Toric Orbifold Chow Rings by Yunfeng Jiang M . S . , Hebei Normal University,China, 2001 A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F Doctor of Philosophy in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Mathematics) The University of Br i t i sh Columbia A p r i l 2007 © Yunfeng Jiang, 2007 Abstract In this thesis we study the orbifold Chow ring of smooth Deligne-Mumford stacks which are related to toric models. We give a new quotient construction of toric Deligne-Mumford stacks defined by Borisov-Chen-Smith such that toric Deligne-Mumford stacks have more representations as quotient stacks. We define toric stack bundles using this new construction and compute their orbifold Chow rings. As an interesting application, we compute the orbifold Chow ring of finite abelian gerbes over smooth schemes. The extended stacky fans we introduced are used to give a new quotient construction of toric Deligne-Mumford stacks. These new combinatorial data have relations to stacky hyperplane arrangements, i.e. every stacky hyperplane arrange-ment determines an extended stacky fan. The hyperplane arrangement determines the topology of the associated hypertoric varieties. We define hypertoric Deligne-Mumford stacks using stacky hyperplane arrangements, generalizing the construc-tion of Hausel and Sturmfels. Their orbifold Chow rings are computed as well. Borisov, Chen and Smith computed the orbifold Chow ring of projective toric Deligne-Mumford stacks. We generalize their formula to semi-projective toric Deligne-Mumford stacks. The hypertoric Deligne-Mumford stack is a closed sub-stack of the Lawrence toric Deligne-Mumford stack associated to the stacky hyper-plane arrangement which is semi-projective, but not projective. We prove that the orbifold Chow ring of a Lawrence toric Deligne-Mumford stack is isomorphic to the orbifold Chow ring of its associated hypertoric Deligne-Mumford stack. This is the orbifold Chow ring analogue of a result of Hausel and Sturmfels. ii Contents Abstract ii Contents iii Acknowledgements vi Co-authorship Statement vii 1 The Orbifold Chow Ring and Toric Deligne-Mumford Stacks . . . 1 1.1 Introduction and Motivation 1 1.2 The Deligne-Mumford Stacks 3 1.3 The Orbifold Chow Groups 8 1.4 Orbifold Cup Product 9 1.5 Gale Duality for Finitely Generated Abelian Groups 15 1.6 Toric Deligne-Mumford Stacks 19 1.7 Finite Abelian Gerbes over Toric Deligne-Mumford Stacks 21 1.7.1 The proof of main results 22 1.7.2 A n example 25 Bibliography 28 iii 2 Simplicial Toric Stack Bundles 31 2.1 Introduction 31 2.2 A New Quotient Representation of Toric Deligne-Mumford stacks. . 35 2.3 The Toric Stack Bundle p ^ ( S e ) 38 2.4 The Orbifold Cohomology Ring 45 2.4.1 The module structure on A * r b ( p A ' ( S e ) ) 45 2.4.2 The orbifold cup product 50 2.4.3 Proof of Theorem 2.1.1 53 2.5 The /i-Gerbe 54 2.6 Application 57 Bibliography 61 3 Hypertoric Deligne-Mumford Stacks 64 3.1 Introduction 64 3.2 The Hypertoric Deligne-Mumford Stacks 70 3.2.1 Stacky hyperplane arrangements 70 3.2.2 Lawrence toric Deligne-Mumford stacks 72 3.2.3 Hypertoric Deligne-Mumford stacks 74 3.3 Properties of Hypertoric Deligne-Mumford Stacks 75 3.4 Substacks of Hypertoric Deligne-Mumford Stacks 79 3.5 Orbifold Chow Ring of M(A) 88 3.5.1 The module structure 88 3.5.2 The orbifold product 94 3.5.3 Proof of Theorem 3.1.1 98 3.6 Applications 100 iv Bibliography 106 4 Semi-projective Toric Deligne-Mumford Stacks 110 4.1 Introduction 110 4.2 Semi-projective Toric Deligne-Mumford Stacks and Their Orbifold Chow Rings 112 4.2.1 Semi-projective toric Deligne-Mumford stacks 113 4.2.2 The inertia stack 114 4.2.3 The orbifold Chow ring 115 4.3 Lawrence Toric Deligne-Mumford Stacks 119 4.3.1 Comparison of inertia stacks 119 4.3.2 Comparison of orbifold Chow rings 124 Bibliography 130 5 Conclusion 133 5.1 Relations Among Chapters 133 5.2 Importance of the Thesis 134 5.2.1 Good reading materials for students 134 5.2.2 Importance of the definitions and results 135 5.2.3 Application of the results 136 5.3 Future Research 137 5.3.1 Twisted orbifold Chow ring 137 5.3.2 Quantum cohomology of gerbes 137 5.3.3 The Crepant resolution conjecture 138 Bibliography 140 Acknowledgements First and foremost I would like to thank my advisor, K a i Behrend, for his unsurpassed generosity, unwavering support in my P h . D studies at U B C . I would like to thank Jim Bryan for teaching me Crepant Resolution Con-jecture, and to Kalle K a r u for teaching me toric varieties. I wish to thank Hsian-Hua Tseng for working together on some topics and invaluable dicussions. I thank my graduate colleages Yinan Song and A m i n Gho-lampour for good discussions about research projects. Finally, this work would not have been possible without my supporting fam-ily, especially my parents and lovely wife. Y U N F E N G J I A N G The University of British Columbia April 2007 vi Co-authorship Statement Chapter 3 and Chapter 4 are joint works with Hsian-Hua Tseng. In Chapter 3, we found the research topic through discussions about the orbifold Chow rings. I had prepared most of the manuscripts including writing and typing. The contents in Chapter 4 came from my idea to generalize the orbifold Chow ring formula to semi-projective toric Deligne-Mumford stacks. I had talked with Hsian-Hua Tseng about the proof details. We worked together about this project eventually. Y U N F E N G J I A N G The University of British Columbia April 2001 vii C h a p t e r 1 The Orbifold Chow Ring and Toric Deligne-Mumford Stacks 1.1 Introduction and Motivation In 1980's Dixon, Harvey, Vafa and Witten [DHVW1], [DHVW2] studied the string theory on orbifolds. Unlike the orbifolds which are generally singular, the string theory on orbifolds are smooth. The physics results motivate the study the stringy properties on orbifolds in mathematics. In 1980's M c K a y studied the simple quotient singularities such as the quotients X = C2/G for which G is a finite group of A,D,E type. For these quotient singularities, there exist unique crepant resolutions Y. M c K a y proved that the Euler and Hodge numbers of the crepant resolutions are equal to the orbifold Euler and Hodge numbers of the orbifold X defined in physics, which is called M c K a y correspondence, see [Mckay]. In 2001 Ruan [R] gave a new conjecture called " Cohomological Hyper-kahler Resolution Conjecture" ( C H R C ) based on the notion of orbifold Chow 1 ring (or Chen-Ruan orbifold cohomology) denned by Chen and Ruan [CR1], which states that the Chow ring of a hyperkahler resolution of an orbifold is isomorphic to the orbifold Chow ring of the orbifold. This conjecture generalized the M c K a y correspondence to the ring structure level. There are many works proving or trying to prove this conjecture, see [FG],[Uribe],[BCS]. The C H R C conjecture involved the computation of orbifold Chow ring which is also interesting in its own right. The orbifold Chow ring