Cédric Bonnafé

Alessandra Sarti
email: sarti@math.univ-poitiers.fr

C Édric

Complex reflection groups

come

Introduction

In this paper we describe a relation between complex reflection groups and K3 surfaces. A relation already appeared recently in the paper [START_REF] Bonnafé | K3 surfaces with maximal finite automorphism groups containing M 20[END_REF] by the authors, where the authors use the reflection group denoted by G 29 in the Shepard-Todd classification [START_REF] Shephard | Finite unitary reflection groups[END_REF] 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 [START_REF] Sarti | Pencils of symmetric surfaces in P 3[END_REF] and of W. Barth and the second author [START_REF] Barth | Polyhedral groups and pencils of K3-surfaces with maximal Picard number[END_REF] 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 ADEsingularities. 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 admint 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. By Shephard-Todd/Chevalley/Serre Theorem [9, Theorem 4.1] there exist four 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 I 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 ring Date: May 9, 2020. The first author is partly supported by the ANR: Projects No ANR-16-CE40-0010-01 (GeRep-Mod) and ANR-18-CE40-0024-02 (CATORE). of all W -invariant derivatives. The aim of the paper is to study the quotient of the zero set of some of these invariant polynomials by some subgroup of W : the derived subgroup W and the group W SL which consists of the matrices of W with determinant 1. It turns out that if the degrees are "well chosen" then the quotient is a K3 surface with ADE-singularities. We give equations for the quotient surfaces in some weighted projective space. More precisely when we consider the group W SL we find that the quotient surfaces are double covers of a weighted projective plane and when the group is W and is different from W SL then the quotient surfaces are complete intersection in a four dimensional weighted projective space. 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. To do this we use several ingredients coming from the geometry of weighted projective spaces and the theory of Springer and Lehrer-Springer on properties of complex reflection groups. Our main Theorem 5.4 is the following (for the description of the reflection groups see the Section 3 and for the definition of K 3 see the Section 5):

Theorem 1.1. Assume that (W, d) ∈ K 3 . Let Γ be the subgroup W or W SL of W and let f be a fundamental invariant of W of degree d such that Z (f ) has only ADE-singularities.

Then Z (f )/Γ 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 or W 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).

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 K3 surfaces with Picard number 20, often called singular K3 surfaces. This will be explained in the sequel to this paper [START_REF]Complex reflection groups and K3 surfaces II. The groups G 29 , G 30 and G 31[END_REF], [START_REF]Complex reflection groups and K3 surfaces III. The group G 28 = W(F 4 )[END_REF], where we aim to complete the qualitative result 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 ). Note that the case where (W, d) = (G 28 , 6) or (G 30 , 12) and Γ = W was already treated by Barth and Sarti [START_REF] Barth | Polyhedral groups and pencils of K3-surfaces with maximal Picard number[END_REF]: these examples will be revisited, and more geometrical informations will be given.

The paper is organized as follows: the 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 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. In Section 5 we describe how to obtain K3 surfaces. In the Table II 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 S. Brochard for useful discussions about the results of Appendix B. We thank also Enrica Floris and Étienne Mann for several interesting discussions.

Notation, preliminaries

If d 1, we denote by µ d the group of d-th roots of unity in C × and we fix a primitive d-th root of unity ζ d (we will use the standard notation i = ζ 4 ). If l 1 ,. . . , l r are non-zero natural numbers, then P(l 1 , . . . , l r ) denotes the corresponding weighted projective space.

We fix an n-dimensional C-vector space V and we denote by P(V ) its associated projective space. If v ∈ V \ {0}, we denote by [v] ∈ P(V ) the line it defines (i.e.

[v] = Cv). The group GL C (V ) acts on the algebra C[V ] of polynomial functions on V as follows:

if g ∈ GL C (V ) and f ∈ C[V ], we write g f (v) = f (g -1 • v). If g ∈ GL C (V ) and ζ ∈ C × , we denote by V (g, ζ) the ζ-eigenspace of g. If v ∈ V (g, ζ) and f ∈ C[V ] g , then (2.1) g (∂ v f ) = ζ∂ v f. If G is a subgroup of GL C (V ), we write P G for its image in PGL C (V ). Recall that a subgroup G of GL C (V ) is called primitive if there does not exist a decompo- sition V = V 1 ⊕• • •⊕V r with V i =
0 and r 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 CS and G(S) = G(CS), where CS denotes the linear span of S. The derived subgroup of G will be denoted by G , and we set G SL = G ∩ SL C (V ). Note that G ⊂ G 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 ∈ GL C (V ), let ζ be a root of unity of order d, let v ∈ V be such that g(v) = ζv and let f ∈ C[V ] g be homogeneous of degree e not divisible by d. Then f (v) = 0. Proof. As f ∈ C[V ] g , we have f (g(v)) = f (v). But f (g(v)) = f (ζv) = ζ e f (v)
because f is homogeneous of degree e. So the result follows from the fact that

ζ e = 1.
If X is a complex algebraic variety and x ∈ X, we denote by T x (X) the tangent space of X at x. If f ∈ C[V ] is homogeneous, we will denote by Z (f ) the projective (possibly non-reduced) hypersurface in P(V ) P n-1 defined by f . Its singular locus will be denoted by Z sing (f ).

Lemma 2.3. Let G be a finite subgroup of GL C (V ), let f ∈ C[V ] G be homo- geneous and let v ∈ V G \ {0} be such that f (v) = 0. Then G acts trivially on T [v] (P(V ))/T [v] (Z (f )).
Proof. Since G is finite, there exists a G-stable subspace E of V such that V = E ⊕ Cv. We fix v ∈ V G \ {0}. Let α ∈ V * be such that α(v) = 1 and α(E) = 0. The affine chart U α of P(V ) defined by α = 0 is identified with v + E and, after translation, is identified with E: through this identification, Z (f )∩U α is the affine surface defined by the polynomial

F ∈ 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, T [v] (P(V )) = E and T [v] (Z (f )) = Ker(F 1 ) (and these identifications are G-equivariant since v ∈ V G ).
But CF 1 is the dual space to E/ Ker(F 1 ): since G acts trivially on CF 1 , this shows that G acts trivially on

E/ Ker(F 1 ) = T [v] (P(V ))/T [v] (Z (f )).
The next result is just an easy generalization of [21, §6].

Corollary 2.4. Let G be a finite subgroup of GL C (V ) such that dim V G = 1 and let f ∈ C[V ] G be
non-zero and homogeneous. We assume that f vanishes at V G , viewed as a point of P(V ). Then V G is a singular point of Z (f ).

Proof. Let v ∈ V G \ {0}. We keep the notation of the proof of the previous Lemma 2.3 (E, α, F , F i ). Since V G = Cv, we have E G = 0 and so, by semisimplicity, we have that (E/ Ker(F 1 )) G = 0. But G acts trivially on E/ Ker(F 1 ) by Lemma 2.3. Therefore, E/ Ker(F 1 ) = 0 so T

[v] (Z (f )) = Ker(F 1 ) = E and so 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 [START_REF] Sarti | Pencils of symmetric surfaces in P 3[END_REF], [START_REF] Bonnafé | Some singular curves and surfaces arising from invariants of complex reflection groups[END_REF]. Let us explain how to proceed.

