2000 Mathematics Subject Classification: 14C05, 14L30, 14E15, 14J35.When the cyclic group G of order 15 acts with some specific weights on affine four-dimensional space, the G-Hilbert scheme is a crepant
resolution of the quotient A^4 / G. We give an explicit description of this resolution using G-graphs

Sebestean, Magda

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Serdica Math. J. 30 (2004), 283–292A SMOOTH FOUR-DIMENSIONAL G-HILBERT SCHEMEMagda SebesteanCommunicated by P. PragaczAbstract. When the cyclic group G of order 15 acts with some specificweights on affine four-dimensional space, the G-Hilbert scheme is a crepantresolution of the quotient A4/G. We give an explicit description of thisresolution using G-graphs.1. Introduction. Let G be a finite group acting faithfully on a quasi-projective smooth scheme X. Consider the quotient X/G, the space of orbits,which is in general a singular scheme. We use a variant of the quotient, toresolve the singularities: we consider not only the set of orbits, but also a wholecollection of zero-dimensional G-invariant subsets of X with the associated ringof global sections of the structure sheaf, isomorphic to the regular representationring of the group G. This is formalized in the notion of a G-Hilbert scheme,introduced in [9]. Apart from the symplectic case treated in [1], the G-Hilbertscheme of a quasi-projective variety X is, in general, a “very” singular variety,especially in higher dimensions. The known cases of smooth G-Hilbert schemesare the minimal resolutions of Klein singularities and the crepant resolutions2000 Mathematics Subject Classification: 14C05, 14L30, 14E15, 14J35.Key words: quotient singularities, crepant resolutions, toric varieties, G-Hilbert scheme,G-graph.284 Magda Sebesteanof the quotient C3/G, with G being a finite subgroup of SL3(C). The three-dimensional case is due to Craw, Ito, Markouchevitch and Roan by a case-by-caseanalysis, and to Bridgeland-King-Reid as a consequence of a more general result(see [2]). Unfortunately, none of the current methods gives an answer to thefollowing question: which finite subgroups of GLn(C), n ≥ 4, admit G-HilbAn asa crepant resolution of the quotient An/G? The only attempt is given in [6], andit is reduced to the two-dimensional case.In order to give some positive answer to the above question, one needs abetter description of the G-Hilbert scheme. Suppose – from now on – that G isa finite abelian subgroup of GLn(C), consisting of diagonal matrices. Then thereis a toric approach to the question given by Nakamura (see [10]).We now introduce some notation needed in the present paper. We recallthat the affine space An is a toric variety associated to the lattice L := Zn andthe cone γ := 〈e1, . . . , en〉, where e1, . . . , en is the standard basis of Rn (cf. [8],§2). Suppose that G acts faithfully on An. Let r denote the order of G. We writeeach (diagonal) matrix g of G in the form g = diag(εa1 , . . . , εan), with ε := e2piir— a fixed rth primitive root of unity. We associate to such a matrix g a vectorvg :=1r(a1, . . . , an) ∈ Qn, and define the lattice:N := L+∑g∈GZvg.We denote by N∨ = HomZ(N,Z) its dual. Then, the quotient An/G is a toricvariety with lattice N and fan reduced to the cone γ. We denote by {f1, . . . , fn}the dual basis of {e1, . . . , en} and by M0 the additive semi-group generated by 0and the fi’s. We define a semi-group homomorphism from M0 to the semi-groupM of all monomials in n variables (endowed with multiplication), by sending fito Xi. In this way, we identify a vector with n non-negative integer coordinateswith a monomial. For a monomial p of C[X1, . . . ,Xn] (or “Laurent monomial” pof C[X1, . . . ,Xn][X−11 , . . . ,X−1n ]), we denote by v(p) the associated vector. LetG∨ be the set of all irreducible characters of the group G. We introduce thefollowing definition:Definition 1.1. Given χ ∈ G∨ and p = Xi11 . . . Xinn a monomial ofC[X1, . . . ,Xn] (or