structure is the classi-cal part of the orbifold Gromov-Witten theory developed by Chen and Ruan [CR2] for arbitrary orbifolds in symplectic category and Abramovich, Graber and Vistoli [AGVI] , [AGV2] in algebraic category. The most interesting feature of this new coho-mology theory is the obstruction bundle in the definition of the orbifold cup product. In the abelian orbifold case, the obstruction bundle is classified in [BCS],[CH] and [Jiangl]. In the general case, it is determined in [JKK]. In this thesis we do the abelian case. Simplicial toric varieties provided good examples for the abelian orbifolds. Generalizing the quotient construction of Cox [Cox] for simplicial toric varieties, Borisov, Chen and Smith [BCS] defined toric Deligne-Mumford stacks using stacky fans and computed their orbifold Chow ring. However, their definition of stacky fan is a little restrictive. In this thesis we generalize their definition and define extended stacky fans. Using this new construction, in Chapter 2 we define toric Deligne-Mumford stacks using extended stacky fans so that toric Deligne-Mumford stacks have more representations as quotient stacks. We define toric stack bundles and compute their orbifold Chow ring. The other outline of the thesis is as follows. In Chapter 3 we study the relation between stacky hyperplane arrangements and extended stacky fans. We 2 use stacky hyperplane arrangements to define hypertoric Deligne-Mumford stacks and study their orbifold Chow ring. In Chapter 4 we compute the orbifold Chow ring of semi-projective toric Deligne-Mumford stacks and use it to study the relation of orbifold Chow rings between Lawrence toric Deligne-Mumford stacks and their associated hypertoric Deligne-Mumford stacks. In Chapter 5 we talk about the relations among chapters and list some future studies. In the current Chapter we introduce the basic definition of orbifold Chow ring and review the construction of toric Deligne-Mumford stacks by Borisov-Chen-Smith. This chapter is outlined as follow. In Section 1.2 we introduce the basic notion and properties of smooth Deligne-Mumford stacks. In Section 1.3 we define the orbifold Chow group of smooth Deligne-Mumford stacks and in Section 1.4 we define the orbifold cup product by studying the degree zero, genus zero orbifold Gromov-Witten invariants. In Section 1.5 we discuss the Gale duality of /? : Z™ —> N for any finitely generated abelian group N. In Section 1.6 we define toric Deligne-Mumford stack and talk about some properties. Finally in Section 1.7 we study finite abelian gerbes over toric Deligne-Mumford stacks. Convention We consider rational coefficient in Chow rings and orbifold Chow rings in this thesis. 1.2 The Deligne-Mumford Stacks Throughout this paper, let X be a projective smooth Deligne-Mumford stack over the complex numbers C with projective coarse moduli space X. In this section, we give the definition of smooth Deligne-Mumford stacks and discuss some general properties of X and fix notations throughout. Let 5 be a 3 scheme, and let Sch/5 denote the category of schemes over 5. Definition 1.2.1 A groupoid over 5 is a category X together with a functor px : X —> Sch/5 such that: (i) If / : X —> Y is an arrow in Sch/5 and rj is an object of X with px(r)) = Y, then there exists an arrow ip : £ —> t] in X such that px(<p) = /; (ii) If cp : £ —> £ a n d ip '• V — • C a r e arrows in X, and h : px{0 —> Pxiv) is such that px(4>) ° h = px(<p), then there is a unique arrow x '• £ V such that ip o x = ^ and pxix) = h. Consider the groupoid X, for any scheme X, let X{X) be the category of schemes over X. Definition 1.2.2 A groupoid X over 5 is a stack if: (i) For any X in Sch/5 and any two objects £i and £2 in X(X), the functor /somx (£1,^2) : Sch/X —> Sets, which associates to a morphism / : Y —> X the set of isomorphisms in X(Y) between and f*& is a sheaf in the etale topology; (ii) Let {Xi —• X} be a covering of X £ Sch/5 in the etale topology. Let X(Xi), and let be isomorphisms in X{X{ x x Xj) satisfying the cocycle condition. Then there is £ G X{X) with isomorphisms ipi : f^. —>• £i, such that </>ij = (*l>i\XiXXXj) 0 (^PjlXiXxXj)'1-Example If G —> S is an affine group scheme, the groupoid BG classifying isomorphism classes of principal G-bundles over schemes is a stack. 4 E x a m p l e Let M be a scheme and G an algebraic group acting on M. The category X = [M/G] which fibred isomorphism classes of principal G-bundles over scheme X 6 S c h / 5 together with a map to M is a groupoid, see the following diagram E — M X. We check from Definition 1.2.2 that X is a stack which we call a quotient stack. The geometry of a stack of the form [M/G] with M a scheme and G an algebraic group is essentially equivalent to the equivariant geometry of M with respect to the G-action. In this thesis all the stacks we consider are of this form for which G is abelian, keeping this interpretation in mind may help the readers unfamiliar with stacks understand this notion. One may also think of a stack as some space whose points can have nontrivial automorphism groups. A point with nontrivial automorphism group is called a stacky point. This point of view helps one understand the notion of morphisms between stacks. R e m a r k A morphism / : X —> y of stacks can be thought of as a map between spaces, together with group homomorphisms fx : Aut(x) —> Aut(f(x)),Vx G X be-tween automorphism groups. A morphism / is called representable if fx is injective for any x G X. D e f i n i t i o n 1.2.3 A stack X is a smooth Deligne-Mumford stack if: (i) The diagonal A is representable, quasicompact and separated; (ii) There is a scheme U and an etale surjective morphism U —> X. Such a morphism U —> X is called an atlas. 5 R e m a r k In the above two examples, the stack BG is Deligne-Mumford if G is finite. A n d the stack [ M / G ] is Deligne-Mumford if the G-action has only finite stabilizers. To a Deligne-Mumford stack X we can associate a coarse moduli space X which is in general an algebraic space [L-MB]. We will often assume that X is a projective scheme. This is related to the second interpretation above: a stack X can be thought of as an additional structure on X describing how X locally looks like a quotient. Moreover, for a morphism X —> y of stacks, there is an induced morphism X —> Y between their coarse moduli spaces. This may be interpreted as forgetting the homomorphisms between automorphism groups. For comprehensive introductions to rigorous foundation of stacks the reader may consult [L-MB] and the Appendix of [V]. A very detailed treatment of the theory of algebraic stacks can be found in the forthcoming book [ B E F F G K ] . We now introduce the inertia stack associated to a stack X,' which plays a central role in the orbifold Chow ring of smooth Deligne-Mumford stacks. D e f i n i t i o n 1.2.4 Let X be a smooth Deligne-Mumford stack. The inertia stack IX associated to X is defined to be the fiber product IX in the following cartesian diagram: IX >• X A X * X x X, where A: X —> X x X is the diagonal morphism. R e m a r k 1. The objects in the category underlying IX can be described as 6 follows: Ob(IX) = {{x,g)\x G Ob{X),g G Aut{x)} = {(x,H,g)\x G Ob(X),H C Aut(x),g a generator of H}; 2. For a stack X, IX is isomorphic to the stack of representable morphisms from a constant cyclotomic gerbe to X, IX S JJ HomRep(B/j,R,X). 