Complex reflection groups and K3 surfaces I

We 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.


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 Z(f ) of an homogeneous fundamental invariant f of W by some subgroup Γ of W : the derived subgroup W and the group W SL = Ker(det) ∩ 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 Z(f )/Γ is a complete intersection in a weighted projective space. More precisely the quotient surface Z(f )/W SL is a double cover of a weighted projective plane whereas if W is different from W SL then Z(f )/W is a complete intersection in a four dimensional weighted projective space. We also explain how to obtain explicit equations for Z(f )/W or Z(f )/W SL .
It turns out that if the degree d of f is "well chosen" and Z(f ) has only ADE-singularities, then Z(f )/Γ 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 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) ∈ 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 whose projective zero set 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).
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 Z(f )/Γ has only ADE singularities. Our proof involves some general facts about singularities (see Appendix B) 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 Γ = W 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 Q and, in particular, all the singular K3 surfaces we obtain are defined over 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.

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 positive integers, 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 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 decomposition 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.
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 ).
Proof. Since G is finite, there exists a G-stable subspace E of V such that V = E ⊕ Cv. 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 hypersurface 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 [Sar01,§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 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 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( For this, it is sufficient to prove that 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. 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.

Reflection groups
We fix a finite subgroup W of GL C (V ) and we set Hypothesis. We assume throughout this paper that 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 [ST54] for such groups: there is an infinite family G(de, e, r) with d, e, r ≥ 0 (they are of rank r if (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 ).

Invariants
By Shephard-Todd/Chevalley/Serre Theorem [Bro10, Theorem 4.1], there exist n algebraically independent homogeneous elements f 1 , . . , 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 ∈ 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 [Bro10, Theorem 4.1]: Also, as W acts irreducibly on V , its center |Z(W )| consists of homotheties, so it is cyclic. Moreover by [Bro10,Proposition 4.6], is naturally graded in such a way that ∂ v has degree −1 for all v ∈ V . By Solomon Theorem [Bro10, Theorem 4.44 and §4.5.4], the graded . . , 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 ) is well-defined 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
is well-defined and induces an isomorphism Moreover, π f induces by restriction an isomorphism

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 [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 ∈ 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 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 Then there exists a unique polynomial P f,Ω in variables x 1 , . . . , x n of respective weights d 1 , . . . , d n such that and a presentation of C[V ] W is given by the relations (3.6). Consequently: Then the map is well-defined and induces an isomorphism Moreover, π f induces by restriction an isomorphism 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 by [Bro10, Remark 3.10 and Proposition 4.4] , which is homogeneous of degree |W /W SL | · |A| if we assign to X i the degree d i , and such that

Proposition 3.11. Assume that the map H → e H is constant on A (and let e denote this constant value, which coincides with
. . , f n , J] and a presentation is given by the single equation (3.10).
is well-defined and induces an isomorphism Moreover, π SL f induces by restriction an isomorphism

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 With this notation, we have 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).  (Springer, Lehrer-Springer). Assume that δ(e) 0. Then: (b) W (e) acts (faithfully) on V (e) as a group generated by reflections.
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∈ d(e) ).
is a line in V , 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, For proving this, let e = |W z k |. We just need to verify that e = d k . But d k divides e and ζ 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 δ(d k ) = 1. This proves that e = d k , as desired.
(a) The family of eigenvalues of w e for its action on the tangent space (b2) Assume that Z(f ) is smooth at z and let k 1 k 0 be such that d ≡ 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 Proof. By permuting if necessary the degrees, we may assume that k 0 = 1. Note that ζ 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 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 As ∂ v k f is homogeneous of degree d − 1, this implies that 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 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 [BS-II], [BS-III], where particular examples will be studied.

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
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 e z . In other words, v ∈ V (w, ζ e z ) and so, by Theorem 3.13(a), we may, and we will, assume that v ∈ V (w e z , ζ e z ) = V (e z ). This shows that Recall from §3.2 that W v is a parabolic subgroup of W and so is generated by reflections. Note the following useful facts: (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 Indeed, P ⊂ W v by construction and, if this inclusion is strict, this means that W v has rank n − 1 or n.

