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The behavior of the Hilbert scheme of points under the derived McKay correspondence Mohyedin Kermani, Ehsan 2013

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The behavior of the Hilbert scheme ofpoints under the derived McKaycorrespondencebyEhsan Mohyedin KermaniB.Sc., Sharif University of Technology, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2013c? Ehsan Mohyedin Kermani 2013AbstractIn this thesis, we completely determine the image of structure sheaves ofzero-dimensional, torus invariant, closed subschemes on the minimal, crepantresolution Y of the Kleinian quotient singularity X = C2/Z/n, under theFourier-Mukai equivalence of categories, between derived category of coher-ent sheaves on Y and Z/n-equivariant derived category of coherent sheaveson C2. As a consequence, we obtain a combinatorial correspondence betweenpartitions and Z/n-colored skew partitions.iiPrefaceAll the research ideas and methods explained in this thesis are the results offruitful discussions between Professor Jim Bryan and Ehsan Mohyedin Ker-mani. The computations and the manuscript preparation were conducedby Ehsan Mohyedin Kermani with invaluable guidance from Jim Bryanthroughout this process.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Finite subgroups of SL(2,C) . . . . . . . . . . . . . . . . . . . 32.2 Kleinian singularities . . . . . . . . . . . . . . . . . . . . . . . 52.3 Minimal, crepant resolution and resolution graph . . . . . . . 72.4 Toric geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.1 Toric varieties as quotients . . . . . . . . . . . . . . . 102.4.2 Resolution of singularities . . . . . . . . . . . . . . . . 102.5 The classical McKay correspondence . . . . . . . . . . . . . . 112.6 On derived category . . . . . . . . . . . . . . . . . . . . . . . 142.7 Hilbert scheme of points . . . . . . . . . . . . . . . . . . . . . 152.8 Main question . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 20ivTable of Contents3.1 Coordinates on the universal closed subscheme of the Hilbertscheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.1 Statement of the main result . . . . . . . . . . . . . . 283.2.2 Computational technique and proof of the result . . . 293.2.3 Combinatorial corollary and illustrations . . . . . . . . 314 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39vList of Tables2.1 Equations of Kleinian singularities . . . . . . . . . . . . . . . 6viList of Figures2.1 ADE Dynkin diagrams . . . . . . . . . . . . . . . . . . . . . . 83.1 Ferrer diagram for m = 5 and ? = (3, 2) . . . . . . . . . . . . 283.2 Ferrer diagram of C[ui, vi] . . . . . . . . . . . . . . . . . . . . 313.3 Ferrer diagram of Mi . . . . . . . . . . . . . . . . . . . . . . . 323.4 Image of the origin . . . . . . . . . . . . . . . . . . . . . . . . 323.5 Attachment possibilities . . . . . . . . . . . . . . . . . . . . . 333.6 Attaching rules . . . . . . . . . . . . . . . . . . . . . . . . . . 333.7 Ferrer diagram of the origin for n = 4, i = 2 and m = 5 with? = (3, 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.8 Ferrer diagram of C[u2, v2]/(u22, u2v22, v32) . . . . . . . . . . . . 343.9 Image of C[u2, v2]/(u22, u2v22, v32) . . . . . . . . . . . . . . . . . 353.10 Z/4-coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.11 Z/4-colored skew partition . . . . . . . . . . . . . . . . . . . . 36viiAcknowledgmentsFirst and foremost, I would like to thank my supervisor Professor Jim Bryanfor all of his help, patience and financial support throughout my study atthe University of British Columbia (UBC). I am indebted to him for hisguidance and invaluable suggestions in writing this thesis. I am grateful toProfessor Kalle Karu for his time in evaluating my thesis and Professor JimCarrell who made it possible for me to join the Mathematics department.I would also like to express wholeheartedly my sincere appreciation to myvery supportive family, my mother ?Mehrzad?, my father ?Hamid? and mysister ?Afsoon? for all of their love and endless encouragement throughoutmy life.At last but not least, my deepest appreciation is for my beloved wife?Shadi? for her continuous love and support.viiiChapter 1IntroductionIn 1980, McKay [14] found an interesting link between representation theoryand algebraic geometry as a bijection between the set of non-trivial irre-ducible representations of a finite group G of SL(2,C) and the componentsof the exceptional divisor of the minimal resolution Y of the Kleinian singu-larity X = C2/G. In particular, he showed that the McKay quiver 2.5 coin-cides with resolution graph 2.3 of Y. Later on, in 1983, Gonzalez-Sprinberg-Verdier [8] formulated McKay?s result in terms of K-theory. That is, theygeometrically constructed an isomorphism between K(Y ) and KG(C2) ?=Rep(G) where KG(C2) is the G-equivariant K-theory on C2 and Rep(G) isthe representation ring of G. Via this isomorphism a non-trivial irreduciblerepresentation corresponds to the structure sheaf of an exceptional com-ponent 2.5. Using the Chern character isomorphism K(Y ) ?= H?(Y ), onerecovers the classical McKay correspondence and it is often described as thefollowing bijectionGeometric basis of H?(Y,Z)?? Set of irreducible representations of G.Finally, in 2001, Bridgeland-King-Reid [1] promoted the above corre-spondence to an equivalence of derived categoriesD(Y )??? DG(C2).The above equivalence is then given by pulling back and pushing forward toand from the universal object and this is called the Fourier-Mukai transform.A key point in Bridgeland-King-Reid is to realize the resolution Y ?? Xas a moduli space, namely Y = GHilb(C2), G-Hilbert scheme which will be1Chapter 1. Introductiondescribed in 2.7.The main question we would like to answer is to completely classify theimage of the Hilbert scheme of points of the minimal, crepant resolutionY under the Fourier-Mukai transform. In other words, we want to investi-gate where the structure sheaves of zero-dimensional closed subschemes ofY (viewed as objects in D(Y )) go under the equivalence. It has been ex-tensively studied where certain sheaves go under Fourier-Mukai equivalencebut always in the other direction [12], i.e. it has been computed where cer-tain sheaves go on the orbifold. Our main result is a complete and explicitdescription of the image of the structure sheaves of torus invariant, closedsubschemes of Y in the case where G ?