On Hypersurface Quotient Singularity of Dimension 4

We consider geometrical problems on Gorenstein hypersurface orbifolds of dimension $n \geq 4$ through the theory of Hilbert scheme of group orbits. For a linear special group $G$ acting on $\CZ^n$, we study the $G$-Hilbert scheme, $\hl^G(\CZ^n)$, and crepant resolutions of $\CZ^n/G$ for $G$=the $A$-type abelian group $ A_r(n)$. For $n=4$, we obtain the explicit structure of $\hl^{A_r(4)}(\CZ^4)$. The crepant resolutions of $\CZ^4/A_r(4)$ are constructed through their relation with $\hl^{A_r(4)}(\CZ^4)$, and the connections between these crepant resolutions are found by the"flop"procedure of 4-folds. We also make some primitive discussion on $\hl^G(\CZ^n)$ for the $G$= alternating group ${\goth A}_{n+1}$ of degree $n+1$ with the standard representation on $\CZ^n$; the detailed structure of $\hl^{{\goth A}_4}(\CZ^3)$ is explicitly constructed.


Introduction
The purpose of this paper is to study some geometrical problems of certain Gorenstein hypersurface orbifolds of dimension 4. The main focus will be on the structure of the newly developed concept of Hilbert scheme of group orbits and its connection with crepant resolutions of the orbifold. For a finite subgroup G in SL n (C), the G-Hilbert scheme, Hilb G (C n ), was first introduced by Nakamura et al [6,8,9,14]; one primary goal aims to provide a conceptual understanding of crepant resolutions of C n /G for n = 3, whose solution was previously known by a computational method, relying heavily on Miller-Blichfeldt-Dickson classification of finite groups in SL 3 (C) [12] and the invariant theory of two simple groups, I 60 (icosahedral group), H 168 (Klein group) [11] ( for the existence of crepant resolutions, see [18] and references therein). For n = 2, Hilb G (C 2 ) is the minimal resolution of C 2 /G, hence it has the trivial canonical bundle [8,9,14]. For n = 3, it was expected that Hilb G (C 3 ) is a crepant resolution of C 3 /G. Recently the affirmative answer was obtained, in [7,15] for the abelian group G by techniques in toric geometry, and in [2] for a general group G by derived category methods bypassing the geometrical analysis of G-Hilbert scheme. With these successful results in dimension 3, a question naturally arises on the possible role of G-Hilbert scheme on crepant resolution problems of orbifolds in dimension n ≥ 4. For n ≥ 4, it is a wellknown fact that C n /G might have no crepant resolutions at all, even for a cyclic group G and n = 4, (for a selection of examples, see e.g., [19]). To avoid many such complicated exceptional cases, we will restrict our attention only to those with hypersurface singularities. In this paper, we will address certain problems on two specific types of hypersurface Gorenstein quotient singularity, C n /G, of dimension n; one is the abelian group G = A r (n) defined in (8) in the main body of the paper, the other group G is the alternating group A n+1 of degree n + 1 acting on C n through the standard representation. In the case G = A r (n), Hilb G (C n ) is a toric variety, hence the methods for toric geometry provide an effectively tool to study its structure. For n = 4, we will give a detailed derivation of the smooth toric structure of Hilb Ar(4) (C 4 ), and construct the crepant toric resolutions of C 4 /A r (4) by blowing-down the canonical divisors of Hilb Ar(4) (C 4 ); in due course the "flop" of 4-folds naturally arises in the process, (see Theorems 3.1, 3.2, 4.1 in the main text of this paper, whose statements were previously announced in [3]). We would expect the concept appeared in the proof of these theorems will inspire certain clue to other cases, not only the A r (n)-type groups, but also for the non-abelian groups A n+1 ( which are simple groups for n ≥ 4). The group A 4 is a solvable group of order 12, also called the ternary trihedral group. The crepant resolution of C 3 /A 4 was explicitly constructed in [1], and the structure Hilb A 4 (C 3 ) over the origin orbit of C 3 /A 4 was analyzed in detail in [6]. Through the representation theory of A 4 , we will give the direct verification that Hilb A 4 (C 3 ) is smooth and a crepant resolution of C 3 /A 4 . Though the conclusion is known by the general result in [2] using qualitative arguments, the object of our detailed analysis aims to reveal that there exist certain common features in determining the structures of G-Hilbert schemes for certain abelian and non-abelian groups G by the computational methods, in hope that the approach could possibly be applied to higher dimensional cases. With this in mind, we will in this paper restrict our attention only to the case A 4 , leave possible generalizations, applications or implications to future work. This paper is organized as follows. In §2, we will summarize the main features of G-Hilbert scheme of group orbits, and results in toric geometry for the need of later discussion. We will also define the group G which we will consider in this paper. The next two sections will be devoted to the discussion of the structure of Hilb G (C 4 ) and crepant resolutions of C 4 /G for G = A r (4). For the simpler terminology to express the ideas, also for the description of geometry of flop of 4-folds, we will consider only the case A 1 (4) in §3 to discuss the structure of Hilb A 1 (4) (C 4 ). The flop relation between crepant resolutions of C 4 /A 1 (4) will be examined in detail through Hilb A 1 (4) (C 4 ). In §4, we will derive the solution of the corresponding problems for G = A r (4) for a general positive integer r, with much more complicated techniques but a method much in tune with the previous section. In §5, we consider the case G = A n+1 acting on C n through the standard representation for n = 3. By employing the structure of the fiber in Hilb A 4 (C 3 ) over the origin orbit of C 3 /A 4 described in [6], we give an explicit construction of the smooth and crepant structure of Hilb A 4 (C 3 ) using finite group representation theory, along a line similar to the previous two sections in a certain sense. Finally we give the conclusion remarks in §6.