Let G be a finite subgroup of GL C (V ), and let H 1 ,. . . , 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 ∈ V H k \ {0} and let Ω k denote the G-orbit of [v k ] in P(V ). Then (2.6) |Ω k | = |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 ⊂ G [v k ] . Conversely, G [v k ] normalizes G v k . But H k ⊂ 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 ) = 0 for all k. We also set

λ k = f 2 (v k )/f 1 (v k ). Then it follows from Corollary 2.4 that (2.7) Z (f 2 -λ k f 1 ) contains Ω k in its singular locus.
It also shows that, if G is defined over a subfield K of C, then the points of Ω k (which are singular points of

Z (f 2 -λ k f 1 )) have coordinates in K. Corollary 2.8. Let G be a finite subgroup of GL C (V ) such that dim V G = 2, let f ∈ C[V ]
G be homogeneous and non-zero and let v ∈ V G \{0} be such that f (v) = 0. Let L be the line P(V G ) and assume that

[v] is a smooth point of Z (f ). Then the intersection of L with Z (f ) is transverse at [v].
Proof. We keep again the notation of the proof of Lemma 2.3 (E, α, F ,

F i ). Since dim V G = 2, this forces dim E G = 1. Since [v] is smooth, this means that F 1 = 0. It then follows from Lemma 2.3 that E = E G ⊕ Ker(F 1 ). But E G = T [v] (L) and Ker(F 1 ) = T [v] (Z (f ))
. This shows the result.

Corollary 2.9. Let G be a finite subgroup of GL C (V ) such that dim V G = 2, and let f ∈ C[V ] G be homogeneous and non-zero. Let L be the line P(V G ) and assume that L ⊂ Z (f ). Then L ⊂ Z sing (f ).

Proof. Let v ∈ V G \ {0} and assume that [v] is a smooth point of Z (f ). The the intersection of L with Z (f ) is not transverse at [v] because L ⊂ Z (f ): this contradicts Corollary 2.8.

Reflection groups

We fix a finite subgroup W of GL C (V ) and we set

Ref(W ) = {s ∈ W | dim(V s ) = n -1}.
Hypothesis. We assume throughout this paper that

W = Ref(W ) .
In other words, W is a complex reflection group. The number codim(V W ) is called the rank of W .

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 [START_REF] Shephard | Finite unitary reflection groups[END_REF] for such groups: there is an infinite family G(de, e, r) with d, e, r 0 (they are of rank r is (d, e) = (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 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 W(H 4 ).

3.1. Invariants. By Shephard-Todd/Chevalley/Serre Theorem [9, Theo. 4.1], there exist n algebraically independent homogeneous elements f 1 , f 2 ,. . . ,

f n of C[V ] W such that C[V ] W = C[f 1 , f 2 , . . . , f n ]. Let d i = deg(f i ).
A family (f 1 , f 2 , . . . , f n ) satisfying the above property is called a family of fundamental invariants of W . Observe that by a result of Marin-Michel [START_REF] Marin | Automorphisms of complex reflection groups[END_REF], these polynomials can be defined over the rational numbers (for more details, see [START_REF]Complex reflection groups and K3 surfaces II. The groups G 29 , G 30 and G 31[END_REF]). A homogeneous element f ∈ 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 , . . . , d n ) is well-defined up to permutation and is called the list of degrees of W : it will be denoted by Deg(W ).

Notation. From now on, and until the end of this paper, we fix a family f = (f 1 , f 2 , . . . , f n ) of fundamental invariants and we set

d i = deg(f i ).
The following equalities are well-known [START_REF] Broué | Introduction to complex reflection groups and their braid groups[END_REF]Theo. 4.1]:

(3.1) |W | = d 1 d 2 • • • d n and | Ref(W )| = n i=1 (d i -1).
Also, as W acts irreducibly on V , its center |Z(W )| consists of homotheties, so it is cyclic. Moreover [9, Prop. ) is welldefined up to permutation and is called the list of codegrees of W : it will be denoted by Codeg(W ).

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

Proposition 3.3. The map

π f : P(V ) -→ P(d 1 , d 2 , . . . , d n ) [v] -→ [f 1 (v) : f 2 (v) : • • • : f n (v)]
is well-defined and induces an isomorphism

P(V )/W ∼ -→ P(d 1 , d 2 , . . . , d n ).
Moreover, π f induces by restriction an isomorphism

Z (f 1 )/W ∼ -→ P(d 2 , . . . , d n ). 3.2. Reflecting hyperplanes. If s ∈ Ref(W )
, then the hyperplane V s is called the reflecting hyperplane of s (or a reflecting hyperplane of W ). We denote by A the set of reflecting hyperplanes of W . If X is a subset of V , then, by Steinberg-Serre Theorem [9, Theo. 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 ∈ 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 \ {1} is the set of reflections of W whose reflecting hyperplane is H, so

(3.4) | Ref(W )| = (e H -1).
We denote by α H an element of V * such that H = Ker(α H ). In particular, if all the reflections have order 2, then

| Ref(W )| = |A |. Finally, note the following equality [9, Rem. 4.4.8] (3.5) |A | = n i=1 (d * i + 1).
If Ω is a W -orbit in A , then we denote by e Ω the common value of the e H 's for H ∈ Ω. We then set

J Ω = H∈Ω α H .
Then there exists a unique polynomial P f ,Ω in variables x 1 ,. . . , x n of respective weights d 1 ,. . . , d n such that (3.6)

J eΩ Ω = P f ,Ω (f 1 , . . . , f n ). Note that P f ,Ω is homogeneous of degree e Ω |Ω|. Then [9, §4.3.2] (3.7) C[V ] W = C[f 1 , . . . , f n , (J Ω ) Ω∈A /W ]
and a presentation of C[V ] W is given by the relations (3.6). This has the following consequence:

Proposition 3.8. Let Ω 1 ,. . . , Ω r denote the W -orbits in A . Then the map

π f : P(V ) -→ P(d 1 , d 2 , . . . , d n , |Ω 1 |, . . . , |Ω r |) [v] -→ [f 1 (v) : f 2 (v) : • • • : f n (v) : J Ω1 (v) : • • • : J Ωr (v)]
is well-defined and induces an isomorphism

P(V )/W ∼ -→ {[x 1 : • • • : x n : j 1 : • • • : j r ] ∈ P(d 1 , . . . , d n , |Ω 1 |, . . . , |Ω r |) | ∀ 1 k r, j eΩ k k = P f ,Ω k (x 1 , . . . , x n )}.
Moreover, π f induces by restriction an isomorphism

Z (f 1 )/W ∼ -→ {[x 2 : • • • : x n : j 1 : • • • : j r ] ∈ P(d 2 , . . . , d n , |Ω 1 |, . . . , |Ω r |) | ∀ 1 k r, j eΩ k k = P f ,Ω k (0, x 2 , . . . , x n )}.
Note that W/W SL is cyclic but W SL is not necessarily equal to the derived subgroup W of W . We have W = W SL if and only if |A /W | = 1. Now, we define the following element

J = H∈A α H = Ω∈A /W J Ω ∈ C[V ].
It is well-defined up to a scalar and homogeneous of degree |A |. It is the generator of the ideal of the reduced subscheme of the ramification locus of the morphism V → V /W . Then [START_REF] Broué | Introduction to complex reflection groups and their braid groups[END_REF]Rem. 3.10 and Prop. 4.4] (3.9)

w J = det(w) -1 J for all w ∈ W . In particular J ∈ C[V ] W SL and J |W/W SL | ∈ C[V ] W . So there exists a unique polynomial P f ∈ C[X 1 , . . . , X n ], which is homogeneous of degree |W/W SL | • |A | if
we assign to X i the degree d i , and such that (3.10)