= Z/n.We find that the structure sheaves of points on Y do go to sheaves onC2 (as opposed to more general objects in DZ/n(C2)), but not necessarilythe structure sheave of subschemes. Instead, they all go to quotients of acertain universal quasi-coherent sheaf which will be constructed in section3.2.Our construction induces as interesting combinatorial correspondencebetween partitions and certain Z/n-colored skew partition. We illustratethis correspondence with examples and diagrams in 3.2.3.2Chapter 2Background2.1 Finite subgroups of SL(2,C)The following is a concise account of the classification of finite subgroupsof SL(2,C). Throughout, we will assume that G is a finite subgroup ofSL(2,C).By taking the standard Hermitian inner product on C2 defined by ?z, w? =z ? w and averaging by G, we arrive at a G-invariant Hermitian inner prod-uct on C2. This implies that G is conjugate to a finite subgroup of SU(2,C).In fact, this argument shows that the classification of finite subgroups ofSL(2,C) is equivalent to the classification of finite subgroups of SU(2,C).We define a surjective group homomorphism SU(2,C)?? SO(3,R) usingthe algebra of quaternions H as follows:Let q = a+bi+cj+dk ? H which can be written in the form q = z1+z2j,where z1 = a + bi and z2 = c + di are complex numbers. Let H1 be thegroup of quaternions with norm 1. There is a natural group isomorphism? : H1 ?? SU(2,C), z1 + z2j 7?(z1 z2?z2 z1)Then we identify R3 with pure quaternions bi + cj + dk and define theaction of H1 ?= SU(2) on R3 byq ? q0 7? q ? q0 ? q?1, q ? H1, q0 ? R3now because the quaternion norm is multiplicative and coincides with Eu-clidean norm on R3, thus p : H1 ? O(3) is a well-defined group homomor-phism.32.1. Finite subgroups of SL(2,C)Moreover, write q ? H1 as q = cos ?+ sin ?q1, where q1 is a pure quater-nion of norm 1. It is also immediate that the action of q on R3 is the rotationdefined by the axis q1 and the angle ?. Using the definition of SO(3,R), wecan define a surjective homomorphism p : H1 ? SO(3,R). The kernel of pis the center of SU(2,C) which is {?1}.Hence there is a short exact sequence of groups1 {?1} SU(2) SO(3)H1 O(3)1?pThe classification of finite subgroups of SU(2,C) is now equivalent tothe classification of finite subgroups of SO(3,R), since any finite subgroup Gof SU(2,C) is mapped to a finite subgroup of G? of SO(3,R), and converselyany finite subgroup G? of SO(3,R) can be lifted to a finite subgroup Gof SU(2,C) by the above diagram. Classification of finite subgroups ofSO(3,R) is very well-known and consists of the three families of groups,namely, the symmetries of a regular polyhedron (tetrahedral of order 12,octahedral of order 24 and icosahedral of order 60), the dihedral groups (oforder 2n) and the cyclic groups (of order n.)By lifting the described subgroups we arrive at1. Cyclic subgroup Cn of order n.2. Binary dihedral group of order 4n,3. Binary tetrahedral group of order 24,4. Binary octahedral group of order 48,5. Binary icosahedral group or order 120.42.2. Kleinian singularities2.2 Kleinian singularitiesIn this thesis, we are interested in working with quotient varieties C2/G sowe should first define what we mean by C2/G and try to find its definingequations.Definition The quotient variety X = C2/G = SpecC[a, b]G is called aKleinian singularity (also known as Du Val singularity, a simple surfacesingularity or rational double point.)We will see that X can be embedded in C3 as a hypersurface with anisolated singularity at the origin, that is, there is only one defining equationin the ring of invariants for various subgroups.The action of SL(2,C) on C2 is defined by left multiplication whichinduces the same action on subgroups of SL(2,C)(c de f): (a, b) 7? (ca+ db, ea+ fb)The induced action on the coordinate ring C[a, b] of C2 is(c de f): p(a, b) 7? p(fa? db,?ea+ cb)and we are looking for the ring of invariants i.e. C[a, b]G ? C[a, b].As an example, when G = Z/n the ring of invariants C[a, b]Z/n can becomputed as followsLet n = e2pii/n, then Z/n is acting on C[a, b] by gn =(n 00 ?1n),gm ? (a, b) 7? (na, ?1n b)The monomials an, bn, ab can be taken as the generators of the ring of in-variants, therefore, C[a, b]Z/n = C[an, bn, ab]. It can be seen that the onlydefining equation in this case is xy ? zn = 0 where x = an, y = bn, z = ab.The actions of other cases are illustrated as follows52.2. Kleinian singularities1. BD4n is a binary dihedral group of order 4n generated by g2n =(2n 00 ?12n), h =(0 ii 0),g2n ? (x, y) 7? (2na, ?12n b), h ? (a, b) 7? (ib,?ia)Remark The group generated by g2n =(n 00 ?1n), h =(0 ii 0), isconjugate to the cyclic group of order n.Similarly for2. BT24 is a binary tetrahedral group of order 24 generated by g2n =(2n 00 ?12n), h =(0 ii 0), k = 11?i(1 i1 ?i)3. BO48 is a binary tetrahedral group of order 48 generated by g8 =(8 00 ?18), h =(0 ii 0), k = 11?i(1 i1 ?i)4. BI120 is a binary icosahedral group of order 120 generated by g10 =(10 00 ?110), h =(0 ii 0), l = 1?5(5 ? 45 25 ? 3525 ? 35 ?5 + 45).The complete account of finding the defining equations in these caseshave been fully illustrated in [3]. The following table contains the equationsof X for various groups up to conjugacy.Conjugacy class Equation Dynkin graphZ/n x2 + y2 + zn = 0 An?1BD4n x2 + y2z + zn+1 = 0 Dn+2BT24 x2 + y3 + z4 = 0 E6BO48 x2 + y3 + yz3 = 0 E7BI120 x2 + y3 + z5 = 0 E8Table 2.1: Equations of Kleinian singularitiesExplicit generator for the ring of invariants C[a, b]G can be found in [3,p.7-13] and the above relations can be checked accordingly.62.3. Minimal, crepant resolution and resolution graph2.3 Minimal, crepant resolution and resolutiongraphDefinition A minimal resolution Y of X is a resolution such that everyother resolution factors through Y.Remark Note that a surface has a unique minimal resolution, however, forhigher dimensional varieties, minimal resolutions are not necessarily unique.In order to construct a minimal resolution Y, we can find an arbitraryresolution for X and contract (?1)-curves to produce Y successively. Theexceptional locus of ? : Y ?? X consists of (?2)-curves Ei intersectingtransversely. If we associate a vertex to each curve Ei and join two vertices ifand only if the corresponding curves intersect in Y, we arrive at the resolutiongraph of X.It is now worth mentioning here that the McKay?s result [14] gives a nicecorrespondence between the resolution graph of a Kleinian singularity C2/Gand the Dynkin diagram of G. The Dynkin diagrams of An?1, Dn+2, E6, E7and E8 are listed below in Figure 2.1.Definition A crepant resolution Y ?? X is a resolution which the pullbackof the canonical divisor KX of X coincides with the canonical divisor KY ofY.Remark The minimal resolution of X = C2/G exists and is crepant due tothe fact that a 2-form f(a, b)da?db on C2/G is invariant under the SL(2,C)action.Remark 2.4 Toric geometryThe Complete account of Toric geometry can be found in [5].Throughout, V is an n-dimensional vector space over R. Recall that asubset C ? V ?