a Laurent monomial), we say that p and χ are associated ifwe have:g · p = χ(g)p, ∀g ∈ G .A smooth four-dimensional G-Hilbert scheme 285Definition 1.2. A subset Γ of M is called a G−graph (cf. [10], Def.1.4) if the following conditions hold:(1) it contains the constant monomial 1;(2) if p is in Γ and a monomial q divides p, then q is also in Γ;(3) the map wt : Γ → G∨ sending a monomial to its associated character (asin Definition 1.1), is a bijection.Denote by Graph(G) the set of all G-graphs.By condition (3) in Definition 1.2, there is a unique monomial of Γ asso-ciated to any character of G∨. Thus, we define a map wtΓ : M → Γ, by sendinga monomial to the unique element of Γ with the same associated character.Definition 1.3. Given a monomial p ofM , we call the fraction p/wtΓ(p) ∈C(X1, . . . ,Xn) the ratio of p with respect to Γ.Finally, we associate to a G-graph Γ, a cone, a semi-group and an idealas follows. We denote by 〈, 〉 the scalar product in Rn. We define a cone in Rnby σ(Γ) := {u ∈ Rn/〈u, v(p/wtΓ(p))〉 ≥ 0,∀p ∈ M}. The dual of a cone σ(Γ)is given by σ∨(Γ) := {v ∈ Rn/〈u, v〉 ≥ 0,∀u ∈ σ(Γ)}. We denote by S(Γ) thesub-semi-group of N∨ generated by vectors v(p/wtΓ(p)), where p runs over theset M . We call I(Γ) the ideal of C[X1, . . . ,Xn] generated by all the monomialsof M that are not in Γ.µ15-HilbC4 scheme and its properties. This section deals withan explicit description of the µ15-HilbC4 scheme. In the sequel, we denote byG = µ15 the cyclic group of order 15, with generator ε := e2pii15 . Let this groupact by weights 1, 2, 4 and 8 on the affine space A4. We identify G with the finiteabelian subgroup of SL4(C) with generator g = diag(ε, ε2, ε4, ε8) and the actionof G on A4 with the natural action of multiplication of an element of A4 by amatrix.The quotient A4/G is a Gorenstein canonical singularity (cf. [12]) andhas only one isolated singularity at the origin. With the notations of Section1, it is a toric variety with lattice N = Z4 + 115 (1, 2, 4, 8)Z and fan the coneγ = 〈e1, e2, e3, e4〉. In order to resolve the singularities, we will provide a simplicialdecomposition of γ into sub-cones such that the resulting variety is G-HilbC4. Weprove that G-HilbC4 is smooth and it has the crepancy property. Here, crepancymeans that the canonical sheaf ωG-HilbC4 and the structure sheaf OG-HilbC4 areisomorphic. We remark that the methods of [2] can’t be applied in this case286 Magda Sebestean— the condition on the fiber product of Theorem 1.1 of the cited paper is notsatisfied. Using Definition 5.2 of [7] and Watanabe’s classification Theorem (seeTheorem 5.3 of the same paper), we see that the group G above doesn’t give riseto a complete intersection singularity. In particular, the techniques of [6] are notapplicable. We are not in the symplectic case, so [1] does not apply either.In what follows, we set x = X1, y = X2, z = X3 and t = X4.Definition 2.1. Given a monomial p = xαyβzγtδ (or a Laurent mono-mial), the weight of p with respect to the group G above is the unique integerw(p) ∈ {0, . . . , 14}, that satisfies α+ 2β + 4γ + 8δ ≡ w(p) (mod 15).Remark 2.2. We recall that an irreducible character χi of G, withi ∈ {0, . . . , 14}, is given by g 7→ εi. So, a monomial p = xαyβzγtδ is associatedto the character χi if and only if its weight is i. Thus, a G-graph is a set of 15monomials of weights 0 to 14, satisfying (1) and (2) of Definition 1.2.Definition 2.3. Given S a finite set of monomials in C[x, y, z, t] and sone of the variables, we set max.p.