3. There is a natural projection q : IX —> X. O n objects we have q{{x,g)) = x. The inertia stack IX is in general not connected, but it is always smooth. We write I X = I I for the decomposition of IX into a disjoint union of connected components. Here T is an index set which represents the conjugate classes of the local group of the stack X, i.e. (<7i) and (52) are equivalent if the local group H2 is a subgroup of the local group Hi. Among all components there is a distinguished one (indexed by 0 G T)XQ = X. There is a natural involution I : IX —> IX defined by I((x,g)) — (a:,*? - 1). E x a m p l e Let X be of the form [M/G] with M a smooth variety and G a finite group. Then the index set T is the set {(g)\g G G} of conjugacy classes of G. In this case the centralizer C(g) acts on the locus M9 of (/-fixed points and m G M corresponds to (m,g). We have X^ = [M9/C(g)] and the distinguished component is [M/C(id)] = [M/G]. The morphism 1^ is an isomorphism between X^ and X(g-iy If G is abelian, then IX = JJ [ M » / G ] . 7 1.3 The Orbifold Chow Groups In this section we define the notion of orbifold cohomology group. From [CR1] (see also [AGV1], [AGV2]), for each component X^ of IX, the age age(X^) is defined as follows: Let (x, g) 6 X{g)- Let rg be the order the g as an element in the local group. The tangent space TXX is decomposed into a direct sum (Bo<i<rg ^ of eigenspaces according to the ^-action, where Vi is the eigenspace with eigenvalue Cr g ;0 < ' < rg-> a n d Cr s = exp(27r^~~~^)• The age is defined to be age{X(g)) := — ^ /• dimcVi. 9 0<l<rg It is easy to see that this definition is independent of choices of (x,g) G <f(9)-We have the following Proposition from the definition of the age. P r o p o s i t i o n 1.3.1 ([CR1], [AGV1J) age(X^) + age(X^g-i^) = dim^X — dim^X^. • D e f i n i t i o n 1.3.2 ([CR1],[AGV1]) The additive orbifold Chow group of the Deligne-Mumford stack X is defined by (s)er R e m a r k 1. In general, the Chow group of a stack can be defined as the Chow group of a geometric realization of the simplicial scheme associated to this stack. For our purpose we define the Chow group of a Deligne-Mumford stack as the Chow group of its coarse moduli space. These two definitions are equivalent if the coefficients are without torsion; 2. The ages on the components of the inertia stack give a grading on the orbifold Chow group. The notion of age comes from physics called "fermion number"or called "degree shifting number" in the sense of Chen and Ruan. Orbifold Poincare pairing: According to [CR1], Section 3.3, the orbifold Poincare pairing is defined as follows: For a G A*(X^),b G A*(X(g-i)), define (a,b)orb := / a A I*b. The orbifold Poincare pairing pairs cohomology classes from a component X^ with classes from the isomorphic component X(g-iy The fact that it is a non-degenerate pairing follows from the fact that the usual Poincare pairing on A*(X^) is non-degenerate. 1.4 Orbifold Cup Product The orbifold cup product is defined by the genus zero, degree zero orbifold Gromov-Witten invariant from which we can define the obstruction bundle. First we recall the basic degree zero orbifold Gromov-Witten theory. D e f i n i t i o n 1.4.1 A n n-pointed orbifold nodal curve (C, £ i , • • • , £ „ ) is a diagram u?=i £ * c — - C c, where (1) C is a proper Deligne-Mumford stack with coarse moduli scheme C; 9 (2) ( C , 7 r ( £ i ) , • • • , 7r(S n)) is an n-pointed nodal curve; (3) Over the node xy = 0 of C, C has an etale chart [xy = 0//j,r] where the action is given by (x,y) i—> (£x,£~^y)', (4) Over a marked point 7r(S,) of C, C has etale chart where the action is given by u i—> £u and is the substack defined by u = 0. After appropriately defining n-pointed twisted nodal curves (and morphisms of n-pointed twisted nodal curves) over an arbitary base scheme, we have the fol-lowing theorem. T h e o r e m 1.4.2 ([Co]). The category of n-pointed twisted nodal curves M°fn is a smooth Artin stack of dimension 3g — 3 + n. Let X be a Deligne-Mumford stack with projective coarse moduli scheme X. D e f i n i t i o n 1.4.3 A n n-pointed twisted stable map (C X, S i , • • • , S n ) is a dia-gram where (1) (C, S i , • • • , S „ ) is an n-pointed orbifold nodal curve; (2) / is representable and the induced map on coarse moduli spaces / is stable. 10 Let Ni(X) be the group of numerical equivalence classes of curves in X, and let N+(X) := Nf(X) be the monoid of effective classes. Then for /? G N}(X) and for g a non-negative integer, one says that (C -^ ->- X, S i , • • • , S n ) has degree /3 and genus g if the stable map / does. After appropriately defining n-pointed orbifold stable maps (and morphisms of n-pointed orbifold stable maps) over an arbitrary base scheme, we have the fol-lowing theorem. T h e o r e m 1.4.4 ([AGV1]). The category of n-pointed orbifold stable maps of genus g and degree (3, MG,N(X, /?), is a proper Deligne-Mumford stack. The coarse moduli space MG,N(X,/3) of MG,N(X,/3) is projective. To define the orbifold cup product, we are only interested in the moduli stack - M o , 3 ( ^ 0 ) of 3-pointed orbifold stable maps of genus zero and degree zero. There is a natural result in [CR1] and [AGVI] . P r o p o s i t i o n 1.4.5 ([CR1],[AGV1J) The moduli stack MQ$(X, 0) is isomorphic to the double inertia stack I2X of X. R e m a r k The objects in the category underlying I2X can be described as follows: Ob(I2X) = {(x,(gi,g2))\x G Ob(X),gi,g2 € Aut(x)} = {(x,(9i,92,93))\x G Ob(X),gi,g2 G Aut(x),g3 = (sitf)) - 1 }, where (31,92) or {g\,g2,g3) represent the conjugacy classes of the 2 or 3 tuples. Like the inertia stack, the double inertia stack I2X is in general not connected as well. We write I*X = JJ X(gi,92,g3) (.91,92,93) € T 2 11 for the decomposition of I2X into a disjoint union of connected components. Here T 2 is an index set which represents the conjugate classes of the conjugate classes triples (51,92,33) in the local group. There are evaluation maps et : I2X —> IX defined by {x,(m,92,93)) '—> (x,(9i)) for 1 < i < 3. Consider the following diagram of universal maps j * (1.1) 3*0,3(* ,0) , where C is the universal curve and / is the universal map. D e f i n i t i o n 1.4.6 ([CR1],[AGV1]) The obstruction bundle E — • MQ,3(X,0) over the double inertia stack is defined by E := Rl-K*f*Tx. R e m a r k If the universal curve C = [D/H] is a global quotient, we can describe E in a more concrete way: write n : D —> M.o,s(X,0) and / : D —> X for the composite map, then E = (Rl7f*f*Tx)H