L’INSTITUTFOURIER DE ANNALES

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A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Samuel BOISSIÈRE & Alessandra SARTI

Contraction of excess fibres between the McKay correspondences in dimensions two and three

Tome 57, n^o6 (2007), p. 1839-1861.

<http://aif.cedram.org/item?id=AIF_2007__57_6_1839_0>

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CONTRACTION OF EXCESS FIBRES BETWEEN THE MCKAY CORRESPONDENCES IN DIMENSIONS TWO

AND THREE

by Samuel BOISSIÈRE & Alessandra SARTI (*)

Abstract. — The quotient singularities of dimensions two and three obtained from polyhedral groups and the corresponding binary polyhedral groups admit natural resolutions of singularities as Hilbert schemes of regular orbits whose exceptional fibres over the origin reveal similar properties. We construct a morphism between these two resolutions, contracting exactly the excess part of the exceptional fibre. This construction is motivated by the study of some pencils of K3 surfaces appearing as minimal resolutions of quotients of nodal surfaces with high symmetries.

Résumé. — Les singularités quotients de dimensions deux et trois obtenues par des groupes polyédraux et les groupes polyédraux binaires correspondants admettent des résolutions de singularités naturelles par les schémas de Hilbert d’orbites régulières, dont les fibres exceptionnelles au-dessus de l’origine révèlent des propriétés similaires. Nous construisons un morphisme entre ces deux réso- lutions, contractant exactement la partie excédentaire de la fibre exceptionnelle.

Cette construction est motivée par l’étude de certains pinceaux de surfaces K3 ap- paraissant comme résolutions minimales de quotients de surfaces nodales à grandes symétries.

1. Introduction

Consider a binary polyhedral groupGe⊂SU(2)corresponding to a polyhedral groupG⊂SO(3,R)through the double-coveringSU(2)→ SO(3,R).

The group Ge acts freely on C²\ {0} and the quotient surface C²/Ge has an isolated singular point at the origin. The exceptional divisor of its minimal resolution of singularities X → C²/Ge is a tree of smooth rational

Keywords:Quotient singularities, McKay correspondence, Hilbert schemes, polyhedral groups.

Math. classification:14C05, 14E15, 20C15,51F15.

(*) The first author was partially supported by a DFG grant.

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curves of self-intersection−2, intersecting transversally, whose intersection graph is an A-D-E Dynkin diagram. The classical McKay correspondence ([15]) relates this intersection graph to the representations of the groupG,e associating bĳectively each exceptional curve to a non trivial irreducible representation of the group: The correspondence in fact identifies the intersection graph with the McKay quiver of the action ofGe onC². Among these irreducible representations we find all irreducible representations of the group G: we call them pure and the remaining ones binary. Since G/{±1} ∼e =G, one can produce a G-invariant cone C²/{±1} −^∼→ K ,→C³ whose quotient K/G is isomorphic to C²/G. In this note, we prove thee following result, conjectured by W. P. Barth:

Theorem 1.1. — There exists a crepant resolution of singularities of C³/Gcontaining a partial resolutionY →K/Gwith the property that the intersection graph of its exceptional locus is precisely the McKay quiver of the action ofGonC³, together with a resolution mapX → Y mapping iso- morphically the exceptional curves corresponding to pure representations and contracting those associated with binary representations to ordinary nodes.

We make this construction in the framework of the Hilbert schemes of regular orbits of Nakamura ([16]) providing, by the Bridgeland-King-Reid theorem ([3]), the natural candidates for the resolutions of singularities in dimensions two and three. We produce a morphismS between these two resolutions of singularities, define our partial resolutionY as the image of this map and study the effect ofS on the exceptional fibres:

G-Hilbe C² S //

e^π

SKKKKKKKK%%%%

KK

K G-Hilb C³

π

Y

+

99s

ss ss ss ss ss

C²/Ge ^∼ //K/G //C³/G

Although the exceptional fibres can be described very explicitly in all cases (see [12]), as a matter of principle our proof avoids any case-by-case analysis: The key consists in a systematic modular interpretation of the objects at issue.

In Sections 2 and 3 we introduce the notation and recall useful facts about clusters and Hilbert schemes of points and clusters. Sections 4-6 give a brief

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survey on polyhedral, binary polyhedral and bipolyhedral groups, their representations and the classical McKay correspondences in dimensions two and three. In Section 7 we define and study the map S (Lemma 7.1, Proposition 7.2). In Section 8 we show that the mapS contracts the curves corresponding to the binary representations and maps the curves corresponding to the pure representations isomorphically to the exceptional curves downstairs (Theorem 8.1) and get Theorem 1.1 as Corollary 8.5. In Section 9, as an example we describe in details the case whenGe is a cyclic group. Finally, Section 10 is devoted to an application to resolutions of pencils of K3 surfaces.

Acknowledgments. We thank Wolf Barth for suggesting the problem and for many helpful comments, Manfred Lehn for his invaluable help dur- ing the preparation of this paper, Hiraku Nakajima for interesting expla- nations and Miles Reid for his careful reading.

2. Clusters

LetV be an-dimensional complex vector space andGa finite subgroup of SL(V). We denote byO(V) := S^∗(V^∨)the algebra of polynomial functions onV, with the induced left actiong·f :=f◦g⁻¹forf ∈ O(V)andg∈G.

We choose a basis X1, . . . , Xn of linear forms on V, denote the ring of polynomials in n indeterminates by S := C[X₁, . . . , X_n] and identify O(V)∼=S. The ringSis given a grading by the total degree of a polynomial, where each indeterminateX_i has degree1. In particular, the action of the groupGonS preserves the degree.

Let mS := hX1, . . . , Xni be the maximal ideal of S at the origin. We denote byS^Gthe subring ofG-invariant polynomials, bym_SG its maximal ideal at the origin and bynG :=m_SG ·S the ideal ofS generated by the non constantG-invariant polynomials vanishing at the origin. The quotient ring ofcoinvariants is by definitionSG:= S/nG.