J |W/W SL | = P f (f 1 , . . . , f n ).
Proposition 3.11. Assume that the map H → e H is constant on A (and let e denote this constant value, which coincides with

|W/W SL |). Then C[V ] W SL = C[f 1 , f 2 , . . . , f n , J
] and a presentation is given by the single equation (3.10). So the map

π SL f : P(V ) -→ P(d 1 , d 2 , . . . , d n , |A |) [v] -→ [f 1 (v) : f 2 (v) : • • • : f n (v) : J(v)]
is well-defined and induces an isomorphism

P(V )/W SL ∼ -→ {[x 1 : • • • : x n : j] ∈ P(d 1 , . . . , d n , |A |) | j e = P f (x 1 , . . . , x n )}.
Moreover, π SL f induces by restriction an isomorphism

Z (f 1 )/W SL ∼ -→ {[x 2 : • • • : x n : j] ∈ P(d 2 , . . . , d n , |A |) | j e = P f (0, x 2 , . . . , x n )}.
3.3. Eigenspaces. We now recall the basics of Springer and Lehrer-Springer theory: all the results stated in this subsection can be found in [START_REF] Springer | Regular elements of finite reflection groups[END_REF], [START_REF] Lehrer | Intersection multiplicities and reflection subquotients of unitary reflection groups. I, Geometric group theory down under[END_REF], [START_REF]Reflection subquotients of unitary reflection groups[END_REF]. Note that some of the proofs have been simplified in [START_REF] Lehrer | Invariant theory and eigenspaces for unitary reflection groups[END_REF]. Let us fix now a natural number e. We set In particular, ζ e is an eigenvalue of some element of W if and only if δ(e) = 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 , ζ e ) = δ(e). We set for simplification V (e) = V (w e , ζ e ) and W (e) = W V (e) /W (V (e)): this subquotient of W acts faithfully on V (e).

∆(e) = {1 k n | e divides d k }, ∆ * (e) = {1 k n | e divides d * k }, δ(e) =
If f ∈ C[V ], we denote by f [e] its restriction to V (e). Note that if i ∈ ∆(e), then f (a) The family of eigenvalues of w e for its action on the tangent space T z (P(V ))

(ζ 1-d k e ) 1 k n . Moreover, if w is such that dim V (w, ζ e ) = δ(e), then w is conjugate to w e . Remark 3.14. Let k ∈ {1, 2, . . . , n} be such that δ(d k ) = δ * (d k ) = 1. Then z k = V (d k ) is a line in V ,
is equal to (ζ -d k e ) k =i0 . (b) Let f ∈ C[V ] W be
homogeneous of degree d and assume that f (v) = 0.

Then:

(b1) If d ≡ d k mod e for all k = k 0 , then Z (f ) is singular at z. (b2) Assume that Z (f ) is smooth at z and let k 1 = k 0 be such that d ≡ d k1
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

T z (Z (f )) is equal to (ζ -d k e ) k =k0,k1 .
Proof. By permuting if necessary the degrees, we may assume that k

0 = 1. Note that ζ 1-d1 e = ζ e . Choose a basis (v 1 , . . . , v n ) of V such that v = v 1 and w(v k ) = ζ 1-d k e
v k for all k ∈ {1, 2, . . . , n} (see Theorem 3.13(e)).

(a) Identify P(V ) with P n-1 (C) through the choice of this basis. Then the action of w e is transported to

w e •[x 1 : x 2 : • • • : x n ] = [ζ e x 1 : ζ 1-d2 e x 2 : • • • : ζ 1-dn e x n ] = [x 1 : ζ -d2 e x 2 : • • • : ζ -dn e x n ]. Since z = [1 : 0 : • • • : 0], this shows (a).
(b) Let us work in the affine chart "x 1 = 1", identified with A n-1 (C) through the coordinates (x 2 , . . . , x n ). The equation of the tangent space T z (Z (f )) is given in this chart by

n k=2 (∂ v k f )(v)x k = 0. By (2.1), w (∂ v k f )(v) = ζ 1-d k e (∂ v k f )(v). But w (∂ v k f )(v) = (∂ v k f )(w -1 (v)) = (∂ v k f )(ζ -1 e v). As ∂ v k f is homogeneous of degree d -1, this implies that ζ 1-d k e (∂ v k f )(v) = ζ 1-d e (∂ v k f )(v).
Therefore, if d ≡ d k mod e for all k ∈ {2, . . . , n}, we get (∂ v k f )(v) = 0 for all k ∈ {2, . . . , n}, and so Z (f ) is singular ar [v]. This shows (b1). Now, if Z (f ) is smooth at z, then there exists k 1 ∈ {2, . . . , n} such that ∂ v k 1 f (v) = 0, and in particular d ≡ d k1 mod e. Since Z (f ) is smooth at z, then there exists k ∈ {2, . . . , n} such that ∂ v k f (v) = 0. This shows that the action of w e on the one-dimensional space T z (P

(V ))/T [v] (Z (f )) is given by multiplication by ζ -d k 1 e = ζ -d e .
The proof of (b2) is complete.

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 Z (f )/Γ in the cases we are interested in (here, f is a fundamental invariant of W and Γ is a subgroup of W ). We provide in this section two different tools that will be used in the sequel to this paper [START_REF]Complex reflection groups and K3 surfaces II. The groups G 29 , G 30 and G 31[END_REF], [START_REF]Complex reflection groups and K3 surfaces III. The group G 28 = W(F 4 )[END_REF], where particular examples will be studied.

4.1.

Stabilizers. The singularity of Z (f )/Γ at the Γ-orbit of z ∈ Z (f ) depends on the singularity of Z (f ) at z and the action of Γ z on this (eventually trivial) singularity. We investigate here some facts about the stabilizers W z and their action on the tangent space T z (Z (f )).

Let f denote a homogeneous invariant of W , let d denote its degree and let v ∈ V \ {0} such that v ∈ Z (f ). We set z = [v] ∈ P(V ). We denote by θ z : W z -→ C × the linear character defined by w(v) = θ z (w)v for all w ∈ W z . Then W v = Ker(θ z ) and we denote by e z = |Im(θ z )|. So there exists w ∈ W z such that θ z (w) = ζ ez . In other words, v ∈ V (w, ζ ez ) and so, by Theorem 3.13(a), we may, and we will, assume that v ∈ V (w ez , ζ ez ) = V (e z ). This shows that

If z ∈ L ∩ Z (f ), then W v = P .
Indeed, P ⊂ W v by construction and, if this inclusion is strict, this means that W v has rank 3 or 4. But it cannot have rank 4 for otherwise W v = W and so v = 0 (which is impossible). And it cannot have rank 3 because Corollary 2.4 would imply that Z (f ) is singular at z, contrarily to the hypothesis. This implies for instance that two points in L ∩ Z (f ) are in the same Γ-orbit if and only if they are in the same N Γ (P )-orbit. Moreover, in this case, we have a P -equivariant isomorphism

(4.3) T [v] (Z (f )) V /V P
(see the proof of Corollary 2.8).

Singularities of double covers.

If n = 4, Γ = W SL and W is generated by reflections of order 2, then the surface Z (f )/Γ is the double cover of a weighted projective plane. Most of (but not all) the singularities of Z (f )/Γ may be then analyzed through the singularities of the ramification locus of this cover. So we fix a double cover π : Y → 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 ramification locus R of π is empty or pure of codimension 1 (i.e. pure of dimension 1). The next well-known fact (see for instance [1, Part III, §7]) will help us in our explicit computations: Proposition 4.4. Let y ∈ Y be such that x = π(y) belongs to R. We assume that x is an ADE curve singularity of the ramification locus R. Then y is an ADE surface singularity of the same type.