Notations. To present our work, we prepare some notations. In this paper, by an orbifold we shall always mean the orbit space of a smooth complex manifold acted on by a finite group. Throughout the paper, G will always denote a finite group unless otherwise stated. We denote The trivial representation of G will be denoted by 1. For a G-module W , i.e., a G-linear representation space W , one has the canonical irreducible decomposition: For an analytic variety X, we shall not distinguish the notions of vector bundle and locally free O X -sheaf over X.

G-Hilbert Scheme, Toric Geometry
In this section, we brief review some basic facts on Hilb G (C n ) ( the Hilbert scheme of G-orbits) and toric geometry necessary for later use, then specify the groups G for the discussion of the rest sections of this paper.
First, we will always assume G to be a finite subgroup of SL n (C). Denote S G := C n /G with the canonical projection, π G : C n → S G , and o := π G ( 0). As G acts on C n freely outside a finite collection of linear subspaces with codimension ≥ 2, S G is an orbifold with non-empty singular set Sing(S G ) of codimension ≥ 2. In fact, the element o is a singular point of S G . By a variety X birational over S G , we will always mean a proper birational morphism σ from X to S G which is biregular between X \ σ −1 (Sing(S G )) and S G \ Sing(S G ), One can form the commutative diagram via the birational morphism σ, Denote F X the coherent O X -sheaf over X obtained by the push-forward of the structure sheaf of one has a canonical morphism, µ * F X ′ −→ F X . In particular, the morphism (1) gives rise to the O X -morphism, Furthermore, all the above morphisms are compatible with the natural G-structure of F X induced from the G-action on C n via (2). Then F X has the canonical G-decomposition of coherent O Xsubmodules: is the residue field at x. Over X \ σ −1 (Sing(S G )), F X is a vector bundle of rank |G| with the regular G-representation on each geometric fiber. Hence (F X ) ρ is a vector bundle over X \ σ −1 (Sing(S G )) with the rank equal to the dimension of V ρ . For x ∈ X, there exists a G-invariant ideal I(x) in C[Z](:= C[Z 1 , · · · , Z n ]) such that the following relation holds, We In particular, for X = S G in (3) For x ∈ X, there exists a direct sum decomposition of C[Z] as G-modules, Here I(x) ⊥ is a finite dimensional G-module isomorphic to C[Z]/I(x). Similarly, we have G-module decompositions for s = σ(x) ∈ S G , so that the relations, I(s) ⊥ ⊂ I(x) ⊥ ⊂ I(s) ⊥ , hold. Note that the above finite dimensional Gmodules with superscript ⊥ are not unique in C[Z] because there is a choice involved, nonetheless we could choose them such that this inclusions are fulfilled. One has the canonical G-decomposition of I(x) ⊥ : I(x) ⊥ = ρ∈Irr(G) I(x) ⊥ ρ , where the factor I(x) ⊥ ρ is isomorphic to a positive finite sum of copies of V ρ . Now we consider the varieties X birational over S G such that F X is a vector bundle. Among all such X, there exists a minimal object, called the G-Hilbert scheme in [8,9,14,15], By the definition of Hilb G (C n ), an element (i.e. closed point) p of Hilb G (C n ) is described by a G-invariant ideal I(= I(p)) of C[Z] of colength |G|, and the fiber of the vector bundle F Hilb G (C n ) over p can be identified with the regular G-module C[Z]/I(p). For simplicity of notations, we shall also make the identification of the element p with its associated ideal I, and write I ∈ Hilb G (C n ) in what follows if no confusion arises. For any other X, the map (1) can be factored through a birational morphism λ from X onto Hilb G (C n ) via σ Hilb , In fact, the ideal I(x) of (3) is a colength |G| ideal in C[Z], by which the map λ is defined. We will denote X G the normalization of Hilb G (C n ), which is a normal variety over S G with the birational morphism from X G onto S G . As every biregular automorphism of S G can always be lifted to one of Hilb G (C n ), hence also to X G , one has the following result.