An ideal I⊂S is called aG-cluster if it is globally invariant under the action ofGand the quotientS/Iis isomorphic as aG-module to the regular representation ofG: S/I∼=C[G]. A closed subschemeZ ⊂Cⁿ is called a G-cluster if its defining idealI(Z)is aG-cluster. Such a subscheme is then zero-dimensional, has length|G|and contains only one orbit. For instance, a freeG-orbit defines aG-cluster.

We are particularly interested inG-clusters supported at the origin. Then I ⊂m_S and in fact this condition is enough to assert that the cluster is supported at the origin. Furthermore, one has automaticallynG⊂I, since

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any non constant functionf ∈n_G not contained inI would induce a new copy of the trivial representation in the quotientS/I, different from that given by the constant functions. Hence we wish to understand the structure of theG-clustersI such thatnG ⊂I⊂mS, equivalent to the study of the quotient idealsI/n_G⊂mS/n_G ⊂S/n_G=S_G, with the exact sequence:

(2.1) 0−→I/nG−→SG −→S/I−→0.

From now on, we assume that there exists a complex reflection group R∈GL(V)containingGsuch that [R:G] = 2. Then mS/nG is a graded finite-dimensional algebra which as a G-module consists exactly of each non trivial representation ρ of G repeated 2 dimρ times: we denote the occurrences of each representationρbyV⁽¹⁾(ρ), . . . , V^{(2 dim}^ρ)(ρ)where each V⁽ⁱ⁾(ρ)is given by homogeneous polynomials modulon_G ([6, 7]).

By the exact sequence (2.1), giving a G-cluster supported at the origin consists in choosing dimρ copies of ρ in mS/nG for each non trivial rep- resentationρ ofG. This gives many choices since any linear combination of someV⁽ⁱ⁾(ρ) and V^(j)(ρ)provides such a copy. The underlying idea is that one does not have to make all these choices to defineI(see §9 for an explicit example).

For such an idealIwithn_G ⊂I⊂m_S, we consider the finite-dimensional G-modules W ⊂ S generatingI in the sense that I =W ·S+nG. Such modules exist by the preceding construction. Among these choices, we consider theminimalones, such that no strictG-submodule of them generate Iin the preceding sense.

IfW is a generator in this sense, then:

I=W ·S+nG =W +mS·W+nG=W +mS·I+nG.

This means that the C-linear map W → I/(m_S ·I+n_G) is surjective.

Also, sinceW is a G-module and since mS·I+nG is G-stable, this map is G-linear. If W is a minimal set of generators, it satisfies in particular W∩(mS·I+nG) ={0}since this intersection would provide aG-submodule whose complementary inW is a smallerG-submodule generatingI. Hence, forW minimal one gets an isomorphism ofG-modulesW ∼=I/(m_S·I+nG).

We set thenV(I) :=I/(mS·I+nG). The set of generators of V(I)may not be uniquely determined, but its structure as a G-module is unique.

The important issue, that will be the core of the classification, will be to determine whetherV(I)is irreducible or not, although it is a minimal set of generators.

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Notation for the two- and three-dimensional cases. When applying the preceding constructions in dimensions two or three, we fix the following notation:

• For n = 2, the polynomial ring is denoted by A := C[x, y], the group byGeand any ideal byI.

• Forn = 3, the polynomial ring is denoted by B := C[a, b, c], the group byGand any ideal byJ.

3. Moduli space of clusters 3.1. Hilbert scheme of points

Let X ⊂ Pⁿ_C be a quasi-projective scheme and N a positive integer.

Consider the contravariant functor from the category of schemes to the category of setsHilb^N_X: (Schemes)→(Sets)given by:

Hilb^N_X(T) :=











Z⊂T×X

(a) Z is a closed subscheme

(b) the morphismZ ,→T×X −→^p T is flat (c) ∀t∈T, Zt⊂X is a closed subscheme

of dimension0and lengthN









 This functor is represented by a quasi-projective schemeHilb^N(X)coming with a universal family Ξ^X_N ⊂ Hilb^N(X)×X. In the sequel, we always denote by pthe projection to the moduli space (here Hilb^N(X)) and by qthe projection to the base (here X). When X is projective, the scheme Hilb^N(X)is projective and comes with a very ample line bundle (for`0):

det p_∗

O_ΞX

N ⊗q^∗OX(`) .

When X = Cⁿ, one gets an open immersion Hilb^N(Cⁿ) ,→ Hilb^N(Pⁿ_C) corresponding to the restriction of the universal family. The restriction of the determinant line bundle gives the very ample line bundledet

p_∗O_ΞCn N

onHilb^N(Cⁿ).

There exists a natural projective morphism from Hilb^N(X)to the symmetric product S^N(X) sending a closed subscheme to the corresponding 0-cycle describing its support, called theHilbert-Chow morphism:

H : Hilb^N(X)−→S^N(X).

By a theorem of Fogarty ([5]), the scheme Hilb^N(X) is connected. For dimX = 2, it is reduced, smooth and the morphismH is a resolution of singularities.

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3.2. Hilbert scheme of regular orbits

We consider the sub-functor G-Hilb_Cn of Hilb^|G|_Cn given by G-Hilb_Cn(T) :=n

Z∈ Hilb^|G|_Cn(T)| ∀t∈T, Z_t⊂Cⁿ is aG-clustero . This functor is represented by a quasi-projective schemeG-Hilb(Cⁿ)called theHilbert scheme ofG-regular orbits, which is a union of some connected components of the subscheme ofG-fixed points

Hilb^|G|(Cⁿ)G

. Further- more, the quotient Cⁿ/G can be identified with a closed subscheme of S^|G|(Cⁿ) and since the support of a G-cluster consists exactly in one orbit through G, the restriction of the Hilbert-Chow morphism factorizes through a projective morphism (see [3, 11, 18]):

H :G-Hilb(Cⁿ)−→Cⁿ/G.