= Rn is convex if given any two elements x1, x2 ? C, then?x1 + (1 ? ?)x2 ? C for any 0 ? ? ? 1 and C is a cone if given x ? C,72.4. Toric geometryAn?1 ? ? ?Dn+2 ? ? ?E6E7E8Figure 2.1: ADE Dynkin diagramsthen ax ? C for all a ? 0. If C ? Rn is a convex cone, then C ? Zn is asemigroup under addition. The convex cone generated by x1, ? ? ? , xk ? V isthe smallest convex cone containing x1, ? ? ? , xk and is defined and denotedby C = ?x1, ? ? ? , xk? = {a1x1 + a2x2 + ? ? ?+ akxk | ai ? 0}.Definition A convex cone is strongly convex if it does not contain a non-zero linear subspace and is simplicial if it can be generated by linearlyindependent vectors.Definition Let y? ? V ? be a linear functional on V. Then y is a supportof a cone C if y(x) ? 0 for all x ? C, and is written as y|C ? 0. ThenHy = {y = 0} is the support hyper plane of C and Hy ? C is called a faceof C. A one-dimensional face is called ray.Definition The dual cone C? of a cone C is the set of all y ? V ? withsupport of C.82.4. Toric geometryDefinition A cone C ? Rn is called rational if it can be generated byrational elements in Qn.Remark If C is strongly convex and rational, then it is generated the firstlattice points on its rays.Lemma 2.4.1 (Gordon). If C ? Rn is a rational cone then the semigroupS = C ? Zn is finitely generated. Consequently, the group ring C[S] is afinitely generated C-algebra.Let N ?= Zn and M = Hom(N,Z) ?= Zn. Suppose that ? ? NR := N ?R ?= Rn is a strongly rational cone. For ?? the corresponding semigroup ???M, the semigroup ring C[???M ] and the corresponding variety SpecC[???M ] are denoted by S?, A? and U?, respectively.Definition A fan ? in NR is a finite, non-empty set of rational stronglyconvex cones in NR, such that1. If ? ? ? and ? ? ? (i.e. ? is a face of ?) then ? ? ?.2. If ?, ? ? ? then ? ? ? ? ?.Definition The toric variety X(?) associated to the fan ? is????U?/{U?1 , U?2 glued along U?1??2}.Definition A morphism of cones is ? : (N1, ?1) ?? (N2, ?2) where ? :N1 ?? N2 is a group homomorphism and ??R : N1?R ?? N2?R maps?1 to ?2. The induced morphism ? : U?1 ?? U?2 is called a toric morphism.Definition A cone ? ? NR generated by ?v1, ? ? ? , vk? is non-singular ifv1, ? ? ? , vk can be extended to a basis of NR.Remark We will use the fact that if ? is a simplicial cone in NR anddim? = n = rkNR, then ? is non-singular if and only if det(v1, ? ? ? , vk) = ?1,and U? is non-singular if and only if ? is non-singular.92.4. Toric geometry2.4.1 Toric varieties as quotientsLet G be a finite group acting on Cn = SpecC[x1, ? ? ? , xn]. For example, letG = ?n? ?= Z/n acting on C2 byn.(x, y) 7? (nx, ?1n y)then X = C2/G = SpecC[x, y]G. Under the action, xayb is invariant ifand only if anxa?bn yb = xayb or a ? b (mod n). Let M ? ? M be thelattice of invariant monomials, and S?? = ? ?M?. Then C[S??] = C[S?]G andU ?? = SpecC[S??] = C2/G.Suppose that N ? = Hom(M ?,Z) and consider the following short exactsequence of abelian groups0 ??M ? ??M ??M/M ? ?? 0and apply the contravariant left exact functor Hom(?,Z) to get0 ?? N ?? N ? ?? Ext1(M ?/M,Z)that is,N ? N ? = {(?, ?)| ?(?, ?),m?? ? Z, ?m? ?M ?} = {(?, ?)|n? or ?+? or n? ? Z},since M ? is generated by (n, 0), (1, 1), (0, n).Theorem 2.4.2 (Fulton [5, p. 34]). Let (?,N) be a singular simplicial conegenerated by v1, ? ? ? , vn. Let N ? ? N be generated as Zv1 + ? ? ? + Zvn, then(?,N ?) is non-singular and in fact U ?? = An. Suppose G = N/N ?, thereforeG acts on An and U? = An/G.2.4.2 Resolution of singularitiesOne of the important features of working with toric varieties is that we caneasily find the explicit defining equations of a resolution of a singular toricvariety by looking at its singular fan.102.5. The classical McKay correspondenceLet X be a singular variety. Recall that a resolution Y for X is aproper, birational map f : Y ?? X where Y is a non-singular variety. Inthe toric case, let ? be a singular fan i.e. there exists a singular cone ? in?. Desingularization of ? is carried out by the so called subdivision process[5, p. 45]. IndeedTheorem 2.4.3 (Fulton [5, p. 48]). Every ? has a non-singular subdivision.In our previous example which we will use it later, the toric fan associatedto the minimal resolution of X consists of n+ 1 rays (then n fans) from theorigin to the points ( jn ,n?jn ) for 0 ? j ? n in C.Each chart Yi of Y (the resolution) corresponds to the fan Ci generatedby the lattice points ( in ,n?in ), (i?1n ,n?i+1n ), where 1 ? i ? n. Then theassociated toric variety isYi = SpecC[x?(n?i)yi, xn?i+1y?(i?1)].2.5 The classical McKay correspondenceLet G be a finite subgroup of SL(2,C), and let ? be the 2-dimensionalrepresentation of G induced by the inclusion G ? SL(2,C). Suppose that{?0, ?1, ? ? ? , ?k} be the set of irreducible representations of G, where ?0 isthe trivial representation. For any 0 ? j ? k, the representation ? ? ?jdecomposes into a direct sum of irreducibles as?? ?j =k?i=0aij?iwhere aij = dimC HomG(?i, ?? ?j).Definition The McKay quiver of G is a directed multi-graph with verticesindexed by irreducible representations and the vertex i is connected to j byexactly aij number of edges.Proposition 2.5.1 ([14]). Using representation theory, we have112.5. The classical McKay correspondence1. aij = aji.2. McKay quiver is a connected quiver.3. aii = 0.4. aij ? {0, 1}.In fact, by the above Proposition, we can view the McKay quiver of Gas a simple, undirected graph called McKay graph and denoted by ??G. Thesubgraph consisting of non-trivial irreducible representations is denoted by?G.Theorem 2.5.2 (McKay [14]). With the above notations, the McKay graph??G is an extended Dynkin graphs of A?D?E? type. Furthermore, the subgraph?G is one of the graphs An, Dn, E6, E7, E8 2.1 which each arises as the res-olution graph of C2/G.This result establishes a one-to-one correspondence between the primedivisors Di of the crepant resolution ? : Y ?? X = C2/G and the non-trivial irreducible representations of G ? SL(2,C).Knowing that the exceptional divisor classes [Di] define a basis forH2(Y,Z),then adding the class of a point, we arrive at a basis for H?