Lemma 2.1 Let Aut(S G ) be the group of biregular automorphisms of S G . Then Hilb G (C n ), X G are Aut(S G )-varieties over S G via Aut(S G )-morphisms. As a consequence, X G is a toric variety for an abelian group G. Now we are going to summarize some basic facts in toric geometry for the later discussion when the group G is abelian, ( for details, see e.g., [5,10,16]) . In this case, we consider G as a subgroup of the diagonal group T 0 of GL n (C n ) with the identification T 0 = C * n . Regard C n as the partial compactification of T 0 , Let T be the torus T 0 /G and consider S G (= C n /G) as a T -space, The combinatorial data of toric varieties are constructed from the lattices of 1-parameter subgroups and characters of tori T, T 0 ,  In what follows, we shall identify M 0 with the group of monomials in variables Z 1 , . . . , Z n via the correspondence: The dual lattice M of N is the sublattice of M 0 , consisting of all G-invariant monomials. Among the varieties X birational over the T -space S G , we shall consider only those X with a T -structure. It has been known that these toric varieties X are represented by certain combinatorial data in toric geometry. A toric variety over S G is described by a fan Σ = {σ α | σ ∈ I} with the first quadrant of R n as its support, i.e., a rational convex cone decomposition of the first quadrant in R n . Equivalently, these combinatorial data can also be described by the intersection of the fan and the standard simplex ∆ in the first quadrant, The corresponding data in △ are denoted by Λ = {△ α | α ∈ I} with △ α := σ α ∩ △. Then Λ is a polytope decomposition of △ with vertices in △∩Q n . Note that for σ α = { 0}, we have △ α = ∅. Such Λ will be called a rational polytope decomposition of △, and we will denote X Λ the toric variety corresponding to Λ. If all vertices of Λ are in N , Λ is called an integral polytope decomposition of △. For a rational polytope decomposition Λ of △, we define Λ(i) : The T -orbits in X Λ are parametrized by n−1 i=−1 Λ(i). In fact, for △ α ∈ Λ(i), there associates a (n − 1− i)-dimensional T -orbit, which will be denoted by orb(△ α ). A toric divisor in X Λ is the closure of an (n − 1)-dimensional orbit, denoted by D v = orb(v) for v ∈ Λ(0). The canonical sheaf of X Λ is expressed by the toric divisors (see, e.g. [5,10,16]), where m v is the least positive integer with m v v ∈ N . In particular, X Λ is crepant, i.e. , ω X Λ = O X Λ , if and only if Λ is integral. On the other hand, the smoothness of X Λ is described by the decomposition Λ to be a simplicial one with the multiplicity one property, i.e., for each Λ α ∈ ∆(n − 1), the elements m v v for v ∈ Λ α ∩ Λ(0) form a Z-basis of N . The following results are known for toric variety over S G (see e.g. [17]): (1) The Euler number of X Λ is given by χ(X Λ ) = |Λ(n − 1)|.
(2) For a rational polytope decomposition Λ of ∆, any two of the following three properties imply the third one: In this paper, we shall consider only two specific series of hypersurface n-orbifold S G for n ≥ 2. The first type can be regarded as a generalization of the A-type Klein surface singularity, the group G is defined as follows, The For a nontrivial character ρ of A r (n), the dimension of I(o) ⊥ ρ is always greater than one. In fact, one can describe an explicit set of monomial generators of I(o) ⊥ ρ . For example, say I(o) ⊥ ρ containing an element Z I with I = (i 1 , . . . , i n ), i 1 = 0 and i s ≤ i s+1 , then I(o) ⊥ ρ is generated by Z K s with K = (k 1 , . . . , k n ) given by here j runs through 1 to n. Note that some of the above n-tuples K might coincide. In particular for r = 1, the dimension of I(o) ⊥ ρ is equal to 2 for ρ = 1, with a basis consisting of Z I , Z I ′ whose indices satisfy the relations, 0 ≤ i s , i s ′ ≤ 1, i s + i s ′ = 1 for 1 ≤ s ≤ n.