is the H invariant subbundle of the usual obstruction bundle of / . This is useful in figuring out some examples. We classify the obstruction bundle for abelian Deligne-Mumford stacks. Let X be an abelian Deligne-Mumford stack, i.e. all the local groups are abelian. Let 12 g — (91,92,93) and ^(g) be a component in the double inertia stack I2X. Let be the obstruction over it defined in 1.4.6. Let e : X^ —> X be the embedding. Then T X \ X ^ } = TXG © N(XG/X), where N(XS/X) is the normal bundle of XG in X. Let H be the group generated by g-[,g2,9z, then C = [D/H], where D is smooth Riemann surface. The group H acts on the normal bundle N(XS/X). Since H is abelian, all the irreducible representations are one dimensional, so N(XS/X) can be represented as direct sum of line bundles based on the representations of H. We write N(XG/X) = © ^ L j . So from the above Remark, we have Let a(gi) + a(g2) + a(g3) = YT=i a «> t n e n « i = 1 or 2 since 0i#203 = L Then we have the following proposition due to [CH]. P r o p o s i t i o n 1.4.7 We have that ( F 1 (D,0D)®Li)H = U ifa>i = 2 and {HX{D, 0D)® E { S ) = (R'KJ*TX) = (H'(D,OD)®TX\XM) II (H1{D,OD)®N{XJX)) Li)H = 0 ifoi = \ . So E(s) - © Li-• P r o p o s i t i o n 1.4.8 ([CRlj) Over the component X ^ g j g 3 y the dimension of the obstruction bundle E(gug2,g3) is given by 3 dimCE(g,,52,93) = d i m c X ( g u g 2 ! g 3 ) + age(X[gi)) - dimcX. (1.2) • 13 Definition 1.4.9 Let e3 = I o e3 : I2X —> IX be the composite map. For a,8 6 A*rb(X), the orbifold cup product is defined by a U o r 6 8 := e 3*(eia U U e(E)), where e(E) is the Euler class of the obstruction bundle. Theorem 1.4.10 ([CRl],[AGVlj) Under the cup product in Definition 1.4-9, the graded Chow group A*rb(X) is a skew-symmetric associative ring. • In the following special case, we can compare the orbifold cup product with the ordinary cup product of A*rb(X). Let q : IX —> X be the natural map defined by (x, (g)) i—> x. Proposition 1.4.11 For a 6 A*(X) and 8 € A*(Xg), the orbifold cup product a ^orb 3 is equal to the ordinary product q*a U 8 in A*rb(X). P R O O F . Using the identification of .Mo,3(<-f,0) with the double inertia stack, the component <*(i,9,9-i) is isomorphic to the component X^ and <-f(9-i) in the inertia stack. So the evaluation map e\= q and e2 = id on the component X(g)- From the definition of e%, we can take the map as the identity on X(gy From the formula (1.2), the obstruction bundle ^ ( i ) 9 , g - i ) has dimension zero. So from the definition of the orbifold cup product in Definition 1.4.9, a l ) o r b 8 = q*all8. • 14 E x a m p l e We consider the weighted projective stack * = P(4,6) := [(A 2 \ {0}) /G m ] with the action (x;y) i—> (tAx;t6y). This is the moduli stack M\,\ of 1-marked elliptic curves. We have a description of the inertia stack: 2 4 I(X) = x \ J x \ j B p 4 [ J B p 6 . i=i i=i The usual cohomology of X is A*(X) = Q[t]/(t2). Let A be the generator of the cohomology of Bfj.4 and B the generator of the co-homology of Bp%. The age of A is ^, and that of of B is ^. We can present the orbifold Chow ring as A * m = Q L Y , A , J ? , T ] orbK ) ( X 2 - 1, AT, BT, A2 - XT, B3 - XT, AB)' 1.5 Gale Duality for Finitely Generated Abelian Groups We review the basic form of Gale duality and its application to toric geometry as in [BCS]. Given n vectors b\, • • • ,bn which span Qd, there is a dual configuration [oi, • • • , an] e q(n~d)xn such that is a short exact sequence; see Theorem 6.14 in [Zagier]. The set of vectors o i , • • • , an in q n ~ d is uniquely determined up to a linear coordinates transformation in Q n d . This duality plays a role in study of smooth toric varieties. Let £ be a fan with n rays such that the corresponding toric variety X ( S ) is smooth. Let N = Z d is the lattice 15 in £ , then the minimal lattice points b\,--- ,bn generating the rays determine a map 3 : Zn —> N. By tensoring with Q, we obtain a dual configuration a\, • • • ,an. Since X ( E ) is smooth, we have a, G Zn~d and the set a i , • • • , a „ is unique up to unimodular (determinant ± 1 ) coordinate transformations of Zn~d. Abbreviating Homz(-,Z) by (•)*, it follows that the set o i , • • • ,an defines a map 3 y : (Zn)* —• Zn~d = Pic{X) and the short exact sequence (1.3) becomes 0 —> N* (Zn)* A Pic(X) 0; see Section 3.4 in [F]. The idea of the notion of toric Deligne-Mumford stack is to generalize the above Gale duality of 3 : Zn —> N for free abelian group N to finitely gener-ated abelian group N. The construction of toric Deligne-Mumford stack under this generalization gives rise to finite abelian gerbe structures over the underlying toric orbifolds coming from the torsion part of the finitely generated abelian group JV. Let N be a finitely generated abelian group with rank d. Let B:Zn -^rN be a map determined by n integral vectors {b\, • • • , bn} in N. Taking Zn and N as Z-modules, from the homological algebra, there exist projective resolutions E and F of Z " and N satisfying the following diagram E ^ Zn 0 F *• N. 16 Let Cone(B) be the mapping cone of the map between E and F. Then we have an exact sequence of the mapping cone: 0 —> F —¥ Cone(B) —> E{1) —> 0, where E[l] is the shifting of E by 1. Since E is projective as Z-modules, so we have the exact sequence by taking the Hom(—,Z) functor 0 — • E[l]* —• Cone(8)* —> F* — • 0. Taking cohomology of the above sequence we get the exact sequence: N* A (Zn)* A Hx{Cone(j3Y) —> Ext^N, Z) —> 0. (1.4) D e f i n i t i o n 1.5.1 Let DG{8) = Hx{Cone{8)*). The map Bw : (Zn)* —> DG(B) is called the Gale dual of the map 8. From [BCS], both DG(B) and By are well defined up to natural isomorphism. Actually we can make this construction more clear. Since N has rank d and Z " is a free Z-module, the projection resolutions can be chosen as: 0 — • Z™ — • 0 = E, 0 — * Z r A Z d + r —> 0 = F, where Q is an integer matrix. Then there is a map from Z " to Z d + r defined by a matrix S which gives the map between E and F. The mapping cone Cone(B) is given by the following complex: 0 _ > Z " + r [ ^ ] Z d + r —> 0 = Cone(B). 