There is a unique irreducible component ofG-Hilb(Cⁿ)containing the free G-orbits and mapping birationally onto Cⁿ/G. This component is taken as the definition of the Hilbert scheme ofG-regular orbits in [16]. By the theorem of Bridgeland-King-Reid [3], ifn63, thenG-Hilb(Cⁿ)is already irreducible, reduced, smooth and the morphismH is a crepant resolution of singularities of the quotient Cⁿ/G. Moreover, H is an isomorphism over the open subset of freeG-orbits. As a byproduct, the two definitions coincide.

As before, the scheme G-Hilb(Cⁿ) has a universal family ZG which is the restriction of the universal family Ξ^C_|G|ⁿ corresponding to the closed immersionG-Hilb(Cⁿ),→Hilb^|G|(Cⁿ). The restriction of the determinant line bundle gives the very ample line bundle det (p_∗O_Z_G) on G-Hilb(Cⁿ) (see [10, §8.1]).

4. Rotation groups

Let SO(3,R) be the group of rotations inR³. Up to conjugation, there are five different types of finite subgroups ofSO(3,R), called polyhedral groups: The cyclic groups Cn, the dihedral groups Dn, the tetrahedral groupT, the octahedral groupOand the icosahedral groupI.

Consider the classical exact sequence:

0−→ {±1} −→SU(2)−→^φ SO(3,R)−→0.

For any finite subgroup G ⊂ SO(3,R), the inverse image Ge := φ⁻¹G is called abinary polyhedral group. It is a finite subgroup ofSU(2)⊂ SL(2,C).

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Consider the second exact sequence:

0−→ {±1} −→SU(2)×SU(2)−→^σ SO(4,R)−→0.

For any binary polyhedral groupG, the direct imagee σ(G×e G)e ⊂SO(4,R)is called abipolyhedral group. In §10, we make use of the following particular groups:

• G6=σ(T ×e Te)of order288;

• G8=σ(O ×e O)e of order 1152;

• G₁₂=σ(eI ×I)e of order7200.

Consider a binary polyhedral group G, the associated polyhedral groupe Gand set τ:={±1}:

0−→τ −→Ge−→^φ G−→0.

This exact sequence induces an injection of the set of irreducible representations ofGin the set of irreducible representations ofG: Ife ρ:G→GL(V) is an irreducible representation ofG, it induces by composition a representation ofGewhich isτ-invariant,i.e.such thatρ(−g) =ρ(g)for allg∈G.e If the representationρadmits a non trivial G-submodule, it is also a none trivialG-submodule after going to the quotientG/τe ∼=G. This shows that the image of the injection:

Irr(G),→Irr(G)e

consists precisely on the irreducible representations which areτ-invariant.

These representations are called pure and the remaining representations are calledbinary. More precisely, ifρ:Ge→GL(V)is an irreducible representation ofG, the subspacee

V^τ:={v∈V|v=ρ(−1)v}

is aG-submodule ofe V. Hence either V^τ =V and the representationρ is pure, orρis binary andV^τ={0}.

For each type of binary polyhedral group, the binary representations are labelled by a “e” and the trivial representation is denoted byχ₀.

5. Graph-theoretic intuition

IfG⊂SL(n,C)is a finite subgroup, it defines a natural faithful represen- tationQofG. Let{V₀, . . . , V_k}be a complete set of irreducible representations ofG, whereV0denotes the trivial one. For each such representation,

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one may decompose the tensor products Q ⊗Vi ∼=

k

M

j=0

V_j^⊕a^i,j

for some non negative integersai,j. Whenai,j=aj,ifor alli, j, one defines theMcKay quiveras the unoriented graph with verticesV₀, V₁, . . . , V_k and ai,j edges between the vertices Vi and Vj. In particular, this quiver may contain some loops. For our purpose, we only consider thereducedMcKay quiver with verticesV1, . . . , Vk and one edge betweenViandVjifi6=j and ai,j 6= 0: This means that we remove from the McKay quiver the vertexV0, all edges starting from it, all loops and all multiple edges. When there is an edge joiningViandVj, the vertices are calledadjacent. All finite subgroups ofSU(2)andSO(3,R)enter in this context.

For each binary polyhedral groupGe⊂SU(2)and its corresponding polyhedral groupG⊂SO(3,R), we draw the reduced McKay quiver with our conventions. For the binary polyhedral groups, we denote by a white vertex the pure representations and by a black vertex the binary ones. We get (see for example [6, 7, 8]) the graphs of Figure 5.1 and 5.2.

In the sequel, we interpret these graphs as the intersection graphs of a family of smooth rational curves meeting transversally. One may then get the following intuition: Looking at the two-dimensional graphs, if one contracts the curves associated to a binary representation (black nodes), then one gets an intersection graph which is precisely the corresponding graph in dimension three!

Another property of the two-dimensional quivers is that no two pure representations and no two binary representations are adjacent. This means that the contraction contracts a disjoint union of−2 curves.

6. Exceptional fibres in dimensions two and three Considering the Hilbert-Chow morphism H : G-Hilb(Cⁿ) −→ Cⁿ/G, our purpose is to describe the exceptional fibre H⁻¹(O) over the origin O ∈ Cⁿ/G in the two- and three-dimensional cases. Note that all finite subgroups ofSL(2,C)orSO(3,R)enter in the context of §2 since they are subgroups of index2 of a reflection group (see [7, §2.7]). Hence we may apply the general procedure for the study of the clusters supported at the origin.