The second type of group G is the alternating group A n+1 (of degree n+1) acting on C n through the standard representation. The representation is induced from the linear action of the symmetric group S n+1 on C n+1 by permuting the coordinate indices, then restricting on the subspace We denote C[Z](:= C[Z 1 , · · · ,Z n+1 ]) the coordinate ring of the affine (n + 1)-space C n+1 , and their elementary symmetric polynomials σ k : are generated by the above σ k s and δ := i<j (Z i −Z j ) with a relation . . , σ n+1 ) for certain polynomial F . In fact, F is a (quasi-)homogeneous polynomial of degree n(n + 1) with the weights of σ k and δ equal to k, n(n+1) 2 respectively. Denoted by s k , d the restriction functions of σ k , δ on V respectively. Then s 1 is the zero function, and V /S n+1 = C n via the coordinates (s 2 , . . . , s n+1 ). The orbifold Then V /S n+1 can be realized as a hypersurface in C n+1 with the equation, where F n (s 2 , . . . , s n+1 ) := F (0, s 2 , . . . , s n+1 ). The polynomial F n (s 2 , . . . , s n+1 ) has a lengthy expression in general. Here we list the polynomial F n for n = 3, 4:

A 1 (4)-Singularity and Flop of 4-folds
We now study the A 1 (n)-singularity with n ≥ 4. The set of N -integral elements in ∆ are given by where v i,j := 1 2 (e i + e j ) for i = j. Other than the simplex ∆ itself, there is only one integral polytope decomposition of ∆ invariant under all permutations of coordinates, and we will denote it by Ξ. Ξ(n − 1) consists of n + 1 elements: △ i (1 ≤ i ≤ n) and ✸, where △ i is the simplex generated by e i and v i,j for j = i, and ✸ is the closure of △ \ n i=1 △ i , equivalently ✸ = the convex hull spanned by v i,j s for i = j. The lower dimensional polytopes of Ξ are the faces of those in Ξ(n − 1). X Ξ has the trivial canonical sheaf. For n = 2, 3, X Ξ is a crepant resolution of S A 1 (n) . For n = 4, one has the following result.
Lemma 3.1 For n = 4, the toric variety X Ξ is smooth except one isolated singularity, which is the 0-dimensional T-orbit corresponding to ✸.
Proof. In general, for n ≥ 4, it is easy to see that for each i, the vertices of ∆ i form a Z-basis of N , e.g., say i = 1, it follows from |A 1 (n)| = 2 n−1 , and det(e 1 , v 1,2 , · · · , v 1,n ) = 1 2 n−1 . Hence X Ξ is non-singular near the T -orbits associated to simplices in ∆ i . As ✸ is not a simplex, orb(✸) is always a singular point of X Ξ . For n = 4, the statement of smoothness of X Ξ except orb(✸) follows from the fact that for 1 ≤ i ≤ 4, the vertices v i,j (j = i) of X Ξ , together with 1 2 4 j=1 e j , form a N -basis. ✷ Remark 3.1 For n ≥ 4, the following properties hold for 0-dimensional T -orbits of X Ξ .
(1) Denote (2) We shall denote x ✸ := orb(✸) in X Ξ . The singular structure of x ✸ is determined by the A 1 (n)-invariant polynomials corresponding to the M -integral elements in the cone dual to the one generated by ✸ in N R . So the A 1 (n)-invariant polynomials are generated by X j := Z 2 j and Y j := Note that for n = 3, the Y j s indeed form the minimal generators for the invariant polynomials, which implies the smoothness of X Ξ . For n ≥ 4, x ✸ is a singularity, not of the hypersurface type. For n = 4, the X j , Y j (1 ≤ j ≤ 4) form a minimal set of generators of invariant polynomials, hence the structure near x ✸ in X Ξ is the 4-dimensional affine variety in C 8 defined by the relations: where i = j with {i ′ , j ′ } the complementary pair of {i, j}.
For the rest of this section, we shall consider only the case n = 4. We are going to discuss the structure of Hilb A 1 (4) (C 4 ) and its connection with crepant resolutions of S A 1 (4) . The simplex ∆ is a tetrahedron, and ✸ is an octahedron; both are acted on by the symmetric group S 4 . The dual polygon of ✸ is the cube. The facets of the octahedron ✸ are labeled by where E is an irreducible divisor isomorphic to the triple product of P 1 , Furthermore for {i, j, k} = {1, 2, 3}, the normal bundle of E when restricted on the fiber P 1 k (≃ P 1 ), for the projection E to P 1 × P 1 via the (i, j)-th factor, is the (−1)-hyperplane bundle: Proof. First we show the smoothness of the toric variety X Ξ * . The octahedron ✸ of Ξ is decomposed into eight simplices of Ξ * corresponding to faces F j , F ′ j of ✸. Denote C j (resp. C ′ j ) the simplex of Ξ * spanned by c and F j (resp. F ′ j ); x C j , x C ′ j are the corresponding 0-dimensional T -orbits in X Ξ * . The smoothness of affine space in X Ξ * near x C j , x C ′ j follows from the N -integral criterion of the cones in N R generated by C j , C ′ j . The coordinate system is given by the integral basis of M which generates the cone dual to the cone spanned by C j ( C ′ j ). As examples, for C 1 , C ′ 2 , the coordinates are determined by the row vectors of the following square matrix: The coordinate functions of X Ξ * centered at x C 1 are given by ( . By the Remark 3.1 (1), one has the smooth coordinate system centered at x ∆ j in X Ξ * . For ∆ 1 , by one has the coordinate system near . Now we are going to show that C[Z]/I(y) is a regular G-module for y ∈ X Ξ * . For an element y in the affine neighborhood of x ∆ 1 with the coordinates The set of monomials, where Similarly, the same conclusion holds for y near with The same argument can equally be applied to all affine charts centered at x ∆ j , x C j , x C ′ j . Therefore we obtain a morphism λ : X Ξ * −→ Hilb G (C 4 ) , with I(λ(y)) = I(y), y ∈ X Ξ * .