17 Then we apply the snake lemma to the following diagram to get the sequence (1.4). o • o • (zd+ry ——• (zd+ry — * o [B,Q]* -> ( Z " + r ) * -(1.5) -»• (Zr)*(<7) • 0. Then DG(/3) = ( Z " + r ) * / J m ( [ B , Q]*) and /? v is the composite map of the inclusion (Z")* «-> (Z"+ r)* and the quotient map (Z"+ r)* —>• (Zn+T)*/Im([B,Q]*). R e m a r k If N is free, i.e. there is no torsion part in the group N. Then from (1.5), the Gale dual /3V is the quotient map (Z n )* — • (Zn)* / Im([B]*) and we have an exact sequence N * P\ Next we give two propositions in [BCS] for later use. P r o p o s i t i o n 1.5.2 Let (3 : Z™ — • N be a map to a finitely generated abelian group N. Then / ? w = fi is and if only if the cokernel of ft is finite. Moreover in this case, ker(/3v) = N*. • P r o p o s i t i o n 1.5.3 Given a commutative diagram 0 • Z™1 >• Z " 2 • Z " 3 • 0 0 ->• iV 2 *• AT3 -> 0, m which the rows are exact and the columns have finite cokernels, then there is a commutative diagram with exact rows: 0 • (Z" 3 )* • (Z" 2 )* • (Z" 1 )* • o -> £>G(/3 3 ) • DG(B2) • DG(fii -+ 0. • 18 1.6 Toric Deligne-Mumford Stacks In this section we introduce toric Deligne-Mumford stacks in the sense of Borisov-Chen-Smith. Let iV be a finitely generated abelian group. Let N —> N be the natural map modulo torsion. Then N is a lattice. Let £ be a simplicial fan in the lattice N with n rays {pi, • • • , pn}- Choose n integer vectors {by, • • • , bn} such that bi generates the ray pi for 1 < i < n. Then we have a map 8 : Z™ —> N determined by the vectors {b\, • • • ,bn}. We require that 8 has finite cokernel. D e f i n i t i o n 1.6.1 ([BCS]) The triple S := ( N , £ , / ? ) is called a stacky fan. We define toric Deligne-Mumford stack from a stacky fan a. Since 8 has finite cokernel, then from Proposition 1.5.2 and 1.5.3, we have the following exact sequences: 0 —>• DG(B)* Zn A N —> Coker(p) —)• 0, (1.6) 0 —»• N* —•> Z n A JDG()8) — » Coker{By) —•> 0. (1.7) Since C x is a divisible as a Z-module, taking i ? o m z ( - , C x ) to the exact sequence (1.7) we get: l ^ p ^ G ^ { C x ) n — > T — > 1 , (1.8) where p = Homz(Coker(8v),Cx) is finite, G = Homz(DG(8),Cx) and T is the d dimensional torus (Cx)d. Let C[z i , • • • , z„] be the coordinate ring of the affine variety A " . Associated to the simplicial fan S, there is an irrelevant ideal J s generated by the elements: / j J z i i C T G s V (1.9) Let Z := A " \ V ( J S ) . Then Z is a quasi-affine variety. The torus ( C x ) " acts on Z naturally since Z is a subvariety of A " . The algebraic group G acts on the variety Z 19 through the map a in the exact sequence (1.8). Then we have a translation groupoid Z x G Z. The associated stack is the quotient stack [Z/G] which is the fibre category of principle G-bundles. Definition 1.6.2 ([BCS]) The toric Deligne-Mumford stack X(E) associated to the stacky fan £ is defined to be the quotient stack [Z/G]. Proposition 1.6.3 ([BCS]) The coarse moduli space of the toric Deligne-Mumford stack X(Tj) is the toric variety X(1j) associated to the simplicial fan S . • In [BCS], the authors proved that the morphism Z x G —> Z x Z defined by the projection and action is a finite morphism so that the quotient stack [Z/G] is Deligne-Mumford. We give a new explanation here. From the exact sequence (1.8), let G = Im(a), then we have an exact sequence 1 —> fj, —->• G —> G —> 1. Since all the groups are abelian, this sequence is a central extension. Then from [DP], the quotient stack [Z/G] is the /x-gerbe over the quotient stack [Z/G] determined by this central extension. Let A , o rfc(S) be the underlying toric orbifold associated to the simplicial fan £ and {6] , • • • , & „ } in N. Remark From Proposition 4.6 in [BN], any Deligne-Mumford stack is a /u-gerbe over an orbifold for a finite group /z. Our results are the toric case of that general result. Proposition 1.6.4 ^for(,(S) is isomorphic to the quotient stack [Z/G]. P R O O F . From the remark before Proposition 1.5.2, we have the following exact sequences 0 —> DGQ5) - 4 Z " A l ^ Cok(p) — • 0; 20 0 —> N* —> (I/1)* •£->• DG{0) — • 0; which are the two exact sequences on page 19 in [F]. So Ad-i{X(E)) = DG(6) and from the construction of Cox [Cox], A' o rft(S) = [Z/G]. • R e m a r k Since the toric Deligne-Mumford stack X(E) is a /u-gerbe over the toric orbifold A ' O J . ^ X ) which is Deligne-Mumford and separated, from the standard stack theory (see [L-MB]), the stack X{Y.) is Deligne-Mumford. 1.7 Finite Abelian Gerbes over Toric Deligne-Mumford Stacks In the last section we know that any toric Deligne-Mumford stack is a finite abelian gerbe over the toric orbifold. In this section we talk about when a finite abelian gerbe over toric Deligne-Mumford stack is again a toric Deligne-Mumford stack. Given a toric Deligne-Mumford stack <-f (S) with stacky fan X . Let v be a finite abelian group, and let Q be a v-gexbe over X{Ti). We give a sufficient condition so that Q is also a toric Deligne-Mumford stack. We have the following theorem: T h e o r e m 1.7.1 Let A' (S) be a toric Deligne-Mumford stack with stacky fan X . If for any rays p\,P2,Pz> there exists a cone a € S such that pi,P2,p3 Q <r, then every v-gerbe Q over #(£) is induced by a central extension 21 i.e., we have a Cartesian diagram: Q BG * BG. In general, the v-gerbe Q is not a toric Deligne-Mumford stack. But if the central extension is abelian, then we have: C o r o l l a r y 1.7.2 If the v-gerbe Q is induced from an abelian central extension, it is a toric Deligne-Mumford stack. 1.7.1 The proof of main results Consider the ideal in (1.9), let V = V ( J s ) . From [Cox], the codimension of V in C " is at least 2. L e m m a 1.7.3 In the simplicial fan S, the following two conditions are equivalent: (a) Codim(V,Cn) > 3; (b) For any rays pi,P2,P3, there exists a cone a G S such that Pi,p2,p3 ^ a. P R O O F , (a) =>• (&): We prove the following claim: If there exist pi,p2,P3 such that for any a G S, there is some pi <£. a, then Codim(V, C") < 3. Since It is easy to see that V{z\) n V{z2) n V(z$) C V. From the condition of the claim, dim(V{z1) n V(z2) fl V(z3)) > n - 3, so dim(V) > n - 3. such that r < 3, and p; ^ cr^ . This means that there exist pi,p2,P3 such that for any a G S, there is some pi a. A contradiction to (6). • (a): Suppose not, then d im(y ) > n — 3. So there exist p i , • • • , pr 22 L e m m a 1.7.4 IfCodim(V,Cn) > 3, then H\Z,v) = H2(Z,v) = 0, where v is the sheaf of abelian groups on C " with etale topology. P R O O F . Consider the following exact sequence: 0 —>• ff£(C",i/) —-> H°{Cn,v) —> H°(Z,v) —+ —> H^v{£n,v) — > J f 1 ( C , » - ^ H \ Z , v ) —»• Since Codim(V,C") > 3, i f y ( C n , v ) = 0 for z = 1,2,3, so from the exact sequence and H1 (C™, i/) = 0 for all z > 0 we prove the lemma. • T h e p r o o f o f T h e o r e m 1.7.1 Consider the following diagram Z *pt [Z/G] BG which is Cartesian. Consider the Leray spectral sequence for the flbration 7r: HP(BG, RqTr*v) => Hp+H[Z/G],v) So = pt x G H"(Z,v) = [H"(Z,v)/G]. When p = 2,q = 0, R°ir,v = v because Z is connected, HP(BG, Rqir*v) = H2(BG, v); When p = l,q = 1, Rlir*v = [H1(Z,v)/G], so Hp(BG,RH*v) = H1 (BG, H1 (Z, u)), but C o d * m ( V , C n ) > 3, from Lemma 2.2, Hl{Z,v) = 0, we have H"{BG,R^ir,v) = 0; When p = 0,g = 2, R2ir,v = [H2(Z,v)/G], so HV(BG,RH*v) = H°(BG,H2(Z, v)), also from Lemma 2.2, H2(Z,v) = 0, we have Hp(BG,Rq7T^) = 0. So we get H2([Z/G},u) =H2(BG,v) 23 Since for the finite abelian group v, the z^-gerbes are classified by the second coho-mology group with coefficient in the group v. Theorem 2.4.1 is proved. • The proof of Corollary 1.7.2 Let X(E) = [Z/G]. The v-gerbe Q over [Z/G] is induced from a u-gerbe BG over BG in the following central extension 1—>v —>G —> G —)• 1, where G is an abelian group. So the pullback gerbe over Z