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. 5.1. Classification. We provide in Table I 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/Z(W ) (which is the group that acts faithfully on P(V )), the order of W and the lists of degrees and codegrees. We also recall their notation in Shephard-Todd classification [START_REF] Shephard | Finite unitary reflection groups[END_REF] as well as their Coxeter name whenever they are real. Recall that G(2, 1, 4) = W(B 4 ) and G(2, 2, 4) = W(D 4 ). Note that the hypothesis on the order of the reflections implies that (5.2)

W |W | |W/Z(W )| |W | Deg(W ) Codeg(W ) G(1, 1, 5) = W(A 4 ) S 5 120
|W SL | = |W | 2 .
In particular, W = W SL if and only if W = G(2, 1, 4), G(4, 4, 4) or G 28 . 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 -→ G means that H is a normal subgroup of G).

(5.3)

G(2, 1, 4) / / s & & G 29 r $ $ G(2, 2, 4) s & & + 8 8 / / G 28 / / G 31 G(4, 4, 4) / / G(4, 2, 4) , : :
5.2. K3 surfaces. Equations of surfaces of the form Z (f )/W or Z (f )/W 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 A (and particularly Corollary A.3) that the canonical sheaf of Z (f )/W or Z (f )/W SL is trivial (provided that 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 Z (f )/W or Z (f )/W 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 Sarti [START_REF] Barth | Polyhedral groups and pencils of K3-surfaces with maximal Picard number[END_REF]: we revisit these cases and simplify some arguments using more theory about complex reflection groups.

We denote by K 3 the set of pairs (W, d) where W is an irreducible complex reflection group of rank 4 and d satisfy one of the following conditions: 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 Z (f )/Γ is a smooth projective K3 surface.

• W = G(1, 1, 5), G(4, 2

Numerical informations.

Before proving this Theorem 5.4, let us make some remarks. By Propositions 3.8 and 3.11, the variety Z (f )/Γ is a weighted complete intersection (see [START_REF] Dolgachev | Weighted projective varieties, Group actions and vector fields[END_REF]Section 3.2] for the definition) in a weighted projective space (it is defined by one or two equations). If Γ = W SL , then Z (f )/Γ is a weighted hypersurface in a weighted projective space of dimension 3 (see Proposition 3.11). If Γ = W , then Z (f )/Γ is a codimension 2 weighted complete intersection in a weighted projective space of dimension 4 (see Proposition 3.8). We give in Table II 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 Z (f )/W as a weighted projective space).

By looking at this Table II, 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) = G(2, 1, 4) , and since invariants of G(2, 1, 4) = W(B 4 ) are invariants for G(2, 2, 4) = 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.

W

G 28 = W(F 4 ) 6 W P (1, 2, 3, 3, 3) 6, 6 P(1, 2, 3)
G 31 20 W = W SL P(2, 1, 2, 5) 10 P(1, 1, 1)
Table II. Weights of ambient projective spaces and degree(s) of equation(s) of Z (f )/Γ 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 SL 4 (C) (see §6.1), it remains 8 non-zero dimensional families of K3 surfaces whose construction is nontrivial.

Hypothesis and notation. From now on, and until the end of this paper, we assume that (W, d) ∈ K 3 , that Γ is the subgroup W or W SL of W , and that f is a fundamental invariant of W of degree d such that Z (f ) has only ADE singularities. We also fix a family f = (f, f 1 , f 2 , f 3 ) of fundamental invariants containing f and we set

d i = deg(f i ).
Proof of the results given in Table II. The proof follows from Table I and a caseby-case analysis. We will not give details for all cases, we will only treat three cases (the reader can easily check that all other cases can be treated similarly).

• Assume that (W, d) = (G 31 , 20) and that Γ = W (= W SL ). Then (d 1 , d 2 , d 3 ) = [START_REF] Bosma | The Magma algebra system. I. The user language[END_REF][START_REF] Gong | A generalized Cramer's rule and characterization of fixed modes[END_REF][START_REF] Springer | Regular elements of finite reflection groups[END_REF] by Table I and

Z (f )/Γ = {[x 1 : x 2 : x 3 : j] ∈ P(8, 12, 24, 60) | j 2 = P f (0, x 1 , x 2 , x 3 )}.
But P(8, 12, 24, 60) = P(2, 3, 6, 15) = P(2, 1, 2, 5), so that I. Note that W = W SL and that there are two W -orbits Ω 1 and Ω 2 of reflecting hyperplanes, which are both of cardinality 12. It then follows from Proposition 3.8, that Z (f )/Γ = {[x 1 : x 2 : x 3 : j 1 : j 2 ] ∈ P(2, 8, 12, 12, 12) | j 2 1 = P f ,Ω1 (0, x 1 , x 2 , x 3 ) and j 2 2 = P f ,Ω2 (0, x 1 , x 2 , x 3 )}. But P(2, 8, 12, 12, 12) = P(1, 4, 6, 6, 6) = P(1, 2, 3, 3, 3), so that

Z (f )/Γ = {[y 1 : y 2 : y 3 : j] ∈ P(2, 1, 2, 5) | j 2 = Q(y 1 , y 2 , y 3 )} for some polynomial Q ∈ C[Y 1 , Y 2 , Y 3 ]. Hence Z (f )/Γ is
Z (f )/Γ = {[y 1 : y 2 : y 3 : j 1 : j 2 ] ∈ P(1, 2, 3, 3, 3) | j 2 1 = Q 1 (y 1 , y 2 , y 3 ) and j 2 2 = Q 2 (y 1 , y 3 , y 4 )} for some polynomials Q 1 and Q 2 in C[Y 1 , Y 2 , Y 3 ]. So Z (f )/Γ is
defined by two equations of degree 6 in P(1, 2, 3, 3, 3), as expected.

Finally, by Proposition 3.8, Z (f )/W P(2, 8, 12) = P(1, 4, 6) 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 Z (f )/Γ 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 ∈ {1, 2, 4}, then G(e, e, 4) has a unique invariant of degree 4 that defines a quartic in P 3 (C), but this invariant is equal to xyzt, and so 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 K 3 .

Remark 5.8. If (W, d) = (G 28 , 6) or (G 30 , 12), and Γ = W , then the above result was obtained by Barth and Sarti [START_REF] Barth | Polyhedral groups and pencils of K3-surfaces with maximal Picard number[END_REF]: the group Γ was denoted by G d in their paper (this must not be confused with Shephard-Todd notation). 5.4. About the families attached to 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 GL 4 (C) with coefficients in µ 2e and such that the product of the non-zero coefficients is equal to 1. Note that this implies that W = W SL .