(s) to be −∞ if S doesn’t contain any mono-mial in s and max{l ∈ N/sl ∈ S} otherwise. We call it the maximal powerassociated to the variable s in the set S.Lemma 2.4. The G-graphs for the group G = µ15 are the following:No. µ15-graph1 (1, x, y, y2, . . . , y7, xy, xy2, . . . , xy6)2 (1, x, y, y2, y3, t, xy, xy2, xy3, xt, yt, y2t, y3t, xyt, xy2t)3 (1, x, y, z, t, xy, xz, xt, yz, yt, zt, xyz, xyt, xzt, yzt)4 (1, x, y, z, z2, z3, xy, xz, xz2, xz3, yz, yz2, yz3, xyz, xyz2)5 (1, x, x2, x3, z, t, xz, x2z, x3z, xt, x2t, x3t, zt, xzt, x2zt)6 (1, x, x2, x3, z, z2, z3, xz, x2z, x3z, xz2, x2z2, x2z3, x3z, x3z2)7 (1, x, x2, . . . , x7, t, xt, x2t, . . . , x6t)8 (1, x, x2, . . . , x14)9 (1, y, y2, . . . , y14)10 (1, y, y2, y3, t, t2, t3, yt, yt2, yt3, y2t, y2t2, y2t3, y3t, y3t2)11 (1, y, z, z2, . . . , z7, yz, yz2, . . . , yz6)12 (1, y, z, t, t2, t3, yz, y2z, y3zyt, yt2, yt3, zt, yzt, y2zt)13 (1, z, t, t2, . . . , t7, zt, zt2, . . . , zt6)14 (1, z, z2, . . . , z14)15 (1, t, t2, . . . , t14)Table 1. List of G-graphs for G = µ15.A smooth four-dimensional G-Hilbert scheme 287P r o o f. The idea of the proof is to consider, for a given G-graph, thepossible choices of max.p.(s), where s is one of the variables x, y, z or t. By (1) inDefinition 1.2, the constant monomial 1 is in the G-graph. In particular, for anyvariable s we have max.p.(s) ≥ 0. By condition (2) of Definition 1.2 and Remark2.2, max.p.(s) is also less than 15 for any variable s.Now, for any G-graph Γ, there are two possible cases: either max.p.(x) =0 – that is x doesn’t occur in Γ, or max.p.(x) > 0 – meaning that Γ contains atleast one monomial in x.We discuss now the case where x occurs in a G-graph Γ. We claim thatin this case max.p.(x) cannot be 2, 4, 5, 6, 8, 9, 10, 11, 12, nor 13. To see this,we argue by contradiction. We suppose that there exists a G-graph Γ such thatwe have max.p.(x) = 2. Then, using (2) of Definition 1.2 and Remark 2.2, we seethat y, z4, t2 cannot occur in any monomial of Γ. In particular, max.p.(y) = 0,max.p.(z) ≤ 3 and max.p.(t) ≤ 1. On the other hand, there has to be a monomialof weight 3 in Γ and the only possible choice is x2z2t. We use (2) of Definition1.2 to deduce that x2z2 and x2t should both belong to Γ. But these monomialsboth have the same weight, which contradicts Remark 2.2 and thus the definitionof a G-graph. Now, if max.p.(x) were 4, 5 or 6, we see in a similar way that thereis no possible choice of monomial of weight 7. When max.p.(x) is one of 8, 9,10, 11, 12 or 13, there are no monomials divisible by y, z or t. This is becausethe monomials x2, x4 and x8 are already in Γ and they have the same weights asy, z and t. But then the G-graph Γ contains less then 15 monomials, which is acontradiction.We discuss now the case max.p.(x) = 1. With the notations of Definition2.1, we have w(y8) = w(z4) = w(t2) = 1. We get max.p.(y) ≤ 7,max.p.(z) ≤ 3and max.p.(t) ≤ 1. A similar discussion as above proves that max.p.(y) equals 1,3 or 7. Now, if max.p.(y) = 7, we get the G-graph Γ1 of Table 1. If max.p.(y) =3, there are no monomials in z and we obtain Γ2. For max.p.(y) = 1, eithermax.p.(z) = 1 and we get Γ3, or max.p.(z) = 3 and no monomial in t is allowed– we get Γ4.For the case max.p.(x) = 3, the G-graph contains no monomial in y andhas max.p.(z) ≤ 3 and max.p.(t) ≤ 1. We obtain the G-graphs Γ5 and Γ6 ofTable 1.The case max.p.(x) = 7 gives max.p.(y) = 0, max.p.(z) = 0, and max.p.(t)≤ 1. The only possible G-graph is Γ7, a “planar” G-graph in x and t.For the last case, i.e. max.p.(x) = 14, we get a “linear” G-graph contain-ing only monomials in the variable x – this is the G-graph Γ8.288 Magda SebesteanNow, if x does not occur in the G-graph, we prove, by a similar argument,that max.p.(y) is one of 1, 3 or 14. We obtain the G-graphs Γ9, Γ10, Γ11, Γ12. Ifneither x, nor y occur, then we get again a “planar” G-graph in z and t — thisis Γ13 and two “linear” graphs – Γ14 only in z and Γ15 only in t. Let Γ be a G-graph as in Table 1. We associate to it a finite set, denotedE(Γ), as follows. If the variable x occurs in Γ, let the vector vg associated to thematrix g be in E(Γ), otherwise include in E(Γ) the vector e1 of the canonicalbase. Similarly, if y is in Γ we take the vector associated to the matrix g8 inE(Γ) and otherwise include the vector e2. For the variable z, we take either thevector associated to the matrix g4 or the vector