We are going to show that the above morphism λ is an isomorphism by constructing its inverse morphism. Let y ′ be an element of Hilb G (C By direct counting, there are twelve such W ′ and the corresponding twelve J 0 's, are exactly those I(x R ) for R ∈ Ξ * (3). The correspondence between W ′ and R by the relation W ′ = I(x R ) † is given as follows: Now we consider an ideal J in C[Z] which defines an element of Hilb G (C 4 ). By the Gröbner basis argument as before, there is a monomial ideal J 0 (= lt(J)) such that J † 0 gives rise to a basis of C[Z]/J, and J 0 = I(x R ) for some R ∈ Ξ * (3). For p ∈ C[Z], the element p + J ∈ C[Z]/J is uniquely expressed in the form, , µ g (p) ∈ C * are the the character values of g on m, p respectively; hence Indeed in the above expression of J, it suffices to consider those ps which from a minimal set of monomial generators of J 0 . Now we are going to assign an element of X Ξ * for a given J ∈ Hilb G (C 4 ). If the monomial ideal J 0 associated to J in our previous discussion is equal to I(x C 1 ), a minimal set of monomial generators of J 0 and the basis representative set J † 0 of C[Z]/J are given by which has the colength at most 8 in C[Z]. Therefore one obtains Moreover, by and Z 2 ∈ J 0 † , one has γ 1 = γ 12 γ 13 γ 4 .
Similarly, one has γ 3 = γ 234 γ 13 , γ 4 = γ 234 γ 14 , Therefore, all γ I s are expressed as functions of γ 12 , γ 13 , γ 14 , γ 234 . This implies J = I(y) for an element y of X Ξ * in the affine neighborhood x C 1 with the coordinate (U i = u i ) by the relations, The above y is defined to be the element λ −1 (J) in X Ξ * for the ideal J under the inverse map of λ. The method can equally be applied to ideals J associated to another monomial ideal J 0 .
We claim that the variables γ ′ 2 , γ ′ 34 , γ ′ 13 , γ ′ 14 form a system of coordinates near I(x C ′ 2 ), i.e., all the γ ′ I s can be expressed as certain polynomials of these four values. Indeed, we are going to show By the description in (21) for C ′ 2 , Z 2 Z 4 is an element in J † 0 , hence represents a basis element of By interchanging the indices 1 and 3, (resp. 1 and 4), in the above derivation and regarding . Thus, γ ′ 2 , γ ′ 13 , γ ′ 13 and γ ′ 34 form the 1 Note that the group G in Section 6.1 of [15] (page 777) is the A1(4) of Theorem 3.1 in this paper. However, we would consider that the statement in [15] about the singular property of Hilb G (C 4 ) by using the structure of I(Γ3)(u) there, is not correct. Indeed, by identifying Z2, Z3, Z4, Z1 with x, y, z, w, and γ ′ 2 , γ ′ 3 , γ ′ 4 , γ ′ 1 , γ ′ 34 , γ ′ 13 , γ ′ 14 with u1, u2, · · · , u7 respectively, the ideal J in our discussion corresponds to I(Γ3)(u) in [15]. Then through the three relations we have obtained here, one can easily verify that all the relations among the uis listed in page 778 of [15] hold.
four independent parameters to describe the ideals J near J 0 = I(x C ′ 2 ) with the regular G-module C[Z]/J. Therefore J = I(y) for y near x C ′ 2 with the coordinates (U ′ i = u ′ i ) via the relations, The previous discussions of three cases can be applied to each of the twelve monomial ideals J 0 's by a suitable change of indices. Hence one obtains an element λ −1 (J) in X Ξ * of an ideal J ∈ Hilb G (C 4 ).