If 1 k 4, we denote by σ 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 ∈ C[x, y, z, t] and l 1 an integer, we set p

[l] = p(x l , y l , z l , t l ) ∈ C[x, y, z, t]. For instance, σ 1 [l] = Σ(x l ). Then (σ 1 , σ 1 [2], σ 1 [3], σ 4
) is a family of fundamental invariants of G(1, 1, 4) S 4 . So there exists a unique polynomial P ∈ C[x 1 , x 2 , x 3 , x 4 ] such that (5.9)

J 2 1 = P (σ 1 , σ 1 [2], σ 1 [3], σ 4
). We do not need here the explicit form of P . If a, b, c ∈ C, we set

F a,b,c = aσ 1 [4] + bσ 4 + cσ 1 [2] 2 = aΣ(x 4 ) + bxyzt + c(Σ(x 2 )) 2 , G a,b,c = σ 1 [6] + aσ 1 [4]σ 1 [2] + bσ 1 [2] 3 + cσ 1 [2]σ 4 .
With this notation, we may, and we will, choose as a family of fundamental invariants of W the family (σ 1 [2e], σ 1 [4e], σ 1 [6e], σ 4 ). Note that the element J ∈ C[x, y, z, t] W defined in §3.2 is equal to J 1 [2e] (up to a scalar). Applying the endomorphism p → p[2e] of C[x, y, z, t] to the formula (5.9) and using Proposition 3.11 gives (5.10) P(V )/W {[x 1 : x 2 : x 3 : x 4 : j] ∈ P(2e, 4e, 6e, 4, 12e)|j 2 = P (x 1 , x 2 , x 3 , x 2e 4 )}, because σ 4 [2e] = σ 2e 4 . 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 P(2e, 4e, 6e, 4, 12e) = P(e, 2e, 3e, 2, 6e) P(1, 2, 3, 2, 6). So it follows from (5.10) that (5.11)

P(V )/W {[x 1 : x 2 : x 3 : x 4 : j] ∈ P(1, 2, 3, 2, 6) | j 2 = P (x 1 , x 2 , x 3 , x 2 4 )}. Now, a fundamental invariant of degree 6e of W if of the form G a,b,c [e]. We deduce from (5.11) that Z (G a,b,c [e])/W {[x 1 : x 2 : x 4 : j] ∈ P(1, 2, 2, 6) | j 2 = P (x 1 , x 2 , -ax 1 x 2 -bx 3 1 -cx 1 x 4 , x 2

5.4.2.

The case where e is even. Assume here that e = 2e for some e 1. Then P(2e, 4e, 6e, 4, 12e) = P(e , 2e , 3e , 1, 6e ) = P(1, 2, 3, 1, 6). So it follows from (5.10) that (5.14) Proof. Assume that 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 Z (F 0,1,c ) forces c = 0. An easy computation then shows that the only singular points of 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 Z (F ), as expected.

P(V )/W {[x 1 : x 2 : x 3 : x 4 : j] ∈ P(1, 2, 3, 1, 6) | j 2 = P (x 1 , x 2 , x 3 , x 4
Le us now assume that a = 0, and even that a = 1. Assume that we have found (b 0 , c 0 ) ∈ C 2 such that Z (F 1,b0,c0 ) 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 = 0. So let us work in the affine chart t = 0 and set F • b,c = F 1,b,c (x, y, z, 1). Let H b,c denote the Hessian matrix of F • b,c . Then (x 0 , y 0 , z 0 , b 0 , c 0 ) belongs to the variety

X = {(x, y, z, b, c) ∈ A 5 (C) | F • b,c (x, y, z) = ∂F • b,c ∂x (x, y, z) = ∂F • b,c ∂y (x, y, z) = ∂F • b,c ∂z (x, y, z) = det(H b,c (x, y, z)) = 0}. Now let π : A 5 (C) → A 2 (C), (x, y, z, β, γ) → (β, γ). A Magma [8] computation shows that π(X ) = {(4, -1/2), (-4, -1/2)}: > 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 ]
This shows that (b 0 , c 0 ) ∈ {(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 Z (F a,b,c ) with (a, b) = (1, 0) was studied in [START_REF]Symmetric surfaces with many singularities[END_REF], 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 1singularities 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 Z (f )/Γ is a K3 surface with ADE singularities, we need to show the following facts:

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

The technical step is to prove (S), namely that Z (f )/Γ 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 Z (f )/Γ is positive. Since Z (f ) has only isolated singularities by (S), it follows from [START_REF] Dimca | Singularities and topology of hypersurfaces[END_REF]Theo. 4.3] that H 1 (Z (f ), C) = 0. Since Z (f ) has only ADE singularities, it is rationally smooth [15, Def. A1]. As it is also projective, one can apply Poincaré duality and so H 3 (Z (f ), C) is the dual of H 1 (Z (f ), C), hence is equal to 0. So Z (f ) has no odd cohomology and since H j (Z (f )/Γ, C) = H j (Z (f ), C) Γ , this shows that Z (f )/Γ has no odd cohomology. So its Euler characteristic is positive. Now it remains to prove (C), namely that the canonical sheaf of X = Z (f )/Γ 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 A. Statements (H1), (H2) and (H4) are easily checked thanks to Table II 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:

• If Γ = W SL , then X is a weighted hypersurface of degree e in some P(l 0 , l 1 , l 2 , l 3 ), and, according to [14, §6.10], X is well-formed if, for all 0 a < b 3, gcd(l a , l b ) divides e. This is easily checked with Table II.

• If Γ = W = W SL , then X is a weighted complete intersection defined by two equations of degree e 1 and e 2 in some P(l 0 , l 1 , l 2 , l 3 , l 4 ), and, according to [14, §6.11], X is well-formed if the following two properties are satisfied:

-For all 0 a < b 4, gcd(l a , l b ) divides e 1 or e 2 .

-For all 0 a < b < c 4, gcd(l a , l b , l c ) divides e 1 and e 2 . Again, this is easily checked with Table II.

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

Singularities of Z (f )/Γ

The aim of this section is to complete the proof of Theorem 5.4, by proving that, under its hypotheses, the surface Z (f )/Γ has only ADE singularities. The first subsection is devoted to the (easy) case where d = 4 while the second subsection deal 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 third section.

6.1. The case 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 Z (f ) is already a K3 surface (with eventually ADE singularities) and we denote by ω a non-degenerate global holomorphic 2-form on the smooth locus (it is well-defined up to a scalar). By hypothesis, Γ ⊂ SL 4 (C), so Γ preserves ω. So Z (f )/Γ inherits a non-degenerate global holomorphic 2-form ω Γ on its smooth locus. Now, let p : X → Z (f ) denote a minimal resolution of Z (f ), and let ω X denote the unique non-degenerate 2-form on X extending ω. Then X is a K3 surface which inherits an action of Γ which stabilizes ω: so this is a symplectic action and so X/Γ is a K3 surface with ADE singularities [START_REF] Nikulin | Finite groups of automorphisms of Kählerian surfaces of type K3, Uspehi Mat[END_REF]Section 5]. Let q : Y → X/Γ denote a minimal resolution. We have a commutative diagram

X p / / Z (f ) Y q / / X/Γ p Γ / / Z (f )/Γ,
where p Γ is induced by p. This shows that p Γ • q : Y → Z (f )/Γ is a symplectic resolution, so Z (f )/Γ is a K3 surface with ADE singularities (and Y is its minimal resolution). This completes the proof of Theorem 5.4 whenever d = 4.