under the map Z —> [Z/G] is trivial. Using the groupoid representation of the stack [Z/G]. The v-gevbe over [Z/G] determines an extension Z x G ^ + Z x G where G is the central extension of G by v. The stack [Z/G] is this v-gerbe Q over [Z/G]. Consider the commutative diagram: " ^ G ' (1-10) G ( C X ) " - ^ ( C X ) ™ where a is the map in (2.3). From (1.10), we have ker(a) = ker(tp) <g> ker(a). So we have the exact sequence like (2.3): v ® n —> G -2+ (C T where T is the torus of the simplicial toric variety X(E). Since the abelian groups G , G and ( C x ) " are all locally compact topological groups, taking Pontryagin duality 24 and Gale dual, we have the following diagrams: 0 • N* • Z " DG{8) • Coker(0y) > 0 id Pip 0 • N* + Z " - ^ U DG(0~) > Coker((0)v) • 0, -> DG(py 0 -> DG(py -+ iV • Coker(p) • 0 id -> Coker(p) -> 0, where p v is induced by <p in (1.10) under the Pontryagin duality. Suppose 0 : Z " —> N is given by {b\, ••• ,bn}, then £ := (N,E,0) is a new stacky fan. The toric Deligne-Mumford stack X(S) = [Z/G] is the v-gerbe Q over X(Z). • R e m a r k From the proof of Corollary 1.2, if a z/-gerbe over A"(S) comes from a gerbe over BG and the central extension is abelian, then we can construct a new toric Deligne-Mumford stack, see the example below. 1.7.2 An example E x a m p l e Let £ be the complete fan of the projective line, N = Z © Z / 3 Z , and 0 : 1 ? —> Z® Z / 3 Z be given by the vectors {&] = (1,0),b 2 = (-1,1)}. Then S = (N, E,P) is a stacky fan. We compute that (/5)v : Z 2 —> DG{0) = Z is given by the matrix [3,3]. So we get the following exact sequence: 1 M3 [3,3V ^X\2 Y cx I ( i . i i ) The toric Deligne-Mumford stack # ( £ ) = [C 2 — {0} /C x ] , where the action is given by X(x,y) = (A 3 x, X3y). So <Y(E) is the nontrivial /i3-gerbe over P 1 coming from 25 the canonical line bundle over IP1. Let Q —> X(Yi) be a /U2-gerbe such that it comes from the /U2-gerbe over BCX given by the central extension 1 M2 — (•)2 1. (1.12) 1. From the sequence (1.11) and (1.12), we have: 1 ^ M 3 ® M 2 ^ C X [ ^ ( C X ) 2 The Pontryagin dual of C x ^ ( C x ) 2 is (/?)v : I? — • Z which is given by the matrix [6,6]. Taking Gale dual we have: 0 : Z 2 — • Z © Z 6 , which is given by the vectors {&] = (1,0), 6 2 = (—1,1)}. Let £ = (iV, S, /?) be a new stacky fan, then we have the toric Deligne-Mumford stack ^f(S) = [C 2 — {0} /C x ] , where the action is given by A(x, y) = (A6a;, X6y). So X(T,) is the canonical /U6-gerbe over If the /^2-gerbe over BCX is given by the central extension C x x / i 2 ^ > C> (1.13) where a is given by the matrix [3,0]. Then from (1.11) and (1.13), we have 1 — > M3 ® M2 where </? is given by the matrix x / x 2 - ^ ( C x ) 2 — X C * 3 0 3 0 . The Pontryagin dual of <p is: (/3')v : Z 2 — • Z © Z 2 which is given by the inverse of the above matrix. Taking Gale dual we get J? : Z 2 — • Z © Z 3 © Z 2 , 26 which is given by the vectors {61 = (1,0,0),6 2 = (-1,1,0)}. So £i' = ( # ' , £ , / ? ' ) a stacky fan. A n d X{^) = [ C 2 — { 0 } / C x x /Z2], where the action is (Ai, A2) • (x,y) (Xlx,Xly). So g' = # ( £ ' ) is the trivial M2-gerbe over # ( £ ) . A n d X{fi) % #(£') 27 Bibliography [ A G V l ] D . Abramovich, T . Graber and A . Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, math.AG/0603151. [AGV2] D . Abramovich, T . Graber and A . Vistoli, Algebraic orbifold quantum product, in Orbifolds in mathematics and physics (Madison, WI,2001), 1-24, Contem. Math. 310, Amer. Math. S o c , 2002. math.AG/0112004. [ B E F F G K ] K . Behrend, D. Edidin, B . Fantechi, W . Fulton, L . Gottsche, and A . Kresch, Introduction to stacks, in preparation. [BN] K . Behrend and B . Noohi, Uniformization of Deligne-Mumford curves, math.AG/0504309. [BCS] L . Borisov, L . Chen and G . Smith, The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc. 18 (2005), no . l , 193-215, math.AG/0309229. [CH] B. Chen and S. 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McKay , Graphs, singularities, and finite groups, AMS, Proc. Symp. Pure Math. Vol . 37, 183-186, 1980. [R] Y . Ruan, Cohomology ring of crepant resolutions of orbifolds, in Gromov-Witten theory of spin curves and orbifolds, 117-126, Contem. Math. 403, Amer. Math . Soc , 2006, math.AG/0108195. [S] R. P. Stanley, Combinatorics and commutative algebra, 2nd ed., Birkhauser Boston, 1996. [Uribe] B . Uribe, Orbifold cohomology of the symmetric product, Comm. Anal. Geom. (to appear), arXiv:math.AT/0109125. [V] A . Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math., 97 (1989) 613-670. [Zagier] G . M . Ziegler, Lectures on polytopes, Graduate Texts in Math. 152, Springer-Verlag, New York, 1995. 30 Chapter 2 Simplicial Toric Stack Bundles i 2.1 Introduction In this chapter we explicitly compute the orbifold Chow ring of toric stack bundles. These are bundles over a smooth base variety B whose fibers are toric Deligne-Mumford stacks of Chapter 1. From [Cox], to a simplicial fan £ with n rays, one can associate a simplicial toric variety X ( £ ) expressed as a quotient Z/G, where Z is an open subset of C n and G is a subgroup of (<CX)". Let T := ( C x ) " / G be the torus acting on X ( E ) . Given a principle T-bundle E ->• B, one can form a fibre bundle EX(E) := E xr X(E) -» B over B with fibers isomorphic to X(E). The cohomology ring of EX(T,) was computed in [SU] . Generalizing Cox's construction, Borisov, Chen and Smith [BCS] constructed toric Deligne-Mumford stacks reviewed in Chapter 1. Let P —> B be a principal ^ h e content, of this chapter has been accepted by Illinois Journal of Mathematics for publication. 31 (C x )"-bundle over a smooth variety B. The group G acts on the fibre product P X( C x)n Z via the map a. Define pX(Y,) to be the quotient stack [(P X( C x)n Z)/G) which we write as P X( C x)n ^ ( S ) . The fibre bundle pX{Yi) —> B is called a toric stack bundle over B whose fibre is the toric Deligne-Mumford stack ^ ( E ) . Both Cox and Borisov-Chen-Smith's construction used the minimal presen-tation of a toric variety (stack) as a quotient. One may expect that toric Deligne-Mumford stacks can be represented as a quotient stack of a larger space Z by a larger group G. For example, the classifying stack B^3 = [pt/ps] is a toric Deligne-Mumford stack in the sense of Borisov, Chen and Smith, where the corresponding stacky fan is (Z3,0,0). The stack 8^13 is isomorphic to the stack [ C x / C x ] , where C x acts on <CX by Aa; —> X3x. Given a line bundle L —> B over B. Applying the construction above yields a /Z3-gerbe [ ( L x x c * C x ) / C x ] over B which is nontrivial if L is. The presentation [ p £ / p 3 ] only produces trivial gerbes. Motivated by the study of gerbes, the above discussion suggests that it is desirable to work with other presentations of toric Deligne-Mumford stacks. For this purpose, we introduce the notion of extended stacky fans. A n extended stacky fan is a triple S e := (TV, S, /? e), where TV and 2 are the same as in the