0)}^{(1)}\neq\varnothing$. So, for proving $(\flat)$, we only need to
prove that

$None$ $l_{0}J_{2,0}^{(1)}dx_{1}\wedge
dx_{2}=l_{1}J_{1,2}^{(0)}dx_{2}^{\prime}\wedge dx_{0}^{\prime}\qquad\text{on
${\mathbb{P}}^{(0)}\cap{\mathbb{P}}^{(1)}$.}$

The variables $(x_{k})_{0\leq k\neq a\leq m}$ and $(x_{k}^{\prime})_{0\leq
k\neq a^{\prime}\leq m}$ are related as follows:

$\begin{cases}x_{0}^{\prime}=\displaystyle{\frac{1}{x_{1}^{l_{0}/l_{1}}}},\\\
\forall 2\leq k\leq m,\leavevmode\nobreak\
x_{k}^{\prime}=\displaystyle{\frac{x_{k}}{x_{1}^{l_{k}/l_{1}}}}.\end{cases}$

Therefore $dx_{0}^{\prime}=-(l_{0}/l_{1})x_{1}^{-1-l_{0}/l_{1}}dx_{1}$ and so

$None$ $l_{1}dx_{2}^{\prime}\wedge
dx_{0}^{\prime}=l_{0}x_{1}^{-(l_{0}+l_{1}+l_{2})/l_{1}}dx_{1}\wedge dx_{2}.$

Moreover, since $F_{j}$ is homogeneous of degree $e_{j}$, we get

$F_{j}^{(0)}(x_{1},x_{2},\dots,x_{m})=x_{1}^{e_{j}/l_{1}}F_{j}^{(1)}(x_{0}^{\prime},x_{2}^{\prime},\dots,x_{m}^{\prime}).$

We deduce that

$\frac{\partial F_{j}^{(0)}}{\partial
x_{k}}=x_{1}^{(e_{j}-l_{k})/l_{1}}\frac{\partial F_{j}^{(1)}}{\partial
x_{k}^{\prime}}$

for all $k\geq 3$ and then

$None$
$J_{1,2}^{(0)}=x_{1}^{((e_{1}+\cdots+e_{m-2})-(l_{3}+\cdots+l_{m}))/l_{1}}J_{2,0}^{(1)}.$

So $(\flat^{\prime})$ follows from $(\flat^{\prime\prime})$ and
$(\flat^{\prime\prime\prime})$ since
$e_{1}+\cdots+e_{m-2}=l_{0}+l_{1}+\cdots+l_{m}$ by (H4). ∎

######  Corollary A.3.

Under the hypotheses _(H1), (H2), (H3), (H4) and (H5)_ , the smooth locus of
the surface $X$ has a trivial canonical sheaf and its minimal resolution is a
smooth K3 surface or an abelian variety.

## Appendix B. Around ADE singularities

The results of this appendix are certainly well-known. Here, we let
${\mathbf{G}}{\mathbf{L}}_{2}({\mathbb{C}})$ act on the ring of formal power
series ${\mathbb{C}}[[t,u]]$ naturally by linear changes of the variables. We
set $B=\operatorname{{\mathrm{Spec}}}{\mathbb{C}}[[t,u]]$ and we denote by $0$
its unique closed point. We set $B^{\\#}=B\setminus\\{0\\}$: it is an open
subscheme of $B$. Since $B$ is normal, it follows from Hartog’s Lemma that

(B.1) ${\mathcal{O}}_{B}(B^{\\#})={\mathbb{C}}[[t,u]].$

######  Lemma B.2.

Let $G$ be a finite subgroup of ${\mathbf{G}}{\mathbf{L}}_{2}({\mathbb{C}})$
containing no reflection and let $\sigma$ be an automorphism of
${\mathbb{C}}[[t,u]]^{G}$. Then $\sigma$ lifts to an automorphism of
${\mathbb{C}}[[t,u]]$.

######  Remark B.3.

If $G$ is a finite subgroup of ${\mathbf{S}}{\mathbf{L}}_{2}({\mathbb{C}})$,
then $G$ contains no reflection. This shows that the above lemma applies to
ADE singularities ${\mathbb{C}}[[t,u]]^{G}$.