However, one has to verify the correspondence λ −1 so defined to be a single-valued map, namely, for a given J with two possible choices of J 0 , the elements in X Ξ * assigned to J through the previous procedure through these two J 0 are the same one. For example, say J = I(y 1 ) = I(y 2 ) for y 1 near We claim that u 1 = 0. Otherwise, both Z 1 and Z 2 Z 3 Z 4 are elements in J with the same G-character κ. Then the κ-eigenspace in C[Z]/J is the zero space, a contradiction to the regular G-module property of C[Z]/J. Hence one has Using the same argument, one can derives u j = v 1 v j for j = 2, 3, 4. These three relations, together with u 1 = v 1 −1 , imply y 1 = y 2 in X Ξ * . For y 2 near x C 1 with (U i = u i ), and y 3 near 1 Z 1 Z 2 are elements in J; furthermore, u 2 , u ′ 1 are non-zero by the fact that only one of Z 1 Z 2 , Z 3 Z 4 could be an element of J. By an argument similar to the one before, one can show As Z 1 Z 2 represents a basis element of C[Z]/J, one has u 1 = u ′ 1 u ′ 2 . The four relations between u i s and u ′ i s imply y 2 = y 3 in X Ξ * . In this way, one can show directly that for a given ideal J with J = I(y) = I(y ′ ) for y, y ′ in X Ξ * , the elements y and y ′ are the same one by the relations of toric coordinates centered at two distinct x R s. Hence we have obtained a well-defined morphism λ −1 from Hilb G (C 4 ) to X Ξ * , then Hilb G (C 4 ) ≃ X Ξ * . By (6), the canonical bundle of X Ξ * . is given by ω = O X Ξ * (E), where E denotes the toric divisor D c , which is a 3-dimensional complete toric variety with the toric data described by the star of c in Ξ * , which is represented by the octahedron in Fig. 1, where the cube in Fig. 1 represents the toric orbits' structure. Therefore E is isomorphic to the triple product of P 1 as in (15). The description of the normal bundle of E restricting on each P 1 -fiber will follow by the direct computation in toric geometry. For example, for the fibers over the projection of E onto (P 1 ) 2 corresponding to the 2-convex set spanned by v 1,2 , v 1,3 , v 3,4 and v 2,4 , one can perform the computation as follows. Let (U 1 , U 2 , U 3 , U 4 ) be the local coordinates near x C ′ 4 dual to the N -basis (2c, v 1,2 , v 1,3 , v 2,3 ), and let (W 1 , W 2 , W 3 , W 4 ) be the local coordinates near x C 1 dual to (2c, v 1,2 , v 1,3 , v 1,4 ). By 2c = v 1,4 + v 2,3 , one has the relations, This shows that the restriction of the normal bundle of E on each fiber P 1 over (U 2 , U 3 )-plane is the (−1)-hyperplane bundle. ✷ Note that the vector bundle F X Ξ * over X Ξ * in Theorem 3.1 carries the regular G-module structure on each fiber with the local frame of the vector bundle provided by the structure of C[Z]/I(x R ) for R ∈ Ξ * (3) with the representative in the list (21). By the standard blowing-down criterion of an exceptional divisor, the property (17) ensures the existence of a smooth 4-fold (X Ξ * ) k by blowing-down the P 1 -family along the projection p k (16) for each k. In fact, (X Ξ * ) k is also a toric variety X Ξ k with Ξ k defined by the refinement of Ξ by adding the segment connecting v k, 4 and v i,j to divide the central polygon ✸ into four simplices, where {i, j, k} = {1, 2, 3}. Each X Ξ k is a crepant resolution of X Ξ (= S G ), and one has the refinement relation of toric varieties : Ξ ≺ Ξ k ≺ Ξ * for k = 1, 2, 3. The polyhedral decomposition in the central core ✸ appeared in the refinements is indicated by the following relation, whose pictorial realization is shown in Fig. 3. The connection between these three smooth 4-folds corresponding to these different ✸ k s can be regarded as the "flop" relation of 4-folds, an analogy to the similar procedure in birational geometry of 3-folds [13]. Each one is a "small" 2 resolution of the 4-dimensional isolated singularity with the defining equation (14). Hence we have shown the following result. In this section, we give a complete proof of a general result as in Theorem 3.2, but on the group A r (4) for all r.  G = A r (4), the G-Hilbert scheme Hilb G (C 4 ) is a non-singular toric variety with the canonical bundle, , where E k s are disjoint smooth exceptional divisors in Hilb G (C 4 ), each of which satisfies the conditions (15) (17). By blowing down E k to P 1 × P 1 via a projection (16) for each k, it gives rise to a toric crepant resolution S G of S G with χ( S G ) = |A r (4)| = (r + 1) 3 . Furthermore, any two such S G s differ by a sequence of flops.