e3. Finally, if t is in Γ let thevector associated to the matrix g2 be in E(Γ), otherwise include the vector e4.For example, the set E(Γ8) is{115(1, 2, 4, 8), e2 , e3, e4}.Lemma 2.5. With the notations at the end of section 1, the cones σ(Γ)are generated by the sets E(Γ).P r o o f. We want to prove that any vector u of a cone σ(Γ) is a linearcombination with positive coefficients of elements of the associated E(Γ). Equiv-alently, the system:(2.1)∑v∈E(Γ)xv · v = u,has to admit a solution with xv positive for any v in E(Γ). For this, we notethat each xv is nothing but the scalar product of u and the vector associatedto the ratio of a monomial generator of I(Γ) (see section 1 for definitions andnotations). We use the definition of σ(Γ) to conclude. Example 2.6. For a more explicit approach, let us actually see whathappens for the G-graph Γ8. The set E(Γ8) is given by the vectors e2, e3, e4and the vector associated to the matrix g – this is vg =115(1, 2, 4, 8). Let u =(u1, u2, u3, u4) be a vector of σ(Γ8). The solutions of the system (2.1) are:xvg = 〈u, 15e1〉,xe2 = 15〈u, e2 − 2e1〉,(2.2)xe3 = 15〈u, e3 − 4e1〉,xe4 = 15〈u, e4 − 8e1〉.A smooth four-dimensional G-Hilbert scheme 289The monomial generators of I(Γ8) are x15, y, z, t. The vectors associated to theratios of those monomial generators are 15e1, e2 − 2e1, e3 − 4e1 and e4 − 8e1.By definition of σ(Γ8), the numbers 〈u, 15e1〉, 〈u, e2 − 2e1〉, 〈u, e3 − 4e1〉 and〈u, e4 − 8e1〉 are positive. Thus the solutions given by (2.2) are also positivenumbers, as wanted. Corollary 2.7. The toric variety obtained by gluing together all the affinepieces Spec[σ∨(Γ) ∩N∨], where Γ runs over Graph(G), is a smooth variety.P r o o f. It is enough to see that every such affine piece is a copy of C4.For this, we use the “smoothness criterion” of [11], Theorem 1.10, page 10. Nowthe result follows because each cone σ(Γ) is generated by the sets E(Γ) of theLemma 2.5 which are a [part of a] basis for the lattice N . Lemma 2.8. With the notations of Corollary 2.7, for any G-graph, wehave S(Γ) = σ∨(Γ) ∩N∨.P r o o f. By definitions of σ(Γ) and S(Γ), the inclusion S(Γ) ⊂ σ∨(Γ)∩N∨follows immediately. Now, for the reverse inclusion, it is enough to prove thatσ∨(Γ)∩N∨ is generated by a set of vectors contained in S(Γ). For this, we proceedas follows. We call a monomial pure if it involves only one of the variables x, y, z ort. We denote by vΓ(s) the vectors associated to the ratios of the pure monomialss that generate I(Γ).We claim that any vector v in σ∨(Γ)∩N∨ is a linear combination with non-negative integer coefficients of the vectors vΓ(s) above. A case-by-case analysisof each I(Γ) shows that this is equivalent to resolving a 4× 4 system. The proofis very similar to that of Lemma 2.5 and the calculations made in the subsequentExample 2.6, so we omit it. Theorem 2.9. Let the cyclic group of order 15 act by weights 1, 2, 4,8 on the affine four dimensional space. Then, the µ15-Hilbert scheme of A4 isa smooth variety and it provides a crepant resolution of the Gorenstein quotientsingularity A4/µ15.P r o o f. We use (iii) of Theorem 2.11 of [10] to see that µ15-HilbA4 isthe variety obtained by gluing together all the SpecC[S(Γ)], when Γ runs overGraph(µ15). By Lemma 2.8, this is the same as the toric variety whose fan isgiven by the cones σ(Γ),Γ ∈ Graph(µ15). We apply Corollary 2.7 to concludethat µ15-HilbA4 is smooth. In particular, the µ15-Hilbert scheme of A4 providesa toric resolution of the quotient A4/µ15, by subdivision of the cone γ into thesub-cones σ(Γ),Γ ∈ Graph(µ15).290 Magda SebesteanNow, for crepancy, we use the equivalence stated in [3], page 656. Forthis, we see that the Euler number of µ15-HilbA4 is given by the number of three-dimensional cones of the associated fan. In this case, this is #Graph(µ15) =#G = 15. Together with smoothness, this gives crepancy. Remark 2.10. Equivalently, we could use Theorem 4.1 of [7] to provethe crepancy of the desingularization