stacky fan S , but 0e -. Z m —> TV is determined by {b\,..., bn} and additional elements {& n+i, • • - , bm} in TV. A n extended stacky fan S e has an underlying stacky fan S . Using S e we define a quotient stack ^ ( S e ) := [Ze/Ge], where Ze = Z x ( C x ) m ~ " and Ge acts on Ze through the homomorphism a e : Ge ——> ( C x ) m determined by the extended stacky fan. We prove that <Y(S e) is isomorphic to the toric Deligne-Mumford stack X{S). So enlarging the presentation from the minimal ones of Cox and Borisov-Chen-Smith is encoded in the extended stacky fan. For example, let TV = Z3, let 0e : Z —> TV be the map defined by 61 = 1 G Z 3 , then E e = (TV,E,/? e ) is 32 an extended stacky fan. (Note that this is not a stacky fan). The toric Deligne-Mumford stack is ^ f ( £ e ) = [ C x / C x ] which is isomorphic to {pt/ps]. Let P —> B be a principal ( C x ) m - b u n d l e . The group Ge acts on the fibre product P X( Cx)m Ze via the map a e . The quotient stack pX(Y^E) := [{P X( C X )m Ze)/Ge] is called a toric stack bundle over B whose fibre is isomorphic to the toric Deligne-Mumford stack X{HE). In [BCS], Borisov, Chen and Smith computed the orbifold Chow ring of toric Deligne-Mumford stacks. The computation in the special case of weighted projective stack was pursued in [Jiangl]. In this paper we compute the orbifold cohomology ring of p X(Y,E). To describe the result, we introduce line bundles for 9 G M = N*. For 6 G M, let x° • ( C x ) m —> <CX be the map induced by 6 o j3e : Z m — r T L . The bundle & — • B is the line bundle P x ^ C . We introduce the deformed ring A*(B)[N]*e = A*{B) <g> Q [ i V ] s e , where Q [ i V ] s e ':= ® c £ A f Qyc, y is a formal variable and A*(B) is the Chow ring of B. The multiplication of Q [ i V ] s e is given by: ( yci+c2 if there is a cone a G S such that c\ G a, c2 G a , (2.1) 0 otherwise. Let 7 J ( p £ e ) be the ideal in A*(B)[N]^e generated by the elements: ( c i ( & ) + £ W y M , (2-2) \ i=i / oeM and A*orb ( p ^ ( £ e ) ) the orbifold Chow ring of the toric stack bundle p ^ ( £ e ) . Theorem 2.1.1 If pX(£e) —> B is a toric stack bundle over a smooth variety B whose fibre is the toric Deligne-Mumford stack X{Y,E) associated to an extended 33 stacky fan E e , then we have an isomorphism of Q-graded rings: The extra data {bn+\, • • • , bm} in the extended stacky fan S e do affect the structure of p X{T,e), but do not affect its orbifold cohomology ring. When the variety B is a point, this formula is the orbifold Chow ring formula of projective toric Deligne-Mumford stacks by Borisov-Chen-Smith. To prove this theorem, we show that twisting by the ( C x ) m -bundle P does not "twist" the components of the inertia stack of the toric Deligne-Mumford stack X(Y,e). Thus we can describe the components of the inertia stack l ( p X(T,e)) of p X(T,e) using Box(Ee), in a manner analogous to [BCS]. This makes it possible to use the similar methods as in [BCS] to determine 3-twisted sectors, obstruction bundles and compute the orbifold Chow ring of p ^ ( £ e ) . As an example, let N be a finite abelian group and /3e : Z —> N any homomorphism, then S e = (N,0,f3e) is an extended stacky fan. The toric Deligne-Mumford stack is X(£e) = Bfx, where /J, — Hom(N,C<). Twisting this toric Deligne-Mumford stack by a line bundle L over a smooth variety B gives a /x-gerbe X over B. So no matter the gerbe is trivial or not, our result gives that H*orb(X, Q) = H*(B, Q) ® H*orb(B^ Q) . This chapter is organized as follows. In Section 2.2 we introduce extended stacky fans and their associated toric Deligne-Mumford stacks. In Section 2.3 we define toric stack bundles and discuss their properties. In Section 2.4 we describe the orbifold Chow ring of toric stack bundles. In Section 2.5 we discuss the /u-gerbe X mentioned above. Finally, in Section 2.6 we give some applications to crepant resolutions. We generalize a result of Borisov, Chen and Smith, showing that the orbifold Chow ring of a toric stack bundle and the Chow ring of its crepant resolution 34 can be put into a fiat family. 2.2 A New Quotient Representation of Toric Deligne-Mumford stacks. In this section we introduce extended stacky fans and construct a new representation of toric Deligne-Mumford stacks. Let J V be a finitely generated abelian group of rank d and J V the lattice generated by J V in the d-dimensional vector space JVQ : = J V ®i Q. Write b for the image of b under the natural map J V —> JV. Let S be a rational simplicial fan in NQ. Suppose pi,... ,pn are the rays in S. We fix b{ € J V for 1 < i < n such that bi generates the cone pi. Let {bn+\, • • • ,bm} C J V . We consider the homomorphism (3e : Z m —> J V determined by the elements { & ] , . . . , b m } . We require that j3e has finite cokernel. D e f i n i t i o n 2.2.1 The triple S e := (JV, S , / 3 e ) is called an extended stacky fan. It is easy to see that any extended stacky fan S e = (JV, S, /?e) naturally deter-mines a stacky fan £ : = (JV, S, /?), where /? : Z n —>• J V is given by {b\,..., bn}. Now since /3 e has finite cokernel, by Proposition 2.2 in [BCS], we have exact sequences: 0 — • DG(/3e)* — • Z m A J V —> Coker((3e) —> 0, 0 —> J V * —> Z m ( ^ \ V DG(/3e) —»• C7o/cer((/3 e) v) —+ 0, where ( / 3 e ) v is the Gale dual of /? e . As a Z-module, C x is divisible, so it is an injective Z-module, and hence from [Lang], the functor Homz(—,Cx) is exact. We get the exact sequence: 1 —> Homz(Coker({f3ey),Cx) —• Homz{DG{pe),C*) —• Homz(Zm,Cx) 35 —• Homz(N*,Cx) — • 1. Let p, := Hom%(Coker((0e)v),Cx), we have the exact sequence: 1 —> p — • G e ( C x ) m —>• T — • 1. (2.3) From [BCS], the toric Deligne-Mumford stack ^f(S) = [Z/G] is a quotient stack, where they use the method of quotient construction of toric varieties [Cox]. Define Ze := Z x ( C x ) m _ ™ , then there exists a natural action of ( C x ) m on Ze. The group Ge acts on Ze through the map ae in (2.3). The quotient stack [Ze/Ge] is associated to the groupoid Ze x G e =} Ze. Define the morphism ip : Ze x G e —> Ze x Ze to be <p(x,g) = (x,g • x). Since Z e = Z x ( C x ) m - n , we can mimic the proof the Lemma 3.1 in [BCS] to show that <p is finite which means that the stack [Ze/Ge] is a Deligne-Mumford stack. L e m m a 2.2.2 The morphism tp : Ze x G e —> Ze x Ze is a finite morphism. D e f i n i t i o n 2.2.3 For an extended stacky fan S e = (N, S , /? e ) , we denote the quo-tient stack [Ze/Ge] by ^ ( S e ) . P r o p o s i t i o n 2.2.4 For an extended stacky fan S e = (N,T,,8e), the stack X(Ee) is isomorphic to the toric Deligne-Mumford stack X(E) associated to the underlying stacky fan S . P R O O F . From the definitions of extended stacky fan S e and stacky fan S , we have the following commutative diagram: • z n — • z m > 1 n —n 0 0" 0 • N • _i fL+ N r 0 -36 From the definition of Gale dual, we compute that DG([3) = Zm~n and /3V is an isomorphism. So from Lemma 2.3 in [BCS], applying the Gale dual and the