Z (f )/W SL {[x 1 : x 2 : x 3 : j] ∈ P(d 1 , d 2 , d 3 , |A |) | j 2 = P f (0, x 1 , x 2 , x 3 )}.
We denote by ρ : Z (f )/W SL -→ Z (f )/W the canonical map, let U denote the smooth locus of P(d 1 , d 2 , d 3 ) and let S denote the set of singular points of

P(d 1 , d 2 , d 3 ). Now, U = π -1 f (U )/W is smooth so ρ -1 (U ) = π -1 f (
∈ {G(2, 1, 4), G 28 , G 30 }. If p k ∈ P(d 1 , d 2 , d 3 ) is singular, then: (a) δ(d k ) = δ * (d k ) = 1. (b) det(w d k ) = 1. (c) p k is an A j singularity of P(d 1 , d 2 , d 3 ) for some j 1.
The proof will be given below. Let us first explain why this lemma might help to check that the points in ρ -1 (S ) are smooth or ADE singularities. So let p k ∈ S and let Ω

k = π -1 f (p k ). Then Ω k = {p ∈ P(V ) | ∀ 1 j = k 3, f j (p) = f k (p) = 0}.
By Lemma 6.3, dim V (d k ) = 1, so we might view V (d k ) ∈ P(V ) as a point z k ∈ Z (f ). We denote by z SL k the image of z k in Z (f )/Γ. By Theorem 3.13(d), we have that Ω k is the W -orbit of z k . But the stabilizer of z k in W is w d k by Remark 3.14, so it is contained in Γ by Lemma 6.3. So the map ρ is étale at z SL k , and so the singularity of Z (f )/Γ at z SL k is equivalent to the singularity of 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 Γ = W SL and (W, d) = (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:

• Type G(2, 1, 4). Assume here that W = G(2, 1, 4 • 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 = G(2, 2, 4) = G(2, 2, 4) SL , the result follows from the previous subsection.

• Type G 28 . Then d ∈ {6, 8}. Then d 1 = 2, d 3 = 12 and d 2 is the unique element of {6, 8} \ {d}. Then

Z (f )/W = {[x 1 : x 2 : x 3 : j 1 : j 2 ] ∈ P(2, d 2 , 12, 12, 12) | j 2 1 = P f ,Ω1 (0, x 1 , x 2 ,
x 3 ) and j 2 2 = P f ,Ω2 (0, x 1 , x 2 , x 3 )}. The group W SL /W has order 2 (we denote by σ its non-trivial element) and it acts on Z (f )/W as follows:

σ([x 1 : x 2 : x 3 : j 1 : j 2 ]) = [x 1 : x 2 : x 3 : -j 1 : -j 2 ].
So one can check in both cases that the ramification locus R of the morphism θ : Z (f )/W -→ Z (f )/W SL is defined by j 1 = j 2 = 0. We only need to prove that R is finite: indeed, if it is finite, then θ is unramified in codimension 1 and Z (f )/W SL has only ADE singularities as it was shown in §6.4.1, so Z (f )/W has only ADE singularities by Lemma B.4. Now,

π -1 f (R) = {p ∈ P(V ) | j 1 (p) = j 2 (p) = f (p) = 0}.
Proof. We set P = P(l 0 , l 1 , . . . , l m ) for simplification. Let U denote the smooth locus of X ∩ P sm . By (H3), X ∩ P sing has codimension 2 in X and so X \ U has codimension 2 in X by (H5). Again by (H5), it is sufficient to prove that U admits a non-degenerate 2-form.

If 0 a m and 1 j m -2, we denote P (a) the open subset of P defined by x a = 0: we identify it with C m /µ la , where the coordinates in C m are denoted by (x 0 , . . . , x a-1 , x a+1 , . . . , x m ), and we set F (a) j (x 0 , . . . , x a-1 , x a+1 , . . . , x m ) = F j (x 0 , . . . , x a-1 , 1, x a+1 , . . . , x m ).

The smooth locus of P (a) will de denoted by P (a) sm and, since P is well-formed by (H1), the above action of µ wa on C m contains no reflection and so P and we want to show that the family of 2-forms (ω

(a)
U,b,c ) (a,b,c)∈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 ∈ {0, 1, . . . , m} and we set U (a) = U ∩ P (a) . Let b, c, b , c ∈ {0, 1, . . . , m} be such that (a, b, c), (a, b , c ) ∈ E . We need to prove that ( ) ω

= • • • = F (a)
m-2 = 0 inside C m . By applying a power of the cyclic permutation (0, 1, . . . , m) to the coordinates, we may (and we will) assume that a = 0 (so that 0 < b < c and 0 < b < c ). Since F (0) j vanishes on X(0) , its differential vanishes also on X, which implies that ∀1 j m -2, m k=1 ∂F (0) j ∂x k dx k = 0 on X(0) .

Then ( ) is an easy application of generalized Cramer's rule [START_REF] Gong | A generalized Cramer's rule and characterization of fixed modes[END_REF].

Second step: glueing affine charts. We denote by ω For simplifying the notation, we will assume that (a, a ) = (0, 1), the general case being treated similarly. We will denote by (x k ) 0 k =a m the coordinates on P (a) and (x k ) 0 k =a m the coordinates on P (a ) . Also for simplifying the notation, we will assume that U (0)

1,2 ∩ U (1)
(2,0) = ∅. So, for proving ( ), we only need to prove that ( ) l 0 J

(1) 2,0 dx 1 ∧ dx 2 = l 1 J (0) 1,2 dx 2 ∧ dx 0 on P (0) ∩ P (1) .

The variables (x k ) 0 k =a m and (x k ) 0 k =a m are related as follows:

       x 0 = 1 x l0/l1 1 , ∀2 k m, x k = x k x l k /l1 1
.

Therefore dx 0 = -(l 0 /l 1 )x -1-l0/l1 1 dx 1 and so ( )

l 1 dx 2 ∧ dx 0 = l 0 x -(l0+l1+l2)/l1 1 dx 1 ∧ dx 2 .
Moreover, since F j is homogeneous of degree e j , we get Corollary A.3. Under the hypotheses (H1), (H2), (H3), (H4), (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.

  |Z(W )| = Gcd(d 1 , d 2 , . . . , d n ). The C[V ]-module Der(C[V ]) of derivatives of the algebra C[V ] is naturally graded in such a way that ∂ v has degree -1 for all v ∈ V . By Solomon Theorem [9, Theo. 4.44 and §4.5.4], the graded C[V ] W -module Der(C[V ]) W of invariant derivatives is free of rank n, hence it admits a homogeneous C[V ] W -basis (D 1 , . . . , D n ) whose respective degrees are denoted by d * 1 ,. . . , d * n . Again, the family (D 1 , . . . , D n ) is not uniquely defined even up to permutation, but the list (d * 1 , d * 2 , . . . , d * n

  |∆(e)| and δ * (e) = |∆ * (e)|. With this notation, we have (3.12) δ(e) = max w∈W dim V (w, ζ e ) .

  [e] i = 0 by Lemma 2.2. Theorem 3.13(Springer, Lehrer-Springer). Assume that δ(e) = 0. Then: (a) If w ∈ W , then there exists x ∈ W such that x(V (w, ζ e )) ⊂ V (e). (b) W (e) acts (faithfully) on V (e) as a group generated by reflections. (c) The family (f [e] k ) k∈∆(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. Deg(W (e)) = (d k ) k∈∆(e) ). (d) We have w∈W V (w, ζ e ) = x∈W x(V (e)) = {v ∈ V | ∀ k ∈ {1, 2, . . . , n} \ ∆(e), f k (v) = 0}. (e) δ * (e) δ(e) with equality if and only if W (V (w e , ζ e )) = 1. (f) If δ * (e) = δ(e), then W (e) = W V (e) = C W (w e ) and the family of eigenvalues (with multiplicity) of w e is equal to

  so we can view it as an element of 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 = w d k . Indeed,let e = |W z k |. Then d k divides e and ζ e is the eigenvalue of some elements of W . So e divides some d j . Therefore, d k divides d j and so d k = d j because δ(d k ) = 1. This proves that e = d k , as desired. Corollary 3.15. Assume that δ(e) = δ * (e) = 0 and let k 0 ∈ {1, 2, . . . , n} be such that d k0 is divisible by e. Let v ∈ V (e) \ {0} and let z = [z].