defined by µ15-HilbA4. This is because thefirst skeleton of any cone σ(Γ), for Γ a µ15-graph, is formed only by elements inthe simplex ∆4 := {(u1, u2, u3, u4) ∈ R4/u1 + u2 + u3 + u4 = 1}, as proved inLemma 2.5.3. Miscellaneous remarks.Remark 3.1. The H-Hilbert scheme of A4 for H the cyclic group oforder 40 acting by weights 1, 3, 9, 27 on A4 fails to provide a smooth crepantresolution for the quotient singularity A4/H. This is mainly because the numberof H-graphs in this case is not equal to the cardinal of the group.Remark 3.2. It is interesting to note that in the situation described inSection 2, the McKay correspondence as stated by Reid holds (see [4] page x fora statement of the conjecture). The Betti numbers bl of a crepant resolution (inthis case µ15-HilbA4) are the cardinalities of the conjugacy classes of elements ofsame “age” l in the group µ15.Remark 3.3. (cf. with [10] “deformation” and [5] “ratio”). We canrecover all the µ15-graphs from a given one, by using ratios of the generatingmonomials of the associated ideal, as follows. We take for example the µ15-graphΓ3 in Table 1. The associated ideal is generated by the monomials x2, y2, z2, t2,xyzt. The corresponding ratios are x2/y, y2/z, z2/t, t2/x, xyzt/1. Let us takeone of those, say z2/t.By repeatedly replacing every occurrence of t in Γ3 by z2, we recover theµ15-graph Γ4. We do the same with the other ratios (note that the ratio xyzt/1provides no µ15-graph), to recover Γ2, Γ5 and Γ12. By repeating the procedure,we get all remaining µ15-graphs of Table 1. The result is given in the figure below.In the Figure 1, the direction of the arrows can be reverted, as follows.We take for example the “bottom” µ15-graph Γ9. The pure monomial x of I(Γ9)has as associated ratio the fraction x/y8. Replacing in Γ9 the monomial y8 andall its occurrences by x, we obtain the µ15-graph Γ1.A smooth four-dimensional G-Hilbert scheme 291Figure 1. Deforming G-graphs for G = µ15.REFERENCES[1] R. Bezrukavnikov, D. Kaledin.McKay equivalence for symplectic res-olutions of singularities. E-print math. AG/0401002.[2] T. Bridgeland, A. King, M. Reid. The McKay correspondence as anequivalence of derived categories. J. Amer. Math. Soc. 14, 3 (2001), 535–554.[3] L. Chiang, S. S. Roan. Orbifolds and finite group representations. Int.J. Math. Math. Sci. 26, 11 (2001), 649–669.[4] A. Craw. The McKay correspondence and representations of the McKayquiver. University of Warwick Ph. D thesis, June 2001.[5] A. Craw, M. Reid. How to calculate A-Hilb C3. Se´minaires et Congre`s6 (2002), 129–154, E-print math. AG/9909085.[6] D. Dais, C. Haase, G. Ziegler. All toric local complete intersectionsingularities admit projective crepant resolutions. Toˆhoku Math. J. (2)53, 1 (2001), 95–107.292 Magda Sebestean[7] D. Dais, M. Henk, G. Ziegler. All abelian quotient C.I.-singularitiesadmit projective crepant resolutions in all dimensions. Adv. Math. 139, 2(1998), 194–239.[8] W. Fulton. Introduction to toric varieties. Annals of Mathematics Stud-ies, vol. 131, Princeton University Press, Princeton, NJ, 1993.[9] Y. Ito, I. Nakamura.McKay correspondence and Hilbert schemes. Proc.Japan Acad. 72A (1996), 135–138.[10] I. Nakamura.Hilbert schemes of abelian group orbits. J. Algebraic Geom.10, 4 (2001), 757–779.[11] T. Oda. Convex Bodies and Algebraic Geometry. Springer-Verlag, Berlin,1988.[12] M. Reid. Young Person’s Guide to Canonical Singularities. Algebraicgeometry, Bowdoin, 1985. In: Proc. Sympos. Pure Math., vol. 46, Part1, Amer. Math. Soc., Providence, RI, 1987, 345–414,E´quipe des Groupes FinisUniversite´ Paris 7 Denis DiderotInstitut de Mathe´matiques de Jussieu2, place Jussieu F-75251, Paris, Francee-mail: sebes@math.jussieu.frReceived March 30, 2004Revised May 28, 2004

A Smooth Four-Dimensional G-Hilbert Scheme

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A Smooth Four-Dimensional G-Hilbert Scheme

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