(Y,Z). Corre-spondingly, adding the trivial representation, we obtainSet of irreducible representations of G?? Basis of H?(Y,Z).The first geometric interpretation of the McKay theorem at the K-theorylevel was given by Gonzalez-Sprinberg and Verdier in 1983 [8]. They asso-ciated the so called tautological vector bundles on Y to each irreduciblerepresentation of G, which will be described quickly as follows;Definition For an irreducible representation ?i : G ?? GL(Vi), let Midenote the OX -module defined byMi := HomC[G](Vi,C[a, b]).122.5. The classical McKay correspondenceBecause Mi is a G-invariant C[a, b]-module, the associated OX -moduleis a coherent sheaf on X.Definition Let Fi := ??Mi/TorsOY where TorsOY is the OY -torsions of??Mi, then the locally free sheaf Fi is called tautological bundle on Y asso-ciated to ?i.By case by case analysis, Gonzalez-Sprinberg and Verdier [8] proved thatthe non-trivial tautological vector bundles {Fi} satisfyc1(Fi) ? [Dj ] = deg(Fi|Dj ) = ?ij ,where c1(Fi) is the first Chern class of Fi and ?ij is the Kronecker deltasymbol.At the K-theory level, they proved that ? : KG(C2) ?? K(Y ) is anisomorphism of abelian groups, where KG(C2) is the Grothendieck ring ofG-equivariant coherent sheaves on C2, K(Y ) is the Grothendieck ring of Yand ? is defined by?(?) := (piY ?(M? pi?C2(?)))Gwhere piY , piC2 are projections of Y ? C2 to Y and C2 respectively, and Mis the structure sheaf of the reduced fiber product (Y ?C2/G C2)red.Moreover, they showed that for KGc (C2) the G-equivariant K-theorywith compact support on C2,KGc (C2) ?= Rep(G)where Rep(G) is the representation ring of G, and the isomorphism is givenby ?(O0??i) = [ODi ] where O0 is the skyscraper sheaf of the origin (0, 0) ?C2 and ?i is a non-trivial irreducible representation of G [2].Bridgeland-King-Reid [1] have expanded the geometric interpretation byenlarging the framework to the derived category of coherent sheaves.132.6. On derived category2.6 On derived categoryOne of the main motivations in constructing derived category for an abeliancategory, such as category of coherent sheaves on a scheme is that for amorphism of schemes ? : Y ? X we are interested in pulling back ??Fand pushing forward ??G of sheaves where F is a sheaf of OX -module andG is a sheaf of OY -module. The issue with these functors is that they arenot exact, that is, the pull back ?? is right exact but not left exact andthe push forward ?? is left exact but not right exact. The first remedywas to introduce right and left ith derived functor Li??, Ri?? respectively,within the category under consideration. However, there are disadvantagesin developing theories using these objects. The main remedy was to firstenlarge the category by adding more objects and then considering morphismswhich behave perfectly at the cohomology and K-theory levels, in order tocapture as much data as possible.The construction of derived category D(A) associated to an abelian cat-egory A will be briefly explained.Like in many mathematical constructions, the aim is to set up a paradigmin which capturing more data becomes more efficient and it also helps sim-plify some previous constructions. Let A be an abelian category. We canobtain the derived category D(A) in the following steps [7]:1. Consider the category of chain complexes Kom(A) in A, where itsobjects are chain complexes and its morphisms are chain maps.2. Identify chain homotopic morphisms in Kom(A), to arrive at the ho-motopy category of chain complexes K(A).3. Construct D(A) by localizing K(A) at the set of quasi-isomorphismsin A.In summary, objects of D(A) are chain complexes and morphisms are chainmaps where two morphisms are equal if and only if they induce isomorphismson cohomology groups, i.e. are quasi-isomorphic.142.7. Hilbert scheme of pointsSome useful examples that we will use later are as follows. The categoryof coherent sheaves on Y and the category of coherent sheaves on Y withcompact support are denoted by Coh(Y ), Cohc(Y ) and the category ofcoherent sheaves on [C2/G] 3 is denoted by Coh([C2/G]) which is isomorphicto CohG(C2) the category of G-equivariant coherent sheaves on C2.Bridgeland-King-Reid [1] result is an equivalence of derived categoriesbetween Db(Coh(Y )) and Db(Coh([C2/G])) where Db is the derived cat-egory of bounded complexes and the equivalence is given by the so calledFourier-Mukai transform ? which we will define and use later.2.7 Hilbert scheme of pointsFor a given non-singular, complex variety (manifold) X, the configurationspace of m points moving around in X is an interesting geometric object.The configuration space of m-ordered points in X is apparently Xm, and theconfiguration space of m-unordered points is SymmX := Xm/Sm, the mthsymmetric product of X, where Sm is the symmetric group of m-letters. Inboth spaces, points that correspond to m-tuples of points that are not pair-wise distinct are of special interests. In Xm, they have non-trivial isotropicgroups with respect to Sm-action and in SymmX they are singular (exceptfor the case where X is one-dimensional.)The smooth compactifications of the configuration space of m-tuples ofdistinct points are important. Fulton-MacPherson [6] constructed a nicecompactification for ordered m-tuples [13].Let X be a smooth quasi-projective scheme of finite type over C. LetZ ? X be a zero-dimensional closed subscheme. The length of Z is thelength of the artinian C-algebra H0(Z,OZ), i.e. l(Z) := dimCH0(OZ). TheHilbert scheme of m points on X is the set of all zero-dimensional closedsubschemes Z ? X of length m and is denoted by Hilbm(X).If x ? Z is a closed point, the multiplicity of x in Z is defined asdimC(OZ,x). We can associate