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Cen • χe1

χ◦1

• χe_n−1 ◦

χn−1

• χen

•χe1

De2`+1 ◦ χ1

•

σe₁ ◦ τ1

•

σe_` ◦ ss ss ss s

KK KK KK τ`K

•χe2

◦χ2

De2` ◦

χ₁ •

σe1

τ◦1

• eσ2

τ_`−1◦ • rr rr rr r

LL LL LL L eσ`

◦χ3

•χe1

Te ◦

χ₁ •

χe2

χ◦₃ • χe3

χ◦₂

◦χ2

Oe •

χe₁ ◦

χ₃ •

χe₃ ◦

χ₄ •

χe₂ ◦ χ₁

◦χ₂

Ie •

χe₂ ◦ χ3

• χe₄ ◦

χ4

• χe₃ ◦

χ1

• χe₁ Figure 5.1. Reduced McKay quivers in dimension 2

The understanding of the exceptional fibre in these cases was achieved by Ito-Nakamura [12, 13] in dimension two and by Gomi-Nakamura-Shinoda [6, 7] in dimension three, by a case-by-case analysis. For the two-dimensional case, there is another proof by Crawley-Boevey [4] avoiding this case-by- case analysis. For any finite groupG,Irr^∗(G)denotes the set of irreducible representations but the trivial one.

Exceptional fibre in dimension two.Let Ge⊂SL(2,C)be a binary polyhedral group and denote by πe: G-Hilbe C²

−→ C²/Ge the Hilbert- Chow morphism. For each non trivial irreducible representation ρ of G,e set:

E(ρ) :={I∈πe⁻¹(O)red|V(I)⊃ρ}.

Theorem 6.1. — ([12, Theorem 3.1])

• EachE(ρ)is a smooth rational curve of self-intersection−2.

• eπ⁻¹(O)red =S

ρE(ρ)and πe⁻¹(O) =P

ρdimρ·E(ρ) as a Cartier divisor (withρ∈Irr^∗(G)).e

• IfI∈E(ρ)andI /∈E(ρ⁰)for allρ⁰6=ρ, then V(I)∼=ρ.

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C_n ◦ χ1

χ◦2

χ_n−2◦ ◦ χ_n−1

D2`+1 ◦

χ₁ ◦

τ1

τ_`−1◦ ◦ τ`

◦χ2

D2` ◦

χ1

τ◦₁ ◦

τ_`−2 ◦ qq qq qq q

MM MM MM τ_`−1M

◦χ3

T ◦

χ1

χ◦3

χ◦2

◦ NN NN NN N χ2

O ◦

χ4

χ◦1

◦ pp pp pp χ3 p

◦ NN NN NN N χ₂

I ◦

χ4

χ◦1

◦ pp pp pp χ₃ p

Figure 5.2. Reduced McKay quivers in dimension 3

• IfI ⊂E(ρ)∩E(ρ⁰), thenV(I)∼=ρ⊕ρ⁰ and the curves E(ρ) and E(ρ⁰)intersect transversally atI.

• The intersection graph of these curves is the reduced McKay quiver of the groupG.e

In particular, a generator V(I) does not contain more than one copy of any irreducible representation, and E(ρ)∩E(ρ⁰)6=∅ if and only if the representationsρandρ⁰ are adjacent.

Exceptional fibre in dimension three. LetG⊂SO(3,R)be a polyhedral group. Denote byπ:G-Hilb C³

−→C³/Gthe Hilbert-Chow morphism. For each non trivial irreducible representationρofG, set:

C(ρ) :={J ∈π⁻¹(O)red|V(J)⊃ρ}.

Theorem 6.2. — ([7, Theorem 3.1])

• EachC(ρ)is a smooth rational curve.

• π⁻¹(O)_red=S

ρC(ρ)(withρ∈Irr^∗(G)).

• IfJ ∈C(ρ)andJ /∈C(ρ⁰)for allρ⁰ 6=ρ, thenV(J)∼=ρ.

• The intersection graph of these curves is the reduced McKay quiver of the groupG.

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6.1. Explicit parameterizations

We explain briefly the explicit parameterizations of the exceptional curves.

This description holds both in dimensions two and three so we do it with our general notation. The example of the cyclic group is treated in §9. As we explained in §2,

mS/nG∼= M

ρ∈Irr(G) ρ6=ρ₀

2 dimρ

M

i=1

V⁽ⁱ⁾(ρ)

whereρ0 denotes the trivial representation. With the exact sequence:

0−→I/nG−→mS/nG −→mS/I−→0,

if one wants to parameterize a (flat) family of clusters overP¹_C, one has to choose, in the trivial sheaf:

O_P1

C⊗ M

ρ∈Irr(G) ρ6=ρ0

2 dimρ

M

i=1

V⁽ⁱ⁾(ρ),

a locally freeG-equivariant sheaf which is a copy of the regular representation on each fibre whose quotient is also locally free. The parameterizations are produced as follows: One choosesone non trivial subbundle

O_P1

C(−1)⊗ρ ,→ O_P1

C⊗(V⁽ⁱ⁾(ρ)⊕V^(j)(ρ))

for some appropriate choice of the indices, and shows that this gives the required family whose pointsIare characterized by their generator:

V(I)⊂P(V⁽ⁱ⁾(ρ)⊕V^(j)(ρ)).

That is: Once one choice has been made, the other choices are automatic, and they always correspond to a trivial subbundle (see §8.4).

7. Geometric construction

LetGe be a binary polyhedral group acting onA=C[x, y]. Set as before τ:=h±1i ⊂Ge andG:=G/τe the associated polyhedral group. We aim to define a morphism

S:G-Hilbe C²

−→G-Hilb C³

inducing a morphism between the exceptional fibres over the origin.