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 branch 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 branch 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 [BPVdV84, 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 branch 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.

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 /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 [ST54] 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 In particular, W W SL if and only if W = G 28 or W = G(2e, e, 4) for some e ≥ 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 −→ G means that H is a normal subgroup of G). (4, 4, 4) / / G(4, 2, 4) , : :

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 the second author [BS03]: 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 is a positive integer satisfying one of the following conditions: (1, 1, 5), G(4, 2, 4) or G 29 , and d = 4.
Theorem 5.4. 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.
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.

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 [Dol82,§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 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 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.
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 non-trivial. 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).
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 the second author [BS03]: the group Γ was denoted by G d in their paper (this must not be confused with Shephard-Todd notation).

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 If p ∈ C[x, y, z, t] and l ≥ 1 is 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 ).
So it follows from (5.10) that

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.
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.
Remark 5.17. Observe that the previous family 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.

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: Indeed, ifX denotes the minimal resolution of Z(f )/Γ and if (S), (E) and (C) are proved, thenX has a trivial canonical sheaf by (S) and (C), so by the classification of smooth algebraic surfacesX is a K3 surface or an abelian surface. But, by (S), the Euler characteristic ofX 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 thatX 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 [Dim92, Theorem 4.3] that H 1 (Z(f ), C) = 0. Since 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 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 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: • 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 [IF00, §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 2.
• 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 [IF00, §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 2.
The proof of Theorem 5.4 is complete, up to the proof of (S). 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) and 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 ).  2, 1, 4), G 28 , G 30 }. If p k ∈ P (d 1 , d 2 , d 3 ) is singular, then: 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 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). 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 1 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.
So one can check that the ramification locus R of the morphism θ : Z(f )/W −→ Z(f )/W SL is defined by j 1 = j 2 = 0 in both cases. 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, 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 H k . Let G = s 1 , s 2 . Then V G = H 1 ∩ H 2 so dim V G = 2. If P(V G ) 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 integers. 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 k is given the degree l k ). 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: (H2) The variety X is a weighted complete intersection, i.e. dim(X) = 2.
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 [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 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.
The smooth locus of P (a) will be denoted by P (a) sm and, since P is well-formed by (H1), the above action of µ l a on C m contains no reflection and so P We now define a 2-form ω e ≡ e 1 + · · · + e m−2 − (l 0 + l 1 + · · · + l m − l a − l b − l c ) mod l a because the variable x a is specialized to 1. So ξ ∈ µ l a acts on J 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 Proving ( ) is a computation in C m and amounts to prove that  (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 onX (0) , its differential vanishes also on X, which implies that Then ( ) is an easy application of generalized Cramer's rule [GA95].
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 Moreover, since F j is homogeneous of degree e j , we get 2,0 . So ( ) follows from ( ) and ( ) since e 1 + · · · + e m−2 = l 0 + l 1 + · · · + 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 GL 2 (C) act on the ring of formal 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 π : B # −→ B # /G is a universal covering: indeed, the morphism B → B/G is ramified only at 0 because G does not contain any reflection. In particular, π • σ is also a universal covering, which means that σ 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 π : Y → 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 ∈ Y and let x = π(y). Then there exists a finite subgroup G of SL 2 (C) such that the completion of the local ring O X,x of X at x is given byÔ X,x C[[t, u]] G . Therefore, the morphism of schemes π y : (SpecÔ Y ,y ) \ {y} −→ B # /G induced by π is unramified by hypothesis, so there exists a morphism of schemes B # −→ (SpecÔ Y ,y ) \ {y} whose composition with π 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 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 by0). 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. Then0 ∈ 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 G. As Γ is of index 2, it is normal and we denote by τ : G → 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 V /G = (V /Γ r )/(G/Γ r ) and V /Γ = (V /Γ r )/(Γ /Γ r ).