Proof. First we define the simplicial decomposition Ξ * of (5) for n = 4, and then we will show that the toric variety X Ξ * is isomorphic to Hilb G (C 4 ). We shall denote an element of N ∩ ∆ by For each v m ∈ N ∩ ∆, there are four hyperplanes passing through v m , and parallel to one of the four facets of ∆. The collection of all such hyperplanes gives rise to a polytope decomposition of ∆, denoted by Ξ, (for r = 2 see the left one of Fig. 4).  For a given v m , v m(i,j) , the hyperplane passing v m in R 4 with the normal vector e i − e j separates ∆ into two polytopes ∆ ′ s, (one of which could possibly be the empty set). We are going to discuss those elements in Ξ containing v m and lying in a non-empty polytope of these two divided ones. For easier description of our conclusion, also for the simplicity of notions, we shall work on a special model case, say i = 2, j = 3, and the nonempty polytope ∆ ′ consisting of those elements in ∆ with non-negative inner-product to e 2 − e 3 , ( no difficulties for a similar discussion will arise on other cases except for a suitable change of indices). The elements in Ξ(3) contained in ∆ ′ with v m as one of its vertices are the following ones: 3) , v m (2,1) , v m (2,4) , 3) , v m(4, 3) , v m (2,1) , v m (4,1) , v m+(−1,1,−1,1) t , m(1,3) , v m (2,4) , v m (1,4) , v m+(1,1,−1,−1) t . (23) Note that ✸ ± are similar by interchanging e 3 and e 4 , ( for the configuration of ∆ u , ∆ d , ✸ + , see the right one of Fig. 4). Both ∆ u , ∆ d are 3-simplices with their vertices forming an integral basis of N , and one facet of each of these 3-simplices is parallel to that of ∆. The toric data of ∆ u , ∆ d give rise to the smooth affine open subsets of X Ξ . The polytope ✸ + (✸ − ) is an octahedron with the center c = v m + e 2 +e 4 −e 1 −e 3 2(r+1) (c = v m + e 1 +e 2 −e 3 −e 4 2(r+1) respectively). We shall mark the octahedron by its center c, and denote it by ✸ c . The affine open subset of X Ξ with the toric data ✸ c is smooth except one isolated singular point x ✸ c , an 0-dimensional toric orbit of the affine toric variety. Hence, one can conclude that Ξ(3) consists of three type of elements: ∆ u , ∆ d or ✸ c . The toric variety X Ξ is smooth except the finite number isolated singularities, x ✸ c s. The structure of X Ξ near a singular element x ✸ c can be determined in the following manner. For a given ✸ c , one can construct a tetrahedron ∆ c inside ∆ with the core ✸ c adjacent to four elements ∆ c are four facets of ✸ c , two of which intersect only at one common vertex, ( there could have two possible ways of forming such ∆ c with the same core ✸ c ). Consider the rational simplicial decomposition Ξ * of ∆, which is a refinement of Ξ by adding c as a vertex with the barycentric simplicial decomposition ✸ c for all c. In fact, the octahedron ✸ c is decomposed into the following eight 4-simplices of Ξ * : , c .
All vertices appeared in the above simplices are elements in N ∩ ∆ except c , while 2c ∈ N . (see Fig. 5) One can determine the singularity structure of the variety X Ξ near x ✸ c by examining the toric orbits associated to ∆ c . The toric data in R 4 for the lattice N and the cone generated by ∆ c are isomorphic to the toric data of the lattice for the group A 1 (4) with the first quadrant cone in Lemma 3.1. Hence as toric varieties, the structure of X Ξ near the singularity x ✸ c is the same as that for A 1 (4). One can apply the result of Theorem 4.1 to describe the local structure of X Ξ * over the singular point x ✸ c of X Ξ . Hence one concludes that X Ξ * is a smooth toric variety with the canonical bundle, ω X Ξ * = O X Ξ * ( ✸ c ∈Ξ(4) E c ), where E c is the toric divisor associated to the vertex c in X Ξ * , and it satisfies the properties (15) (17). By (7) and the structure of E c , one obtains the desired crepant resolutions S Ar(4) by blowing-down each E c to P 1 × P 1 as in Theorem 3.2. and different crepant resolutions are connected by flop relation. It remains to show X Ξ * ≃ Hilb G (C 4 ), and the total number of ✸ c s is equal to r(r+1)(r+2) 6 . As in the proof of Theorem 3.1, we first construct a regular morphism λ from X Ξ * to Hilb G (C 4 ) by examining I(y) for y ∈ X Ξ * in terms of toric coordinates. For R ∈ Ξ * (3), we denote x R := orb(R) ∈ X Ξ * . For the simplicity of notions, we again work on some special 3-simplices as the model cases, whose argument can equally be applied to all elements in Ξ * (3). We consider the 3-simplices of X Ξ * contained in the first three polytopes in (23), and they are: ∆ u , ∆ d of (23) and the eight simplices of (24) with c = v m + e 2 +e 4 −e 1 −e 3