###### Proof.

First, note that $B$ has a trivial fundamental group by [Gro71, exposé I,
théorème 6.1]. As it is regular of dimension 2, its open subset $B^{\\#}$ has
also a trivial fundamental group by [Gro71, exposé X, corollaire 3.3].
Therefore, the natural map

$\pi:B^{\\#}\longrightarrow B^{\\#}/G$

is a universal covering: indeed, the morphism $B\to B/G$ is ramified only at
$0$ because $G$ does not contain any reflection. In particular,
$\pi\circ\sigma$ is also a universal covering, which means that $\sigma$ lifts
to an automorphism of $B^{\\#}$ since $B^{\\#}/G$ is connected. Taking global
sections and using (B.1) yields the result. ∎

######  Lemma B.4.

Let $\pi:Y\to X$ be a finite morphism of normal surfaces which is unramified
in codimension $1$. We assume moreover that $X$ has only ADE singularities.
Then $Y$ has only ADE singularities.

###### Proof.

Let $y\in Y$ and let $x=\pi(y)$. Then there exists a finite subgroup $G$ of
${\mathbf{S}}{\mathbf{L}}_{2}({\mathbb{C}})$ such that the completion of the
local ring ${\mathcal{O}}_{X,x}$ of $X$ at $x$ is given by
$\hat{{\mathcal{O}}}_{X,x}\simeq{\mathbb{C}}[[t,u]]^{G}$. Therefore, the
morphism of schemes

$\pi_{y}:(\operatorname{{\mathrm{Spec}}}{\hat{\mathcal{O}}}_{Y,y})\setminus\\{y\\}\longrightarrow
B^{\\#}/G$

induced by $\pi$ is unramified by hypothesis, so there exists a morphism of
schemes

$B^{\\#}\longrightarrow(\operatorname{{\mathrm{Spec}}}{\hat{\mathcal{O}}}_{Y,y})\setminus\\{y\\}$

whose composition with $\pi_{y}$ is a universal covering of $B^{\\#}/G$ (see
the proof of Lemma B.2).

Consequently, there exists a subgroup $H$ of $G$ such that

$(\operatorname{{\mathrm{Spec}}}{\hat{\mathcal{O}}}_{Y,y})\setminus\\{y\\}=B^{\\#}/H.$

Taking global sections and applying Hartog’s Lemma together with (B.1) yields
that ${\hat{\mathcal{O}}}_{Y,y}={\mathbb{C}}[[t,u]]^{H}$. ∎

Recall that, if $G\subset{\mathbf{G}}{\mathbf{L}}_{2}({\mathbb{C}})$, then the
only point of ${\mathbb{C}}^{2}/G$ that might be singular is the $G$-orbit of
$0$ (denoted by $\bar{0}$). The next result is certainly well-known:

######  Lemma B.5.

Let $G$ be a finite subgroup of ${\mathbf{G}}{\mathbf{L}}_{2}({\mathbb{C}})$
which is generated by $\operatorname{Ref}(G)$ and let be a subgroup of $G$ of
index $1$ or $2$. Then $\bar{0}\in V/\Gamma$ is smooth or an ADE singularity.

###### Proof.

We argue by induction on the order of $G$, the case where $|G|=1$ being
trivial. Also, if $\Gamma=G$, then $V/\Gamma$ is smooth so we may assume that
$\Gamma\neq G$. As is of index $2$, it is normal and we denote by
$\tau:G\to{\boldsymbol{\mu}}_{2}$ the unique morphism such that
$\Gamma=\operatorname{{\mathrm{Ker}}}(\tau)$. Let r be the subgroup of
generated by reflections belonging to . It is a normal subgroup of $G$ and

$V/G=(V/{}_{r})/(G/{}_{r})\qquad\text{and}\qquad
V/\Gamma=(V/{}_{r})/(\Gamma/{}_{r}).$

Two cases may occur:

  * $\bullet$

If ${}_{r}\neq 1$, then $V/{}_{r}$ is isomorphic to a vector space on which
$G/{}_{r}$ acts linearly as a reflection group since $V/G$ is smooth (see also
[BBR02, proposition 3.5]). So the result follows from the induction
hypothesis.

  * $\bullet$

If ${}_{r}=1$, then $\tau(s)\neq 1$ for all $s\in\operatorname{Ref}(G)$. This
implies that all reflections of $G$ have order $2$. Indeed, if
$s\in\operatorname{Ref}(G)$, then $\tau(s^{2})=1$ so $s^{2}$ cannot be a
reflection, hence is equal to $1$. This shows in particular that
$\tau(s)=\det(s)$ for all $s\in\operatorname{Ref}(G)$ and so $\tau(w)=\det(w)$
for all $w\in W$. In particular,
$\Gamma\subset{\mathbf{S}}{\mathbf{L}}_{2}({\mathbb{C}})$ and the result
follows.