the cycle |Z| to any Z which correspondsto the underlying set counted with multiplicities. That is, |Z| is a point inSymmX and is defined by152.7. Hilbert scheme of points|Z| :=?x?XdimC(OZ,x) ? x ? SymmXThe Hilbert-Chow morphism is then? : Hilbm(X) ?? SymmXwhich sends Z to |Z|.Briefly, lets discuss why Hilbm(X) has a scheme structure. Let X bedefined as above. A flat family of proper subschemes in X parameterizedby scheme S is a closed subscheme Z ? S ? X such that the projectionZ ?? S is flat and proper. For a closed point s ? Z, denote the fiber of Zover s by Zs. Now, given such a family and a morphism f : S? ?? S, thefamily Z ? := (f ? idX)?1(Z) ? S? ?X is flat and proper over S?. Thus wehave defined a functorhilb(X) : Schop ?? Setwhich associates to S the set of all flat families of proper subschemes in Xparameterized by S, where Sch and Set are categories of schemes and sets,respectively.For any proper subscheme Z ? X the Hilbert polynomial of Z is definedby PZ(n) := ?(OZ ?OX(nH)) where H is an ample Cartier divisor and ?is the Euler characteristic. It can be seen that for a flat family Z ? S ?Xthe function s 7? PZs ? Q[T ] where s ? S, is locally constant. This meansthat given a polynomial P, the functorhilbP (X) : S ?? {Z ? S?X| Z is proper and flat over S, P (Zs) = P,?s ? S}is an open and closed subfunctor of hilb(X). In particular, the functorhilbmX associated to the constant polynomial P = m ? N, parameterizesall zero-dimensional subschemes of length m. In generalTheorem 2.7.1 (Grothendieck [9]). For a quasi-projective (projective) scheme162.7. Hilbert scheme of pointsX, the functor hilbP (X) is representable by a quasi-projective (projetive)scheme HilbP (X).Using Yoneda?s lemma, we can deduce that there exists a universal sub-schemeZP ? HilbP (X)?X,flat over HilbP (X), such that for any Z ? hilbP (X)(S), there is a uniquemorphism f : S ?? HilbP (X) with Z ?= (f ? idX)?(ZP ).Theorem 2.7.2 (Fogarty [4]). Let X be a smooth connected quasi-projectivesurface, the the Hilbert scheme Hilbm(X) is smooth and connected of dimen-sion 2m.We are interested in Hilbert scheme of points on C2 which has an ele-mentary description as followsHilbm(C2) = {Iideal? C[a, b] | dimCC[a, b]/I = m}.Remark Consider the Hilbert-Chow morphism Hilbm(C2) ?? C2/Sm whichsends a zero-dimensional closed subscheme Z of length m on C2 to its asso-ciated effective divisor |Z| in C2/Sm. Let Hilb|G|(C2)G denote the G fixedpoint set of Hilb|G|(C2). In fact, Hilb|G|(C2)G parameterizes G-invariant sub-schemes, so the Hilbert-Chow morphism restricts to a mapHilb|G|(C2)G?? C2/G ? Sym|G|(C2).This map is a bijection on the set of points parameterizing free orbits of Gand hence is a birational map. Moreover, Hilb|G|(C2)G is non-singular by[11, Lemma 9.1].If we define GHilb(C2), the G-Hilbert scheme, to be the component ofHilb|G|(C2)G which contains the free orbits of G, we find thatGHilb(C2) ?? C2/Gis a resolution of singularities by [11, Theorem 9.3].172.8. Main question2.8 Main questionAs discussed previously, we have denoted the irreducible component ofHilb|G|(C2)G containing free G-orbits by GHilb(C2). Points of GHilb(C2)correspond to G-invariant subschemes Z ? C2 with H0(Z,OZ) ?= Reg,where Reg is the regular representation ring of G.We can generalize GHilb(C2) as follows. The scheme whose points cor-respond to G-invariant subschemes Z ? C2 with H0(Z,OZ) ?= ?, for anyrepresentation ? of G. Precisely,GHilb?(C2) = {Iideal? C[a, b] | I is G invariant and C[a, b]/I ?= ?}.Therefore, when ? = Reg, we obtain GHilbReg(C2) = GHilb(C2).For reasons which will become clear later on, we change our notationsGHilb(C2) and GHilbmReg(C2) to HilbReg([C2/G]) and HilbmReg([C2/G]),respectively.As mentioned before, Y, the minimal resolution, can be identified withGHilb(C2) and is clearly isomorphic to Hilb1(Y ). Therefore, the main ques-tion we would like to answer is related to a generalization of the describedisomorphismHilb1(Y ) ?= HilbReg([C2/G]).That is, we would like to study the relation between the Hilbert scheme ofm points of Hilb1(Y ) i.e. Hilbm(Hilb1(Y )) = Hilbm(Y ) and HilbmReg([C2/G]).From the moduli space point of view, Hilbm(Y ) can be considered asa subspace of the space of objects of D(Y ) parameterizing one term com-plexes consisting of the structure sheaves of closed subschemes of length mon Y. Likewise, HilbmReg([C2/G]) is parameterizing finite substackes of thequotient stack [C2/G], thus HilbmReg([C2/G]) is also parameterizing someobjects of [C2/G].We can observe that the birational morphism from Y 99K X induces abirational morphism182.8. Main question? : Hilbm(Y ) 99K HilbmReg([C2/G])for m > 1, (we will explain the exact definition of ? in the next chapter)because m distinct points of Y away from the exceptional set map to mdistinct points in X\{0} where the preimage of m points of X\{0} is in fact,m distinct Z/n-orbits in C2 \ {0} under the projection C2 \ {0} ?? X \ {0}.Now the question isQuestion: What is the image of ??In other words, what sort of objects in D[C2/G] are being parameterizedby ?(Hilbm(Y ))?We will provide an explicit answer to our question to the case of G = Z/nin the next chapter.19Chapter 3The main resultsIn this chapter, we will use the described framework and concepts in chap-ter one to introduce new concepts in order to explicitly answer our mainquestion.From now on, G = Z/n. Let X = C2/Z/n = SpecT where T =C[x, y, z]/(xy? zn) for n ? 2, and Y ?= GHilb(C2) be the minimal (crepant)resolution of X and let X = [C2/Z/n] be the quotient stack whose coarsespace is X. Let Z ? Y ?X be the universal closed subscheme associated tothe Hilbert scheme Y. We have the following diagramZ?Y XXp? q?? p?iwhere p? is finite and flat, q?, ? are birational and p?i is finite.The Fourier-Mukai transform [1] ? from D(Y ) the (bounded) derivedcategory of coherent sheaves on Y, and D(X) the (bounded) derived categoryof coherent sheaves on the stack resolution of X, is defined by? = Rq?? ? Lp?? : D(Y ) ?? D(X)Note that, since p is already exact we have Lp?? = p?? and in our case,according to Bridgeland-King-Reid [1] ? is an equivalence of categories.20Chapter 3. The main resultsIn this thesis, we will be working with derived category of coherentsheaves with compact support so from now on by D(Y ) we mean the de-rived category of coherent sheaves with compact support on Y.For the computational purposes, we use the equivalence of categoriesbetween D(X) and the (bounded) derived category of Z/n-equivariant co-herent sheaves on C2, i.e. DZ/n(C2). Thus, not only we can replace X by C2,but we can also use the fact that Z? = [Z/G], where Z is the universal closedsubscheme of Y = GHilb(C2) due to the isomorphism D(Z?) ?