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Since A^τ = C[x², y², xy], we consider the following composite of ring morphisms, withB =C[a, b, c]:

(7.1) σ: B ////B

hab−c²i ^∼ //A^τ //A

where the identification is defined by a =x², b = y², c = xy. The action ofGe onA induces an action ofGon A^τ. Using the identification, we can define an action ofG on the coordinates a, b, c, inducing an action on B with the property that the coneK=hab−c²iisG-invariant.

Let I be an ideal of A and J :=σ⁻¹(I) the corresponding ideal ofB. Observe the following property of the mapσ:

Lemma 7.1. — If I is a G-cluster ine A, then J is a G-cluster in B. Furthermore, ifI is supported at the origin, then so isJ.

Proof. — IfIis aG-cluster, thene A/I∼=C[G]. Since the groupe τis finite, we have isomorphisms:

B/J∼=A^τ/I^τ ∼= (A/I)^τ∼=C[G]e^τ∼=C[G],

henceJ is aG-cluster inB. Furthermore, note thatσ⁻¹mA=mB hence if Iis aG-cluster supported at the origin, one hase I⊂m_Aand thenJ ⊂m_B, which implies thatJ is also supported at the origin (see §2).

This construction defines set-theoretically a map between the two moduli spaces of clustersS:G-Hilbe C²

−→G-Hilb C³

byS(I) =def J.

Proposition 7.2. — The mapS is regular, projective, and induces a morphism between the exceptional fibres.

Proof. —

Step 1.To get that the map S is regular, we show that it is induced by a natural transformation between the two functors of points

G-Hilbe _C2(·) =⇒G-Hilb_C3(·).

LetT be a scheme andZ∈G-Hilbe _C2(T). ThenZ⊂T×C²is a flat family ofG-clusters overe T and the mapZ ,→T×C²isτ-equivariant (for a trivial action onT). It induces a family

Z/τ ,→T ×(C²/τ),→T×C³

where the quotientC²/τ is considered as the conehab−c²iinC³. If T is a point, this is precisely our set-theoretic construction: ifZ is given by an idealI,Z/τ is given by the idealI^τ.

To show thatZ/τ ∈G-Hilb_C3(T), we have to prove that this family is flat overT. Since this problem is local inT, we may assume thatT is an affine

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scheme, say T = SpecR. Then the family Z is given by a τ-equivariant quotientR⊗AQso that the compositionR ,→R⊗_CAQmakesQ a flatR-module. The familyZ/τ is then given by the quotient

R ,→R⊗_CB R⊗_CA^τQ^τ,

where the quotientR⊗_CBR⊗_CA^τ is induced by tensorization of the quotientB A^τ. We have to show that this makesQ^τ a flatR-module.

By hypothesis, the functorQ⊗R−in the category ofR-modules is exact.

Since τ is finite, the functor (−)^τ is also exact in this category, and we note that the functorQ^τ⊗R−is the composition of this two functors since Q^τ⊗RN = (Q⊗RN)^τ for anyR-moduleN. Hence the functorQ^τ⊗R− is exact, which means that the family is flat.

Step 2.The composite of ring morphisms (7.1) gives an equivariant ring morphism

C[a, b, c] ^σ //

G

WW C[x, y]

Ge

WW

inducing a surjective map at the level of the invariants:C[a, b, c]^G C[x, y]^Ge, hence a closed immersion:

η: C²/Ge= SpecC[x, y]^Ge−→SpecC[a, b, c]^G=C³/G.

Taking more care of the coneK=C²/τ, the equivariant morphism C² //

Ge

XX C²/τ //

G

VV C³

G

XX

induces the morphismη between the quotients:

η: C²/Ge ^∼ // C²/τ

G //C³/G

sending the originO ∈C²/Ge to the originO∈C³/G and by definition of S the following diagram is commutative:

G-Hilbe C² S //

eπ

G-Hilb C³

π

C²/Ge ^η //C³/G

This implies thatS induces a morphism between the exceptional fibres:

S:eπ⁻¹(O)→π⁻¹(O).

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Step 3.We prove that the morphism S is proper by applying the val- uative criterion of properness. LetK be any field over CandR ⊂K any valuation ring with quotient fieldK. Consider a commutative diagram:

SpecK ^φ //

i

G-Hilbe C²

S

SpecR ^ψ //G-Hilb C³

We have to show that there exists a unique factorization SpecK ^φ //

i

G-Hilbe C²

S

SpecR ^ψ //

φ˜rrrrrr99 rr rr

G-Hilb C³

making the whole diagram commute.

By modular interpretation, the data of the morphism φ consists of an idealI⊂K[x, y]such thatK[x, y]/I∼=C[G]e ⊗_CKandK[x, y]/I isK-flat (it is here trivial sinceKis a field). Similarly, the data of the morphismψ consists in an idealJ ⊂R[a, b, c] such thatR[a, b, c]/J ∼=C[G]⊗_CR and R[a, b, c]/J isR-flat. The commutativityS◦φ=ψ◦imeans the following.

Consider the diagram of ring morphisms induced by natural extension of scalars and base-change from the mapσ:

R[a, b, c] _ ^σ^R //

R[x, y] _

K[a, b, c] ^σ^K //K[x, y]

Then the commutativity condition means thatσ_K⁻¹(I) =J·K[a, b, c].