The proof of the lemma is complete. ∎

######  Remark B.6.

We explain here why the general result stated in Lemma B.5 is in some sense
optimal. Let $d\geq 3$. Then there exists a reflection group $G$ in
${\mathbf{G}}{\mathbf{L}}_{2}({\mathbb{C}})$ admitting a normal subgroup such
that $W/\Gamma$ is cyclic of order $d$ and ${\mathbb{C}}^{2}/\Gamma$ admits a
non-simple singularity. Take for instance
$G={\boldsymbol{\mu}}_{d}\times{\boldsymbol{\mu}}_{d}$ embedded through
diagonal matrices, and $\Gamma\simeq{\boldsymbol{\mu}}_{d}$ embedded through
scalar multiplication. Then $\bar{0}$ is not an ADE singularity of
${\mathbb{C}}^{2}/\Gamma$.

In the same spirit, there exists a reflection group $G$ in
${\mathbf{G}}{\mathbf{L}}_{2}({\mathbb{C}})$ admitting a normal subgroup such
that $G/\Gamma\simeq{\boldsymbol{\mu}}_{2}\times{\boldsymbol{\mu}}_{2}$ and
such that $V/\Gamma$ admits a non-simple singularity. Take for instance
$G=G(4,2,2)$ and $\Gamma={\mathrm{Z}}(W)$. Then $G/\Gamma$ is indeed
isomorphic to ${\boldsymbol{\mu}}_{2}\times{\boldsymbol{\mu}}_{2}$ and is
isomorphic to ${\boldsymbol{\mu}}_{4}$ acting through scalar multiplication.
So $\bar{0}$ is not an ADE singularity of ${\mathbb{C}}^{2}/\Gamma$.

######  Corollary B.7.

Let $X$ be a surface with only ADE singularities and let $G$ be a group acting
on $X$ such that $X/G$ is smooth. Let be a subgroup of $G$ of index $2$. Then
$X/\Gamma$ has only ADE singularities.

###### Proof.

Let $x\in X$. It is sufficient to show that $X/{}_{x}$ has an ADE singularity
at the image of $x$. Note that we know that $X/G_{x}$ is smooth at the image
of $x$. Also, x has index $1$ or $2$ in $G_{x}$. This shows that we may, and
we will, assume that $G=G_{x}$ (and so $\Gamma={}_{x}$).

By hypothesis, there exists a subgroup $H$ of
${\mathbf{S}}{\mathbf{L}}_{2}({\mathbb{C}})$ such that the complete local ring
$\hat{{\mathcal{O}}}_{X,x}$ is isomorphic to ${\mathbb{C}}[[t,u]]^{H}$. Let us
identify ${\hat{\mathcal{O}}}_{X,x}$ with ${\mathbb{C}}[[t,u]]^{H}$, so that
the group $G$ acts on ${\mathbb{C}}[[t,u]]^{H}$. By Lemma B.2, the action of
an element $g\in G$ on ${\mathbb{C}}[[t,u]]^{H}$ lifts to an automorphism
${\tilde{g}}$ of ${\mathbb{C}}[[t,u]]$: we fix such a ${\tilde{g}}$ for all
$g\in G$. Note that $\\{{\tilde{g}}h\leavevmode\nobreak\ |\leavevmode\nobreak\
h\in H\\}$ is the set of all lifts of $g$ to ${\mathbb{C}}[[t,u]]$. Let

${\tilde{G}}=\\{{\tilde{g}}h\leavevmode\nobreak\ |\leavevmode\nobreak\ g\in
G\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ h\in H\\}.$

Then ${\tilde{G}}$ is a group (as ${\tilde{g}}h{\tilde{g}}^{\prime}h^{\prime}$
is a lift of $gg^{\prime}$ so belongs to ${\tilde{G}}$) and we have an exact
sequence

$1\longrightarrow H\longrightarrow{\tilde{G}}\longrightarrow G\longrightarrow
1.$

Let ${\tilde{\Gamma}}$ denote the inverse image of in ${\tilde{G}}$. Note that
$({\mathbb{C}}[[t,u]]^{H})^{G}={\mathbb{C}}[[t,u]]^{\tilde{G}}$ and
$({\mathbb{C}}[[t,u]]^{H})={\mathbb{C}}[[t,u]]^{\tilde{\Gamma}}$.

Now, by hypothesis, $({\mathbb{C}}[[t,u]]^{H})^{G}$ is regular. This shows
that ${\tilde{G}}$ acts as a reflection group on the tangent space of
$\operatorname{{\mathrm{Spec}}}{\mathbb{C}}[[t,u]]$ at its unique closed
point, and so the result follows from Lemma B.5 because ${\tilde{\Gamma}}$ has
index $1$ or $2$ in ${\tilde{G}}$. ∎

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