= DZ/n(Z), toreplace Z? by Z. Then, the above diagram is altered intoZY C2Xp q? piand? = Rq? ? Lp? : D(Y ) ?? DZ/n(C2) ?= D(X)We want to study the image of torus invariant structure sheaves of zero-dimensional subschemes (sheaves supported at points) on Y, considered asone term complexes in D(Y ), under ?. As discussed in the previous chapter,Y has n? 1 exceptional divisors (with n coordinate charts Yi) Ei ?= P1, andsince Y \ {exceptional divisors} is isomorphic to X \ {0}, the image of thosesheaves (one term complex of sheaves with compact support) supportedcompletely outside of the exceptional divisors Eis, will be sent into sheaves,therefore, we are left to investigate the image of sheaves supported on theexceptional curves.To simplify the problem, we will consider torus-invariant, zero-dimensional,length m sheaves supported at the origin of the coordinate chart Yi. We willuse the fact that torus-invariant, zero-dimensional, length m sheaves sup-213.1. Coordinates on the universal closed subscheme of the Hilbert schemeported at the origin correspond to length m monomial ideals of the ringof regular functions on each chart and the latter corresponds to the par-tition of m which is of combinatorial interest. We also go back and forthbetween length m torus-invariant, zero dimensional structure sheaves on Yiand length m torus-invariant, zero-dimensional subschemes of Yi.Since our computations rely on coordinates, in the first step we shoulduse toric geometry techniques in order to write the defining equations oneach coordinate chart of Y as follows3.1 Coordinates on the universal closedsubscheme of the Hilbert schemeFor our computational purposes, we need coordinate charts on the universalclosed subscheme Z. To obtain such coordinates, we prove the followingtheoremTheorem 3.1.1. With the same objects as above, one can isomorphicallyreplace Z on each chart with (Yi ?X C2)red for 1 ? i ? n. Equivalently,Z ?= (Y ?X C2)red.Proof. By the definition of the universal closed subscheme, a point in Z isa pair (Z, p) where Z ? C2 is a G-invariant subscheme with H0(OZ) ?= Reg(the regular representation of G) and p ? Z.By the definition of the fiber product, a point in Y ?X C2 is a pair(y, q) where y ? Y, q ? C2 and ?(y) = pi(q). Since Y ?= GHilb(C2), apoint y ? Y corresponds uniquely to a G-invariant closed subscheme Z ?C2 with H0(OZ) ?= Reg. The condition ?(y) = pi(q) is then the conditionq ? Z. Thus, we see that Z and Y ?X C2 coincide pointwise, so to provethe isomorphism Z ?= (Y ?X C2)red, it suffices to show that Z is reduced.Because Z ?? Y is the universal family, Z is flat over Y and thus theassociated points on Z are pulled back from the associated points of Y [16,p. 632]. Since Y is smooth, the only associated point is itself and so likewisefor Z. Then, since Z is generically reduced, hence is globally reduced.223.1. Coordinates on the universal closed subscheme of the Hilbert schemeHence, we have the following diagramXYi C2(Yi ?X C2)redBy definition of the fiber product, we obtain the defining equations forYi ?X C2 as follows.LetSi = C[ui, vi]R = C[a, b]T = C[x, y, z]/(xy ? zn)where ui = x?(n?i)yi, vi = xn?i+1y?(i?1) andYi = SpecSiC2 = SpecRX = SpecTTherefore,Yi ?X C2 = Spec(Si ?T R)and can be easily seen thatYi ?X C2 = SpecC[ui, vi, a, b]/(an ? un?i+1i vn?ii , bn ? ui?1i vii, uivi ? ab)for 1 ? i ? n.233.1. Coordinates on the universal closed subscheme of the Hilbert schemeProposition 3.1.2. With the same above notations(Yi ?X C2)red ?= SpecC[ui, vi, a, b]/(ai ? uibn?i, bn?i+1 ? viai?1, uivi ? ab)for 1 ? i ? n.Proof. First, we show that I ? J ??I whereI = (an ? un?i+1i vn?ii , bn ? ui?1i vii, uivi ? ab),andJ = (ai ? uibn?i, bn?i+1 ? viai?1, uivi ? ab).The inclusion I ? J is obvious becausean ? un?i+1i vn?ii = an?i(ai ? uibn?i) + ui(an?ibn?i ? un?ii vn?ii ),bn ? ui?1i vii = bi?1(bn?i+1 ? viai?1) + vi(ai?1bi?1 ? ui?1i vi?1i ),andan ? un?i+1i vn?ii , bn ? ui?1i viiare zero in C[ui, vi, a, b]/J.For the second inclusion J ??I, we claim that(ai ? uibn?i)n, (bn?i+1 ? viai?1)n ? I.In fact,(ai ? uibn?i)n =n?k=0(?1)n?k(nk)akiun?ki b(n?i)(n?k).243.1. Coordinates on the universal closed subscheme of the Hilbert schemeWhen ki > (n? i)(n? k) or k > n? i each monomial can be written inthe form(ab)(n?i)(n?k)aki?(n?i)(n?k)un?ki = (uivi)(n?i)(n?k)an(k+i?n)un?kiin C[ui, vi, a, b]/I. Likewise, when ki ? (n ? i)(n ? k) or k ? n ? i we canwrite each monomial of the form akiun?ki b(n?i)(n?k) into(ab)kiun?ki b(n?i)(n?k)?ki = (uivi)kiun?ki bn(n?i?k)in C[ui, vi, a, b]/I.Therefore, using the relations an = un?i+1i vn?ii and bn = ui?1i vii, we obtain(ai ? uibn?i)n =?0?k?n?i(?1)n?k(nk)ui(n?i+1)i vi(n?i)i+?n?i<k?n(?1)n?k(nk)ui(n?i+1)i vi(n?i)i= ui(n?i+1)i vi(n?i)in?k=0(?1)n?k(nk)= ui(n?i+1)i vi(n?i)i (1? 1)n= 0in C[ui, vi, a, b]/I.Similarly,(bn?i+1 ? viai?1)n =n?k=0(?1)n?k(nk)bk(n?i+1)vn?ki a(n?k)(i?1),and when k(ni + 1) > (n ? k)(i ? 1) or k > i ? 1 the monomial can berewritten in the form253.1. Coordinates on the universal closed subscheme of the Hilbert schemebk(n?i+1)?(n?k)(i?1)(ab)(n?k)(i?1)vn?ki = bn(k?i+1)(uivi)(n?k)(i?1)vn?kiand when k(ni + 1) ? (n? k)(i? 1) or k ? i? 1,(ab)k(n?i+1)an(i?1?n)vn?ki = (uivi)k(n?i+1)an(i?1?n)vn?kiin C[ui, vi, a, b]/I. Then using the relations an = un?i+1i vn?ii and bn =ui?1i vii, we arrive at(bn?i+1 ? viai?1)n =?0?k?i?1(?1)n?k(nk)u(i?1)(n?i+1)i vi(n?i+1)i+?i?1<k?n(?1)n?k(nk)u(i?1)(n?i+1)i vi(n?i+1)i= u(i?1)(n?i+1)i vi(n?i+1)i (1? 