(4. 1 )

 1 W z = W v w ez .Recall from §3.2 that W v is a parabolic subgroup of W and so is generated by reflections. Note the following useful facts: (a) Let m = |Z(W )| (recall from (3.2) that m = gcd(Deg(W ))). Then m divides e z , because µ m = Z(W ) ⊂ W z . (b) If δ(e z ) = δ * (e z ), f (v) = 0 and Z (f ) is smooth at z, then the eigenvalues of w ez on the tangent space T z (Z (f )) are given by Corollary 3.15. (c) Let P be a parabolic subgroup of W of rank n -2 and assume that Z (f ) is smooth. Then dim V P = 2 and so L = P(V P ) is a line in P(V ). Then L intersects Z (f ) transversally by Corollary 2.8, so |L ∩ Z (f )| = d because f has degree d. Moreover, (4.2)

, 4 )Theorem 5 . 4 .

 454 or G 29 , and d = 4. • W = G(2e, 2e, 4) with e odd, and d ∈ {4e, 6e}. • W = G(4e, 4e, 4), and d = 4e. • W = G(2, 1, 4), and d ∈ {4, 6}. • W = G 28 , and d ∈ {6, 8}. • W = G 30 , and d = 12. • W = G 31 , and d = 20. Assume that (W, d) ∈ K 3 . Let Γ be the subgroup W or W SL of W and let f be a fundamental invariant of W of degree d such that Z (f ) has only ADE singularities.Then Z (f )/Γ is a K3 surface with ADE singularities.

  |A | = 60 by (3.1) and (3.4). It then follows from Proposition 3.11 that

  defined by an equation of degree 10 in P(2, 1, 2, 5), as expected. Finally, by Proposition 3.11, Z (f )/W P(8, 12, 24) = P(2, 3, 6) P(2, 1, 2) P(1, 1, 1), as expected. • Assume that (W, d) = (G 28 , 6) and Γ = W . Then (d 1 , d 2 , d 3 ) = (2, 8, 12) by Table

  a,b,c [e])/G(2e, 2e, 4) = Z (G a,b,c )/G(2, 2, 4) . Similarly, a fundamental invariant of degree 4e of W is of the form F a,b,c [e] with (a, b) = 0 and (5.13) Z (F a,b,c [e])/G(2e, 2e, 4) = Z (F a,b,c )/G(2, 2, 4) . This shows that the varieties Z (G a,b,c [e])/G(2e, 2e, 4) and Z (F a,b,c [e])/G(2e, 2e, 4) do not depend on e.

5 . 4 . 3 .

 543 4 )}. Now, a fundamental invariant of G(4e , 4e , 4) of degree 4e is of the form F a,b,c [e ] and a similar argument as before shows that (5.15) Z (F a,b,c [e ])/G(4e , 4e , 4) = Z (F a,b,c )/G(4, 4, 4) . Again, the variety Z (F a,b,c [e ])/G(4e , 4e , 4) does not depend on e . Complements. Note for future reference (see §6.3) the following fact: Lemma 5.16. If Z (F a,b,c ) is irreducible, then it is smooth or has only A 1 singularities.

  U )/Γ contains only ADE singularities by Corollary B.7 (because Γ = W SL has index 2 in W ). Hence, it remains to show that the points in ρ -1 (S ) are smooth or ADE singularities. Let p 1 = [1 : 0 : 0], p 2 = [0 : 1 : 0] and p 3 = [0 : 0 : 1] in P(d 1 , d 2 , d 3 ). Then S ⊂ {p 1 , p 2 , p 3 }. The following fact is checked by a case-by-case analysis: Lemma 6.3. Assume that d = 4 and W

( 8 ) 8 = ζ -24 8 = 1 20 = ζ -60 20 = 1

 888120201 = 1 = δ(12) = δ * (12). Also, Theorem 3.13(f) implies that det(w 8 ) = ζ 4-d-d1-d2-d3 and det(w 12 ) = ζ 4-d-d1-d2-d3 12 = ζ -24 12 = 1. • Type G 30 . Assume here that W = G 30 . Then d = 12 and (d 1 , d 2 , d 3 ) = (2, 20, 30). But P(2, 20, 30) = P(1, 10, 15) = P(1, 2, 3) so S = {p 2 , p 3 }, so d k ∈ {20, 30}. Note also that p 2 is an A 1 -singularity while p 3 is an A 2singularity. It follows fom Table I that δ(20) = δ * (8) = 1 = δ(30) = δ * (30). Also, Theorem 3.13(f) implies that det(w 20 ) = ζ 4-d-d1-d2-d3 and det(w 30 ) = ζ 4-d-d1-d2-d3 30 = ζ -60 30 = 1. The proof of Lemma 6.3 is complete. 6.4.2. The case where Γ = W = W SL . This case can only occur if W = G(2, 1, 4) or G 28 .

.

  unramified locus of the morphism C → C µ wa . Let J (a) = ∂F (a) j /∂x k 1 j m-2,0 k =a m denote the Jacobian matrix of the family (F (a) 1 , . . . , F (a) m-2 ). If b, c are two different elements of {0, 1, . . . , m} which are different from a, we denote by J (a) b,c the (m -2) × (m -2) minor of J (a) obtained by removing the two columns numbered by b and c. We set P (a) sm,b,c = {p ∈ P (a) sm | J Let us first explain why this defines a 2-form on P (a) sm,b,c . This amounts to show that ω (a) b,c is invariant under the action of µ la on the variables (x k ) 0 k =a m given by ξ• x k = ξ l k x k . But, if M is a monomial in J (a) b,c of degree e in the variables (x k ) 0 k =a m , then (#) e ≡ e 1 + • • • + e m-2 -(l 0 + l 1 + • • • + l m -l a -l b -l c ) mod l abecause the variable x a is specialized to 1. So ξ ∈ µ la acts on J (a) b,c by multiplication by ξ l b +lc by (H4). So it acts trivially on ω (a) b,c . We now denote by ω (a) U,b,c the restriction of ω

  ,b , so we have to make some choice. We denote by E the set of triples (a, b, c) of elements of {0, 1, . . . , m} such that a < b < c or c < a < b or b < c < a. Then again U = (a,b,c)∈E U

.

  Proving ( ) is a computation in C m and amounts to prove that ( ) J (a) b ,c dx b ∧ dx c = J (a) b,c dx b ∧ dx c on the variety X(a) defined by F (a) 1

U

  the glueing of the 2-forms ω (a) U,b,c , where b, c are such that (a, b, c) ∈ E . Let a, a ∈ {0, 1, . . . , m}. We need to prove that ( ) ω (a) | U (a) ∩U (a ) = ω (a ) | U (a) ∩U (a ) .

F ( 0 )jl1 1 F ( 1 ) 1 J

 0111 (x 1 , x 2 , . . . , x m ) = x ej /j (x 0 , x 2 , . . . , x m ). We deduce that ∂F = x ((e1+•••+em-2)-(l3+•••+lm))/l1 . So ( ) follows from ( ) and ( ) since e 1 +• • •+e m-2 = l 0 +l 1 +• • •+l m by (H4).

Table I .