We are looking for a morphism φ˜such that φ˜◦i =φ and S ◦φ˜=ψ, i.e.for an idealIe⊂R[x, y]such thatR[x, y]/Ie∼=C[G]e ⊗_CRandR[x, y]/eI is R-flat, satisfying the conditions Ie·K[x, y] = I and σ_R⁻¹(eI) = J. A natural candidate isIe=def I∩R[x, y]. We have to prove that it satisfies all the conditions and that it is unique for these properties. Denote by ν:K− {0} → H the valuation with values in a totally ordered groupH, satisfying the properties:

ν(x·y) =ν(x) +ν(y)andν(x+y)>min(ν(x), ν(y)) forx, y∈K\ {0}

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and such thatR={x∈K|ν(x)>0} ∪ {0}. Recall that Ris by definition integral and that aR-module is flat if and only if it is torsion-free (see for instance [1, 9]).

i.It is already clear that Ie·K[x, y] ⊂ I. Let P = P

i,jpi,jxⁱy^j ∈ I andp∈ {pi,j} an element of minimal valuation. Ifν(p)>0, then P ∈I.e Otherwise all coefficients ofp⁻¹P have positive valuation and sop⁻¹P ∈I.e SoP =p·(p⁻¹P)∈Ie·K[x, y], hence the equality.

ii. By commutativity of the above diagram, σ⁻¹_R (eI) =σ⁻¹_R (I∩R[x, y])

=σ⁻¹_K (I)∩R[a, b, c]

= (J·K[a, b, c])∩R[a, b, c].

It is clear thatJ ⊂(J·K[a, b, c])∩R[a, b, c]. LetP∈(J·K[a, b, c])∩R[a, b, c], decomposed asP =P

`U`·V` withU` ∈J andV`∈K[a, b, c]. As before, there exists a coefficientq in all V_` of minimal valuation, and we assume ν(q)<0(otherwise there is no problem). Thenq⁻¹P ∈J. By assumption, theR-moduleR[a, b, c]/J is torsion-free, so the multiplication byq⁻¹∈R is injective. This means thatP ∈J.

iii. By definition, we have anR-linear inclusionR[x, y]/I ,e→K[x, y]/I, which shows thatR[x, y]/Ieis torsion-free, hence flat. It inherits an action of Geand sinceK[x, y]/I∼=C[G]e ⊗_CK, there exists a subrepresentationV of C[G]e such thatR[x, y]/Ie∼=V ⊗_CR(this uses the flatness, see [13, Lemma 9.4]). By the isomorphism ofR-modulesR[x, y]/Ie⊗RK ∼=K[x, y]/I, the representationV is such thatV⊗RK=C[G]⊗e _CK, which forcesV ∼=C[G].e iv.The uniqueness follows from the condition I˜·K[x, y] = I since we already noted thatI∩R[x, y] = ( ˜I·K[x, y])∩R[x, y] = ˜I, so our natural candidate is the only possibility.

Step 4. To conclude, remark that any proper morphism between two quasi-projective varieties is automatically projective.

8. Contracted versus non contracted fibres

Theorem 8.1. — Consider the restriction of the morphismS to a re- duced curveE(ρ). Then:

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(1) If the representation ρ is pure, then S maps isomorphically the curveE(ρ)onto the curveC(ρ).

(2) If the representationρis binary, thenS contracts the curveE(ρ) to a point.

Proof. — Let E(ρ) be any exceptional curve. Since the morphism S sends this curve to the tree of curves π⁻¹(O), the image lies in some irreducible componentC and the restricted morphism S:E(ρ) → C is a proper morphism. We prove that:

• If the representationρis binary, then the morphismS:E(ρ)→C contracts the curve to a point;

• If the representation ρis pure, then C =C(ρ)and the restricted morphismS:E(ρ)→C(ρ)is an isomorphism.

The parameterizations of the two curves E(ρ) and C define a proper morphismf whose properties reflect those of the restriction ofS:

P¹_C ^∼_φ //

f

E(ρ)⊂G-Hilbe C²

S

P¹_C ^∼_ψ //C⊂G-Hilb C³

We know (see [9, II.6.8,II.6.9]) that either the morphism f contracts the curve to a point, or it is a finite surjective morphism. The basic idea to determine which case occurs is to take an ample line bundleO_P1

C(a)on the target (witha >0): If the morphismf contracts the curve to a point, then f^∗O_P₁_C(a)is trivial, otherwisef^∗O_P1

C(a)∼=O_P1

C(deg(f)·a)is ample.

The natural candidate for an ample line bundle over the curve C is the determinant det(p_∗O_Z(C)) obtained by restriction of the universal family Z(C) := ZG|_C.

The parameterizationP¹_C

−→φ G-Hilbe C²

of the curveE(ρ)corresponds to a flat familyZ

Ge(ρ)⊂P¹_C×C²which is the restriction toE(ρ)of the universal familyZ

Ge overG-Hilbe C²

. The direct imagep_∗O_Z

Ge(ρ) is a vector bundle of rank|G|e overP¹_Cequipped with an action ofGeinducing the regular representation on each fibre. It admits an isotypical decomposition over the irreducible representations ofGe and we recall the well-known explicit decomposition:

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Lemma 8.2. — p_∗O_Z

Ge(ρ)∼= O_P1

C(1)⊕ O^⊕^dimρ−1

P¹_C

⊗ρ⊕ M

ρ⁰∈Irr(^G)e

ρ⁰6=ρ

O^⊕^dimρ⁰

P¹_C ⊗ρ⁰

Proof of the lemma. — This is an equivalent form of [14, §2.1] or [11, Proposition 6.2(3)]. We recall briefly the argument. Since this bundle is a quotient ofO_P1

C⊗A(see §6.1), it is generated by its global sections, hence it is a sum of line bundlesO_P1

C(a)fora>0. Sincedeg(p_∗O_Z

Ge(ρ)) = 1(see [8]), all line bundles are trivial but one, of degree one.

In particular, note that det(p_∗O_Z

eG(ρ))∼=O_P1

C(dimρ)is the ample determinant line bundle in dimension two. By the functorial definition of the morphismS, the compositeP¹_C

−→φ G-Hilbe C²−→S G-Hilb C³

parame- terizes the flat familyZ

Ge(ρ)/τwith structural sheafO_Z

Ge(ρ)/τ= O_Z

Ge(ρ)

τ

and one gets:

f^∗(det(p_∗O_Z(C))) = det (p_∗O_Z

Ge(ρ))^τ .