1)n= 0in C[ui, vi, a, b]/I.Now, it is enough to show that J is self-radical i.e. J =?J. We will beusing the fact a projective map is flat if and only if the Hilbert polynomialof its fibers coincide for every fiber [16, corollary 24.7.2] and [15, theorem1.4]. We will show that the fibers of the mapq : SpecC[ui, vi, a, b]/J ?? SpecC[ui, vi]are finite sets and thus the Hilbert polynomial of each fiber is constant andis equal to the length of that fiber.First of all, q is a finite morphism, therefore, is quasi-finite, i.e. its fibers arefinite sets. It is also projective, because we can embed SpecC[ui, vi, a, b]/Jinto ProjC[ui,vi]C[a, b, c] by homogenizing J to J, for example, when i ? n?iJ = (aicn?2i ? uibn?i, bn?i+1 ? viai?1cn?2i+2, uivic2 ? ab).263.2. Main resultTo see the number of elements of fibers is constant on every fiber, weclaim that 1, a, ? ? ? , ai?1, b, ? ? ? , bn?i is a C-basis of C[ui, vi, a, b]/J, for fixedui, vi. Any monomial element of C[ui, vi, a, b] is of the form albkuri vsi . If l ? ior k ? n? i+ 1 we can use the relations ai = uibn?i and bn?i+1 = viai?1, toreduce l, k to the cases where l < i and b < n? i+ 1. Furthermore, for fixedui, vi the monomial albkuri vsi is considered only having a, b variables and ifk ? l ? i?1, we can replace albk by al?k(uivi)k using the relation ab = uiviand by bk?l(uivi)l when l ? k ? n? i.3.2 Main resultAs described earlier, let I be a monomial ideal sheaf of a length m sub-scheme supported at the origin of the coordinate chart Yi where OYi/I isthe structure sheaf of the corresponding torus-invariant, zero dimensional,closed subscheme of length m supported at the origin of Yi. We have thefollowing short exact sequence0 ?? I ?? OYi ?? OYi/I ?? 0if we apply ? to the above short exact sequence we will obtain the longexact sequence0 ?? h0(?(I)) ?? h0(?(OYi))f?? h0(?(OYi/I)) ???? h1(?(I)) ?? h1(?(OYi)) ?? h1(?(OYi/I)) ?? 0Proposition 3.2.1. ?(OYi/I) is a sheaf, i.e. one term complex of sheaves.Proof. The support of OYi/I is zero dimensional, p is a finite morphism alsoq maps the finite set Supp(OYi/I) to C2, therefore, ?(OYi/I) has no highercohomology, that is, Rq?(p?(OYi/I)) = q?(p?(OYi/I)). Hence, ?(OYi/I) isa sheaf.273.2. Main result3.2.1 Statement of the main resultAs we proceed, it is necessary to mention that the ideal sheaves of torusinvariant subschemes of C2 are associated to monomial ideals of C[a, b] andmonomial ideals also correspond to partitions. The illustration of the latercorrespondence is the following.Each partition ? = (?1, ?2, ? ? ? , ?d) (where ?1 ? ?2 ? ? ? ? ? ?d ? 1)corresponds to a monomial idealI? = (v?1 , uv?2 , ? ? ? , ud?1v?d , ud)on the chart Yi. We can associate the so called Ferrer diagram to the mono-mial C-basis of C[ui, vi]/I?. For example, let m = 5 and ? = (3, 2) themonomial ideal is (v3i , uiv2i , u2i ) and its C-basis Ferrer diagram is depictedbelow.1uiviviv2iuiv3iuiv2iu2iFigure 3.1: Ferrer diagram for m = 5 and ? = (3, 2)Let ? be a partition of m and let Z?,i ? Yi be a zero-dimensional, torusinvariant, closed subscheme of length m supported at the origin of the chartYi with the structure sheaf OZ?,i whose ideal sheaf is determined by ?. Wewould like to describe the image under ?, that is, ?(OZ?,i) ? CohZ/n(C2),where CohZ/n(C2) is isomorphic to the category of finitely generated Z/n-invariant, C[a, b]-modules.Theorem 3.2.2. The image of C[ui, vi] the associated module to OYi under? = q? ? p? is the following R-module283.2. Main resultMi :=?ktkR/(ai?1tk ? bn?i+1tk?1, ?k ? 1 and aitk ? bn?itk?1, ?k ? 0)where R = C[a, b].Remark Obviously the above R-module is not finitely generated becauseOYi is not of compact support.The main result isTheorem 3.2.3.?(OZ?,i) = Mi/J?,iwhere J?,i is the R-submodule of Mi determined combinatorially by ? by theprocedure described in 3.2.3.3.2.2 Computational technique and proof of the resultThe underlying computational technique that we have exploited is demon-strated in the following diagramSi-modulesSi ?T R-modulesR-modulesp? q?where T = C[x, y, z]/(xy? zn) and an Si = C[ui, vi]-module is considered asa C[ui, vi]?C[x,y,z]/(xy?zn)C[a, b]-module through the way C[ui, vi] sits insideC[ui, vi]?C[x,y,z]/(xy?zn) C[a, b] by p? and then using the existing relations itis sent to a C[a, b]-module under q?.Proof. 3.2.2 Define a surjective R-module homomorphism293.2. Main resultMi ?? q?(p?(C[ui, vi]))sending t0 7? 1, tk 7? ?(vki ) and t?k 7? ?(uki ) for k ? N. It is well-definedbecause of these relations ai = uibn?i and bn?i+1 = viai?1. The injectivityboils down to the fact that the mentioned relations together with uivi = abexhaust all the possible relations among ui, vi, a and b as can be seen fromProposition 3.1.2.Proof. 3.2.3 Let I?,i be the ideal sheaf of OZ?,i , that is, OZ?,i = OYi/I?,i.Let I?,i be the associated ideal of Si to I?,i. Then, we have the followingshort exact sequence of Si-modules0 ?? I?,i ?? Si ?? Si/I?,i ?? 0Since p is a flat morphism, pulling back the above sequence by p preservesthe exactness and it will be0 ?? I?,i ?T R ?? Si ?T R ?? Si/I?,i ?T R ?? 0Now, because q? is not right exact, applying q? to our sequence we will get0 ?? q?(I?,i?T R) ?? q?(Si?T R) ?? q?(Si/I?,i?T R) ?? R1q?(I?,i?T R)where R1q?(I?,i ?T R) is the first right derived functor of q?(I?,i ?T R).Lemma 3.2.4. R1q?(I?,i ?T R) = 0.Proof. In other words, we want to show that R1q?(I?,i?OX OC2) = 0, whichholds, since q?(I?,i ?OX OC2) is a quasi-coherent OC2-module on an affinevariety.303.2. Main result3.2.3 Combinatorial corollary and illustrationsFirst of all, to get more sense of how we found Mi (the image of C[ui, vi]under ?) in 3.2.2 and later the combinatorial consequent, we represent theFerrer diagram of Mi. The method we used was thematically the same inall the coming illustrations.At the outset, the C-basis of C[ui, vi] is {uri vsi | r, s ? 0} with the Ferrerdiagram1viv2iv3iuiuiviuiv2iuiv3iu2iu2i viu2i v2iu2i v3iu3iu3i viu3i v2iu3i v3i? ? ?? ? ?? ? ?? ? ?. ..............Figure 3.2: Ferrer diagram of C[ui, vi]under ? it goes to the Ferrer diagram Figure 3.3. and the discovered rulewhich is: we move to the right by multiplication to a and to the up bymultiplication to b.To understand ?