 I Irreducible complex reflection groups of rank 4 generated by reflections of order 2

			120	60	2, 3, 4, 5 0, 1, 2, 3
	G(e, e, 4), e 2	24e 3	24e 3 /gcd(e, 4)	12e 3	e, 2e, 3e, 4 0, e, 2e, 3e -4
	G(2e, e, 4), e 1	384e 3 192e 3 /gcd(e, 4) 96e 3	2e, 4e, 6e, 8 0, 2e, 4e, 6e
	G 28 = W(F 4 )	1 152	576	288	2, 6, 8, 12 0, 4, 6, 10
	G 29	7 680	1 920	3 840	4, 8, 12, 20 0, 8, 12, 16
	G 30 = W(H 4 )	14 400	7 200	7 200	2, 12, 20, 30 0, 10, 18, 28
	G 31	46 080	11 520	23 040	8, 12, 20, 24 0, 12, 16, 28

  6.2. The case where Z (f )/W is smooth. Assume in this subsection, and only in this subsection, that Z (f )/W is smooth. By examining TableII, this occurs only if (W, d) = (G(2e, 2e, 4), 6e) or (G 31 , 20). In both cases, Γ = W = W SL is of index 2 in W , and so the result follows from Corollary B.7.6.3.The case where 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.[START_REF] Barth | Polyhedral groups and pencils of K3-surfaces with maximal Picard number[END_REF]. 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 is even 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 §6.1.6.4. 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 = 4 and W ∈ {G(2, 1, 4), G 28 , G 30 }.6.4.1.The case where Γ = W SL . Assume in this subsection, and only in this subsection, that Γ = W SL . In this case, Γ is a subgroup of index 2 of W .

		Recall from
	Propositions 3.3 and 3.11 that
	(6.1)	Z (f )/W P(d 1 , d 2 , d 3 )
	and
	(6.2)

  ). Then d = 6 and (d 1 , d 2 , d 3 ) = (2, 4, 8). But P(2, 4, 8) P(1, 2, 4) P(1, 1, 2), so S = {p 3 } and p 3 is an A 1 singularity. Moreover, d 3 = 8 and it follows from Table I that δ(8) = δ * (8) = 1. Also, Theorem 3.13(f) implies that the eigenvalues of w 8 are (ζ -5 8 , ζ -1 8 , ζ -3 8 , ζ 8 ), so det(w 8 ) = 1. • Type G 28 . Assume here that W = G 28 . If d = 6, then (d 1 , d 2 , d 3 ) = (2, 8, 12): but P(2, 8, 12) = P(1, 4, 6) = P(1, 2, 3) so S = {p 2 , p 3 } and so d k ∈ {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): but P(2, 6, 12) = P(1, 3, 6) = P(1, 1, 2) so S = {p 3 } and so d k = 12 (and note that p 3 is an A 1 singularity). It follows fom Table I that δ(8) = δ *

We only need to prove that π -1 f (R) is finite. First, let

Then the irreducible components of H are lines of the form P(H 1 ∩ H 2 ), where H 1 ∈ Ω 1 and H 2 ∈ Ω 2 . This means that we only need to prove that such a line cannot be entirely contained in Z (f ). So, let H 1 ∈ Ω 1 and H 2 ∈ Ω 2 and let s k denote the reflection of W whose reflecting hyperplane is

is entirely contained in Z (f ), it then follows from Corollary 2.9 that it is contained in Z sing (f ): but this contradicts the fact that Z (f ) has only ADE singularities.

The proof of Theorem 5.4 is complete.

Appendix A. Surfaces in weighted projective spaces Let m 3 and let l 0 , l 1 ,. . . , l m be positive natural numbers. We denote by x 0 , x 1 ,. . . , x m the coordinates in the weighted projective space P(l 0 , l 1 , . . . , l m ) and we fix m -2 polynomials F 1 ,. . . , F m-2 in the variables x 0 , x 1 ,. . . , x m which are homogeneous of degree e 1 ,. . . , e m-2 (where x i is given the degree l i ). We consider the variety

Let P sm (resp. P sing ) denote the smooth (resp. singular) locus of P(l 0 , l 1 , . . . , l m ). We assume throughout this section that the following hold: (H1) The weighted projective space P(l 0 , l 1 , . . . , l m ) is well-formed, i.e. gcd(l 0 , . . . , l j-1 , l j+1 , . . . , l m ) = 1 for all j ∈ {0, 1, . . . , m}. (H2) The variety X is a weighted complete intersection, i.e. dim(X) = 2. (H3) The variety X is well-formed, i.e codim X (X ∩ P sing ) 2. (H4)

(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 C m+1 is smooth outside the origin [11, §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), (H5), the smooth locus of the surface X has a non-degenerate 2-form.

Appendix B. Around ADE singularities

The results of this appendix are certainly well-known. Here, we let GL 2 (C) act on the ring of formal power series C[[t, u]] naturally by linear changes of the variables. We set B = Spec C[[t, u]] and we denote by 0 its unique closed point. We set Recall that, if G ⊂ GL 2 (C), then the only point of C 2 /G that might be singular is the G-orbit of 0 (denoted by 0). The next result is certainly well-known: Lemma B.5. Let G be a finite subgroup of GL 2 (C) which is generated by Ref(G) and let Γ be a subgroup of G of index 1 or 2. Then 0 ∈ V /Γ 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 Γ = G, then V /Γ is smooth so we may assume that Γ = W . As Γ is of index 2, it is normal and we denote by τ : W → µ 2 the unique morphism such that Γ = Ker(τ ). Let Γ r be the subgroup of Γ generated by reflections belonging to Γ. It is a normal subgroup of G and

Two cases may occur:

• If Γ r = 1, then V /Γ r is isomorphic to a vector space on which G/Γ r acts linearly as a reflection group since V /G is smooth (see also [START_REF] Bessis | Quotients et extensions de groupes de réflexion[END_REF]Prop. 3.5]). So the result follows from the induction hypothesis.

This implies that all reflections of G have order 2. Indeed, if s ∈ Ref(G), then τ (s 2 ) = 1 so s 2 cannot be a reflection, hence is equal to 1. This shows in particular that τ (s) = det(s) for all s ∈ Ref(G) and so τ (w) = det(w) for all w ∈ W . In particular, Γ ⊂ SL 2 (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 3. Then there exists a reflection group W in GL 2 (C) admitting a normal subgroup Γ such that W/Γ is cyclic of order d and C 2 /Γ admits a non-simple singularity. Take for instance W = µ d × µ d embedded through diagonal matrices, and Γ µ d embedded through scalar multiplication. Then 0 is not an ADE singularity of C 2 /Γ.

In the same spirit, there exists a reflection group W in GL 2 (C) admitting a normal subgroup Γ such that W/Γ µ 2 × µ 2 and such that V /Γ admits a nonsimple singularity. Take for instance W = G(4, 2, 2). Then W can be generated by three reflections s 1 , s 2 , s 3 (of order 2), which lie in three different conjugacy classes. Take Γ = W , s 1 s 2 s 3 . Then the tangent space of V /Γ at 0 has dimension 5, so 0 cannot be an ADE singularity.

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 Γ be a subgroup of G of index 2. Then X/Γ has only ADE singularities.

Proof. Let x ∈ X. It is sufficient to show that X/Γ 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, Γ x has index 1 or 2 in G x . This shows that we may, and we will, assume that G = G x (and so Γ = Γ x ).

By hypothesis, there exists a subgroup H of SL