Now, as we noticed in §4, taking the invariants underτ keeps invariant the pure representations and kills the binary ones. Hence:

• If the representationρis binary, then:

p_∗O_Z

eG(ρ)

τ

∼= M

ρ⁰∈Irr(G)

O^⊕^dim^ρ⁰

P¹_C ⊗ρ⁰ hencedet(p_∗O_Z

Ge(ρ))^τ∼=O_P1

C is trivial;

• If the representationρis pure, then:

p_∗OZ

Ge(ρ)

^τ

∼= O_P¹

C(1)⊕ O^⊕^dim^ρ−1

P¹_C

⊗ρ⊕ M

ρ⁰∈Irr(G) ρ⁰6=ρ

O^⊕^dim^ρ⁰

P¹_C ⊗ρ⁰

hencedet(p_∗O_Z

Ge(ρ))^τ∼=O_P1

C(dimρ)is ample.

This achieves the first part of the proof. It remains to show that in the case of a pure representationρ, the target curve isC=C(ρ)and that the finite surjective morphismf is an isomorphism. We do it by hand. A point I∈E(ρ)is characterized by the choice of V(I) and genericallyV(I)∼=ρ.

For a pure representationρ, the polynomials defining V(I)are even so:

V(I^τ) =V ((A·V(I) +nA)^τ)⊃V(I)

so generically V(I^τ) = V(I) (only changed by a = x²,b = y²,c = xy).

This means thatC =C(ρ)and ifI 6=J ∈E(ρ), then V(I)6=V(J)hence

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the images are also different, so the morphism is generically injective. This

concludes the proof.

As a byproduct of our argument, we get the following equivalent in dimension three of Lemma 8.2 which, to our knowledge, does not appear explicitly in the literature:

Corollary 8.3. — For any finite subgroupG⊂SO(3,R)and any non trivial representationρ ofG, the restriction of the tautological bundle to the exceptional curveC(ρ)decomposes as:

p_∗O_Z_G_(ρ)∼= O_P1

C(1)⊕ O^⊕^dim^ρ−1

P¹_C

⊗ρ⊕ M

ρ⁰∈Irr(G) ρ⁰6=ρ

O^⊕^dim^ρ⁰

P¹_C ⊗ρ⁰.

Proof. — The same argument as in the proof of Lemma 8.2 shows that this bundle in generated by its global sections. The bĳectivity of the morphismf on the curves associated to pure representations (in the notation of the proof of Theorem 8.1) implies thatdet(p_∗O_Z_G_(ρ))∼=O_P1

C(dimρ), hence in the isotypical decomposition there is only one non trivial line bundle, of degree one, and we already know by the explicit parameterizations that the isotypical component corresponding toρis not trivial.

Remark 8.4. — In the decomposition of Lemma 8.2, the presence of a uniqueO_P1

C(1) corresponds to the choice of the line V(I) in a projective spaceP(ρ⊕ρ)as explicitly described in §6.1. The fact that no other ample bundle occurs reflects the property that once one choice has been made, the other generators of the ideal do not involve the choice any more, as one can easily notice from the explicit computations of [13, §13,§14] (see

§9 in this paper for an example). In the three-dimensional case, the same situation occurs by Corollary 8.3.

We get now Theorem 1.1, presented in the introduction, as a consequence of Theorem 8.1:

Corollary 8.5. — The imageY :=S(G-Hilbe C²

)projects onto the quotientK/G, inducing a partial resolution of singularities containing only the exceptional curves corresponding to pure representations. The mor- phism S: G-Hilbe C²

−→ Y is a resolution of singularities contracting the excess exceptional curves to ordinary nodes.

Proof. — The projection π:Y −→ C³/G factors throughK/G by construction ofY. The other assertions result from Theorem 8.1. The excess curves contract to ordinary nodes since, as one checks with Figures 5.1 and 5.2, each excess(−2)-curve is contracted to a different point.

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Remark 8.6. — It is not difficult to see that in factY =G-Hilb(K).

9. Example: The cyclic group case Let the cyclic groupCe_n∼=Z/(2n)Zact onC²with generator:

ξ 0 0 ξ⁻¹

withξ=e⁽²ⁿ⁾^2πi.

The choice of coordinates made in §7 implies that the groupCn ∼=Z/nZ acts onC³ with generator:





ξ² 0 0 0 ξ⁻² 0

0 0 1



.

The irreducible representations of the cyclic group Cen are given by the matrices (ξⁱ), i = 0, . . . ,2n−1. For i even, they are also the irreducible representations ofC_n. There are thennpure andnbinary representations.

We setχi := ρ2i and χei = ρ2i+1 for i = 0, . . . , n−1. By Theorem 8.1, the exceptional curves on Ce_n-Hilb C²

corresponding to the binary representations are contracted byS to a node on S(Cen-Hilb C²