(OZ?,i) better, first note that the origin of Yi, that is,C[ui, vi]/(ui, vi) maps to C[a, b]/(ai, bn?i, ab) which its Ferrer diagram is L-shaped depicted in Figure 3.4.313.2. Main resultt0ai?1t1 = bn?i+1t0t1ait0 = bn?it?1t?1...? ? ?...? ? ?...? ? ?. ... ... ..Figure 3.3: Ferrer diagram of Mi1 uivi1 a ai?1bbn?i?1aibn?iab? ? ?...Figure 3.4: Image of the originThen, for any monomial ideals I in a chart Yi, to find the image under ?,323.2. Main resultwe first find the Ferrer diagram of the monomial C-basis of C[ui, vi]/I andstick to the following set of rules:We start with the box 1 (the origin). Either there is a box attached toits right hand side or on top of it1 1Figure 3.5: Attachment possibilitiesand the image will be constructed by attaching the L-shaped image of theorigin to the right hand side of the leftmost box or to the top of it as follows...? ? ?...? ? ?...? ? ?...? ? ?Figure 3.6: Attaching rulesand we continue this process for other boxes in the Ferrer diagram of C[ui, vi]/I.For example, let n = 4, i = 2 and m = 5 with ? = (3, 2). Thus, on Y2,C[u2, v2]/(u2, v2) is mapped to C[a, b]/(a2, b3, ab) and in order to find theimage of C[u2, v2]/(u22, u2v22, v32) we should follow the above rules step bystep.First, determine the image of the origin333.2. Main result1 u2v21 u2 u22v2v22v32u2v2Figure 3.7: Ferrer diagram of the origin for n = 4, i = 2 and m = 5 with? = (3, 2).then find the Ferrer diagram of the C-basis of C[u2, v2]/(u22, u2v22, v32)1u2v2v2v22u2v32u2v22u22Figure 3.8: Ferrer diagram of C[u2, v2]/(u22, u2v22, v32)and finally, use the attaching rules to construct the Ferrer diagram of theimage shown below in Figure 3.9. Therefore, the image is(a4, a2b2, ab5, b8)/(a6, a5b, a4b4, a2b6, ab9, b11)which is also isomorphic as an R = C[a, b]-module toM2/J?,2where J?,2 is a submodule of M2 with relations coming from the monomialsa6, a5b, a4b4, a2b6, ab9, b11.343.2. Main result1a2b2a4a5bab5a2b6b8ab9b11a4b4a6Figure 3.9: Image of C[u2, v2]/(u22, u2v22, v32)Moreover, we have established the subsequent combinatorial resultCorollary 3.2.5. The above procedure establishes a correspondence betweenpartitions of m and Z/n-colored skew partitions of mn.We color the boxes according to the initial coloring of the image of theorigin as can be seen in the coming example.Figure 3.10: Z/4-coloring353.2. Main resultFigure 3.11: Z/4-colored skew partition36Chapter 4ConclusionRecall that the question we set out to answer was : ?Where do the structuresheaves of zero-dimensional, closed subschemes on Y go under the Fourier-Mukai equivalence D(Y ) ?? DG(C2)??We have used explicit coordinate charts on Y to fully answer this ques-tion in the case where G = Z/n and the subschemes are torus invariant. Wehave heavily exploited the combinatorics of the C-basis of a monomial idealassociated to the ideal sheaf of a zero-dimensional, torus invariant closedsubscheme of Yi, in the associated Ferrer diagram. Our method involvedfirst determining the Ferrer diagram of the image of the origin of each chartYi and then by the combinatorial attaching rules, constructing the imageusing the Ferrer diagram of the image of the origin. We have also explainedhow to construct the image from Ferrer diagrams in an example. Never-theless, in order to obtain a unifying picture, we have found the universalR-module Mi on each chart, so that the formulation of our result becomeseasier and later, we have proven that the image is obtained by taking thequotient of Mi by the ideal determined from our combinatorial rules.It remains to address our question in the case of general zero-dimensionalclosed subschemes of Y or even when G is not necessarily cyclic. One pos-sible approach is related to Haiman?s result [10] which gives a system ofcoordinates for Hilbert scheme in general, and our method could be applieddirectly to get a more general result.Moreover, we believe that our combinatorial outcome has not been pushedas much as it could. As discussed, the combinatorial Fourier-Mukai givesus the correspondence between partitions of m and Z/n-colored skew parti-tions of mn. Also, since Hilbm(Y ) and HilbmReg([C2/Z/n]) are holomorphicsymplectic, their Betti numbers coincide, therefore, they do have the same37Chapter 4. ConclusionEuler characteristic. Meanwhile, we know that the Euler characteristic isthe number of torus fixed points. Thus, there is a correspondence between mpartitions of total size n and Z/n-colored partitions of size mn with m boxesof each color. Hence, there should be a set of rules by which given a Z/n-colored partition of mn with m boxes of each color, one can reconstructa Z/n-colored skew partition of mn and vice versa. Our correspondencecomes close to this: we associate to m partitions of size n, m Z/n-coloredskew partitions with a total of m boxes of each color. This suggests thatthere should be some way of combining our skew partitions into a singleZ/n-colored partition with m boxes of each color. Discovering this processis a project for the future.38Bibliography[1] Bridgeland-King and Reid. The McKay correspondence as an equiva-lence of derived categories. J. Amer. Math. Soc., 14(3):535?554 (elec-tronic), 2001.[2] Craw. The McKay correspondence as the representation of McKayquiver. PhD thesis, June 2001.[3] Dolgachev. McKay correspondence. October 26, 2009 version.[4] Fogarty. Algebraic families on an algebraic surface. Amer. J. Math,90:511?521, 1968.[5] Fulton. Introduction to Toric varieties. 1993.[6] Fulton-MacPherson. A compactification of configuration spaces. Ann.of Math. (2), 139(1):183?225, 1994.[7] Gelfand-Manin. Methods of homological algebra. 2003.[8] Gonzalez-Sprinberg and Verdier. Construction gomtrique de la cor-respondance de McKay. Annales scientifiques de l?cole Normale Su-prieure, 16(3):409?449, 1983.[9] Grothendieck. Techniques de construction et thormes d?existence engomtrie algbrique. IV Les schmas de Hilbert. Sminaire Nicolas Bour-baki, 6(221):249?276, 1960/61.[10] Haiman. t, q-Catalan numbers and the Hilbert scheme. Discrete Math.,193(1-3):201?224, 1998. Selected papers in honor of Adriano Garsia(Taormina, 1994).39Bibliography[11] Ito-Nakamura. Hilbert schemes and simple singularities. In New trendsin algebraic geometry (Warwick, 1996), volume 264 of London Math.Soc. Lecture Note Ser., pages 151?233. Cambridge Univ. Press, Cam-bridge, 1999.[12] Kapranov-Vasserot. Kleinian singularities, derived categories and Hallalgebras. volume 316, pages 565?576. 2000.[13] Lehn. Lectures on Hilbert schemes.[14] McKay. Graphs, singularities, and finite groups. In The Santa CruzConference on Finite Groups (Univ. California, Santa Cruz, Calif.,1979), volume 37 of Proc. Sympos. Pure Math., pages 183?186. Amer.Math. Soc., Providence, R.I., 1980.[15] Osserman. The Hilbert polynomial and degree.[16] Vakil. Foundations of algebraic geometry. June 11, 2013 version.40

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