2(r+1)
. The affine toric coordinates for X Ξ * are determined by the integral basis of M in the simplicial cone dual to the one in N generated by the corresponding 3-simplex. By computation, the affine coordinate systems corresponding to these 3-simplices are as follows: Here the indices i, j, k, s indicate the four 3 by permuting 1, 2, 3, 4 , and we shall adopt this convention for the rest of this proof if no confusion will arise. Define the following eigen-polynomials of G for β ∈ C and integers l with 0 ≤ l ≤ (r + 1), Let y be an element of X Ξ * . For y near x ∆u with coordinates (V , the ideal I(y) has the generators, , I(y) has the generators: il = u l ) 1≤l≤4 , I(y) has the generators: For y near x C ′c i with the coordinates (U ′(c) il = u ′ l ) 1≤l≤4 , I(y) has the generators: (28) The centers of the above affine charts have the monomial ideals , say the one near x ∆u , I(x ∆u ) is obtained by setting v l = 0 in (25), hence an monomial ideal. There are exactly (r + 1) 3 monomials not in I(x ∆u ), i.e., |I(x ∆u ) † | = (r + 1) 3 . For y near x ∆u , by using (25) and employing the Gröbner basis techniques and the toric data, one obtains the colength of I(y) in C[Z] satisfying the relation, colength(I(y)) ≤ colength(I(x ∆u )) = (r + 1) 3 ; this implies colength(I(y)) = (r + 1) 3 . By which it determines an element λ(y) ∈ Hilb G (C 4 ). One can also show the colength of I(y) equal to (r + 1) 3 for y in other affine charts using (26) (27) (28). The same conclusion holds for y in any affine coordinate neighborhood centered at x R for R ∈ Ξ * (3), and one obtains an element λ(y) in Hilb G (C 4 ), by which the morphism λ : X Ξ * −→ Hilb G (C 4 ) is defined. Now we are going to show that λ is an isomorphism. For n ∈ Z, we denote n the unique integer satisfying the relation, n ≡ n (mod r + 1) , 0 ≤ n ≤ r .
We first determine the G-invariant monomial ideals J 0 in Hilb Ar(4) (C 4 ). For a such J 0 , the set J † 0 := W 0 \ (W 0 ∩ J 0 ) forms a basis of a G-regular representation space. Denote l i the smallest integer with Z l i i ∈ J 0 ; l ij the smallest one with (Z i Z j ) l ij ∈ J 0 for i = j, and so on. By 1 ∈ J 0 , and i.e. I(o) ⊂ J 0 , and the following relations hold, 1 ≤ l ijk ≤ l ij ≤ l i ≤ r + 1 .
By J ⊥ 0 ⊂ I(o) ⊥ , and (9) for the description of the G-eigenspace of I(o) ⊥ , (Z j Z k Z s ) r+1−l i is the only monomial u ∈ I(o) † with u ∼ Z l i i , which implies (Z j Z k Z s ) r+1−l i ∈ J † 0 and (Z j Z k Z s ) r+2−l i ∈ J 0 , hence l jks = r + 2 − l i . By a similar argument, one has l ks = r + 2 − l ij . Hence we have l i + l jks = l ij + l ks = r + 2 .
Therefore γ 1 = γ 12 γ 134 and γ 12 = γ 123 γ 124 . Set v ′ where 4 j=1 z j = 0. The irreducible representation of G on C 3 (= V ), denoted by 3, has the following matrix forms for generators of G, , and 1 * is the G-character determined only by the (123)-value * . Using the coordinates (z i ) 3 i=1 of C 3 , the generators of G-invariant polynomials in C[Z] are: Note that the above variables are related to s 2 , s 3 , s 4 , d in (11) n=3 by the relations, Y 1 = −8s 2 , Y 2 = −8s 3 , Y 3 = 16s 2 2 − 64s 4 , X = 64d. The G-invariant polynomial relation (11) with F 3 in (12) becomes Lemma 5.1 Among m k s (1 ≤ k ≤ 10), the following tree diagram holds: Note that the J 0 's presented in (38) are characterized as the ideals in Hilb G (C 3 ) with monomial polynomial generators in C[Z]. All the above four elements lie over o ∈ S G under the morphism σ Hilb of (4). By the analysis in §2.5 of [6], σ −1 Hilb (o) consists of a tree of three smooth rational curves, L + l + L ′ . Here are the locations of J 0 s in σ −1 Hilb (o): Fig. 6). We are going to show that every J in Hilb G (C 3 ) can be deformed to . Therefore h(J) is a homogeneous ideal in Hilb G (C 3 ). Note that σ Hilb (h(J)) = o. By (2.4) in [6], h(J) ∈ {x 0 , x ′ 0 } ∪ l. Hence h(J) and J can be deformed to an element in (38). Now we are going to determine the local structure near these four central elements in Hilb G (C 3 ).
For J near the element x ∞ in Hilb G (C 3 ), we have