)whereas the curves corresponding to the pure representations are in 1 : 1 correspondence with the exceptional curves downstairs (see Figure 9.1). In this section, we check this by a direct computation.

``````

#

# c

c c

c χe1

χe2

χe3

χe4

χ1

χ₂

χ₃

? S

aa aa

aa!!!!!!!a aa

aa a χ1

χ2

χ3

Figure 9.1. Contracted fibres forCe4

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The ring of invariantsC[x, y]^Ceⁿis generated byx²ⁿ,y²ⁿ,xyandC[a, b, c]^Cⁿ is generated by c, aⁿ, bⁿ, ab. We recall the description of the exceptional curves ofCe_n-Hilb C²

following [13, Theorem 12.3]. We sort the basis of the algebra of coinvariants with respect to each irreducible representation:

{1},{x, y²ⁿ⁻¹}, . . . ,{xⁱ, y²ⁿ⁻ⁱ}, . . . ,{x²ⁿ⁻¹, y}.

To choose a clusterI/nA supported at the origin amounts in choosing one copy of each non trivial representation,i.e.for alli= 1, . . . ,2n−1a point (pi:qi)∈P¹_C defining the ideal by the generators:

hp₁x−q₁y²ⁿ⁻¹, . . . , p_ixⁱ−q_iy²ⁿ⁻ⁱ, . . . , p_2n−1x²ⁿ⁻¹−q_2n−1yi.

But the point is that one only needsone choice. Assume that there exists an indexisuch that p_iq_i 6= 0, and take the smallest iwith this property.

Set p = pi, q = qi and v = pxⁱ−qy²ⁿ⁻ⁱ. Then since xy is invariant, xⁱ⁺¹, . . . , x²ⁿ⁻¹ ∈ I/nA and y²ⁿ⁻ⁱ⁺¹, . . . , y²ⁿ⁻¹ ∈ I/nA so all our other choices were trivial, andV(I) =C·v. More formally, we parameterized the exceptional curveE(ρi)by a subbundle:

O_P1

C(−1)⊗ρi⊕M

j6=i

O_P1

C⊗ρj ,→M

j

(O_P1 C⊕ O_P1

C)⊗ρj.

If there is no such index, letxⁱbe the minimal power ofxin the choice: To find once each non trivial representation one has to choosey²ⁿ⁻ⁱ⁺¹and the minimal set of generatorsV(I) =C·xⁱ⊕C·y²ⁿ⁻ⁱ⁺¹contains two adjacent representations.

Otherwise stated, a Cen-cluster at the origin takes the form:

Ij(p:q) :=hpx^j−qy^2n−j, xy, x^j+1, y^2n−j+1i for16j62n−1,(p:q)∈P¹_C andE(ρ_j) ={I_j(p:q)}.

By the same method, aCn-cluster at the origin takes the form:

Jk(s:t) :=hsa^k−tb^n−k, c, a^k+1, b^n−k+1, abi for16k6n−1,(s:t)∈P¹_Cand C(χk) ={Jk(s:t)}.

With the construction (7.1) we have to compute σ⁻¹(Ij(p:q)). Setting

¯ σ:B

hab−c²i −→A, it is equivalent to computeσ¯⁻¹(Ij(p:q)). First we computeIj(p:q)^τ∈A^τ. We distinguish two cases:

• jeven,i.e.j= 2j⁰,j⁰= 1, . . . , n−1. We have inA^τ=C[x², y², xy]:

I_j(p:q)^τ=I_j(p:q) =hp(x²)^j

0

−q(y²)^n−j

0

, xy,(x²)^j

0+1

,(y²)^n−j

0+1

i

¯

σ⁻¹(Ij(p:q)) =hpa^j⁰−qb^n−j⁰, c, a^j⁰⁺¹, b^n−j⁰⁺¹i=Jj⁰(p:q).

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• jodd,i.e.j= 2j⁰+1,j⁰= 0, . . . , n−1. Note thatxy∈I_j(p:q)^τand (x²)^j⁰⁺¹,y^n−j⁰ ∈I_j(p:q)^τ, butpx^2j⁰⁺¹−qy^2n−2j⁰⁻¹ ∈/ I_j(p:q)^τ. Soσ¯⁻¹(Ij(p:q)) =ha^j⁰⁺¹, b^n−j⁰, ci. Since:

¯

σ⁻¹(I_j(p:q)) =J_j⁰(0 : 1) =J_j⁰₊₁(1 : 0), one has¯σ⁻¹(Ij(p:q))∈C(ρj⁰)∩C(ρj⁰+1).

The curvesE(ρ_j)with j even correspond to the pure representations and are not contracted byS as the previous computation shows. The curves with j odd correspond to the binary representations and are contracted byS.

10. Application: Pencils of symmetric surfaces The polynomial invariants of the bipolyhedral groups G6, G8 and G12

are studied by the second author in [17]: The first non trivial invariant other than a power of the quadric Q:x²+y²+z²+t² = 0 is a homogeneous polynomialS_n of degree n. Consider then the following pencil of Gn-symmetric surfaces in P³_C:

Xn(λ) ={Sn+λQ^n/2= 0}, λ∈C.

The general surface X_n(λ) is smooth and for each n there are precisely four singular surfaces in the corresponding pencil. The singularities of these surfaces are ordinary nodes forming one orbit throughG_n (see [17]).

Consider now the pencil of quotient surfaces in P³_C Gn: {Xn(λ)/Gn}, λ∈C.

These quotient surfaces have only A-D-E singularities and the minimal resolutions of singularitiesYn(λ)→ Xn(λ)/Gnare K3 surfaces with Picard number greater than19(see [2]). For the four nodal surfaces in each pencil, a careful study of the stabilizers of the nodes shows that, ifX denotes one of these nodal surfaces, the image of the node on X/Gn ⊂ P³_C

Gn is a particular quotient singularity locally isomorphic to C²/Ge ⊂ C³/G for some polyhedral groupGexplicitly computed (see [2, §3, Proposition 3.1]):

• forn= 6:C₃,T;

• forn= 8:D2, D3, D4,O;

• forn= 12:D₃, D₅,T,I.

Therefore, Theorem 1.1 gives locally a group-theoretic interpretation of the exceptional curves of the K3 surfacesY_n(λ)over the particular singularities of the nodal surfaces.

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Manuscrit reçu le 27 juin 2005, révisé le 29 mars 2006, accepté le 11 janvier 2007.

Samuel BOISSIÈRE

Université de Nice Sophia-Antipolis

Laboratoire J.A.Dieudonné UMR CNRS 6621 Parc Valrose

06108 Nice (France)

[email protected]

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Alessandra SARTI

Johannes Gutenberg Universität Mainz Institut für Mathematik

55099 Mainz (Deutschland) [email protected]