Homz( — ,Cx) functor to the above diagram we get: 1 • G Ge • ( C x ) m ~ " • 1 a ae a (2-4) 1 > ( C x ) n • ( C x ) m > (C*)m-n > 1. We define the morphism ipo : Z —> Ze = Z x ( C x ) m ~ ™ to be the inclusion defined by z H—> (z,l). So (</?o x <pi,<po) : (Z x G =S Z) —• (Ze x Ge =$ Z e ) defines a morphism between groupoids. Let ip : [Z/G] —> [Ze/Ge] be the morphism of stacks induced from (ipo x ipi,ipo). From the above commutative diagram we have the following commutative diagram: Z x G ^ Z e x G e (s,t) (5,4) Z x 2 *- Ze x Z e . In (2.4), 5 is an isomorphism, which implies that the left square in (2.4) is carte-sian. So the above commutative diagram is cartesian and tp : [Z/G] —> [Ze/Ge] is injective. Given an element (z\,..., z n , zn+i,..., zm) G Ze, there exists an element ge g ( C X ) m - n s u c h t h a t g e . (z^ . . . , Z n + 1 ) . . . = ( 2 ] , . . . , 2:„, 1, . . . , 1). From (2.4), <?e determines an element in Ge, so tp is surjective. We conclude that the stacks X(Ee) and X(Y,) are isomorphic. • R e m a r k In view of Proposition 2.2.4, we call X(Ee) the toric Deligne-Mumford stack associated to the extended stacky fan £ e . Let X(E) be the simplicial toric variety associated to the simplicial fan £ in the extended stacky fan S e . We have the following corollary: 37 C o r o l l a r y 2.2.5 Given an extended stacky fan S e , the coarse moduli space of the toric Deligne-Mumford stack ^ ( S 6 ) is also the simplicial toric variety -X"(£). As in [BCS], for each top dimensional cone a in S, denote by Box(a) to be the set of elements v G N such that v = ] C P i c < 7 a A f ° r some 0 < a; < 1. The elements in Box(a) are in one-to-one correspondence with the elements in the finite group N(a) = N/Na, where N(a) is a local group of the stack X(Ee). If r C a is a low dimensional cone, we define BOX(T) to be the set of elements in v G N such that v = ^2PicT aib~i, where 0 < Oj < 1. It is easy to see that BOX(T) C Box(a). In fact the elements in BOX(T) generate a subgroup of the local group N(a). Let Box(Ee) be the union of Box(a) for all d-dimensional cones a G S. For v \ , . . . , v n G TAT, let . . . ,tJ„) be the unique minimal cone in £ containing v\,... ,vn. 2.3 The Toric Stack Bundle p ^ ( S e ) . In this section we introduce the toric stack bundle p A ' ( S e ) and determine its twisted sectors. Let P —> B be a principal ( C x ) m - b u n d l e over a smooth variety B. Through the map ae in (2.3), Ge acts on the fibre product P X ( C x ) m Ze. D e f i n i t i o n 2.3.1 We define the toric stack bundle pX(?je) — • B to be the quo-tient stack ^ ( f f ) : = [ ( P x ( P ) m f ) / G £ ] . (2.5) Let cj) '• <^m — • <^m be the map given by i—>• for 1 < i < n, and ej i—> ej + Yli=i for n + 1 < j < m, where a] E Z . Then from the following 38 commutative diagram -> 17 -> 0 -> 0 id 0 -> N id ->• 0 we obtain a new extended stacky fan S e = (N,T,,0e), where the extra data in £ e are b'n+1 = + < + 1 6 ; , • • • , b'm = bm + V - = ] ^ 6 * . The map 0 gives a map C n x ( C x ) m - n —> Cn x ( < C x ) m _ T l which is the identity on the first factor and given by (f> on the second factor. Since the map in the above diagram doesn't change the fan in the extended stacky fans, we have a map <pa : P X( Cx)m Ze —> P X(Cx)m Ze. We may use the same argument as that in Proposition 2.3 to prove that pX(Yie) = PX(Se). This means that we can always choose the extra data {bn+\, • • • , bm} so that bj — £)™=] aibi for j = n + 1, • • • ,m and 0 < aj < 1. These extra data are actually in the Box(Ee). E x a m p l e By above, the extra data can be chosen to lie in Box(Ee). In this example we prove that they can not be put into the torsion subgroup of N. Let N = 1 and b\ = 2,b2 = —2. Then S = {61,62} is a simplicial fan in NQ. Let S e = ( J V ; £ , / 3 e ) , where 0e : Z 3 — • Z is determined by {61,62,63 = 1}- We compute that DG(0e) = 1? and the Gale dual (/? e) v : Z 3 —> Z 2 is given by the 1 1 0 matrix . The toric Deligne-Mumford stack is A"(S e ) = [(C 2 - {0}) x - 1 0 2 C X / ( C X ) 2 ] , where the action is given by (Xi,X2)(x,y,z) = (AiAjT 1 • x, A] • y, A2, • z). We get ^ ( S e ) = P 1 x [ C x / C x ] = P 1 x Bfi2. Now let S e = (N,Z,0~e), where 0e : Z 3 —> Z is determined by {61,62,63 = 0}, then we compute that DG(0e) = Z 2 0 Z 2 39 1 1 0 and the Gale dual (/? e) v : Z 3 — • Z 2 © Z 2 is given by the matrix 0 0 1 0 0 0 The toric Deligne-Mumford stack is x ( z e ) = [(C 2 - {0}) x C X / ( C X ) 2 x p2], where the action is given by ( A i , X 2 , ^ ( x , y , z ) = (X\ • x,X-\ • y,X2 • z). We get ( £ e ) = [PVA*2] = P 1 x Bp2. Let B = P 1 and P = <CX © <CX © Q(-l)x, then p ^ ( S e ) is a nontrivial p2-geibe over P 1 x P 1 coming from the line bundle C9 P i x pi (0, — 1). Let Q = C ( n i ) x © 0(n2)x © 0 ( n 3 ) x , then ^ ^ ( S 6 ) is the trivial ^ 2-gerbe over the F - b u n d l e E over P 1 . So pX(Y,e) is not isomorphic to QX'Y?) for any Q . From Corollary 2.2.5, A"(S e ) has the coarse moduli space X(E) which is the simplicial toric variety associated to the simplicial fan S. From the exact sequence in (2.3), a ( C x ) m - b u n d l e over B naturally determines a T-bundle over B. Let E —> B be the principal T-bundle induced by P, then we have the twists pXred(Ee) —>• B with fibre the toric orbifold Xre(iCEe) and EX(T,) —> B with fibre the simplicial toric variety X ( E ) , where pXred{T,e) := [ ( P x p ) ™ Ze)/G% EX{'S) := B x r I ( S ) , and G e = 7m(a e ) in (2.3). We obtain the exact sequence: 1 —• p — • G e G e —> 1. (2.6) From [DP], we have: Proposition 2.3.2 pXi^T) is a p-gerbe over F Xred[Yje) for a finite abelian group p. Because any toric stack bundle is a /x-gerbe over the corresponding toric orbifold bundle and can be represented as a quotient stack, we have the following propositions: 40 Proposition 2.3.3 The simplicial toric bundle EX(T,) is the coarse moduli space of the toric stack bundle p X(*£,e) and the toric orbifold bundle pVfred(£e) • P R O O F . The toric stack bundle p X(Y,e) is a /x-gerbe over the simplicial toric orbifold bundle p Xre(i(Tie) for a finite abelian group fi. The stacks pX(Y:e) = [(P X(c*)m Ze)/Ge] and pXred('Ee) — [ ( P x p j m Z ' J / G ' j are quotient stacks. Taking geometric quotient, we have the coarse moduli space (P X(Cx)m Ze)//Ge = (P x Ze)//(Cx)m x Ge. From Corollary 2.4 X ( E ) = = Ze//Ge, so £ x r (Ze//Ge) = (P x ( C x ) m T ) x T (Ze//Ge) = (P x Ze)//(Cx)m x G e . From the universal geometric quotients in [KM], is the coarse moduli space of pX{-£e) and pXred{Y,*). • Proposition 2.3.4 The toric stack bundle FX(Ee) is a Deligne-Mumford stack. P R O O F . From (2.5), p ^ ( S e ) = [(P x ( € x ) m Ze)/Ge] is a quotient stack, where Ge acts trivially on P. The action of Ge on Ze has finite, reduced stabilizers because the stack [Ze/Ge] is a Deligne-Mumford stack, so the action of Ge on P X(Cx)m Z e also has finite, reduced stabilizers. From Corollary 2.2 of [Ed], pX{T,e) is a Deligne-Mumford stack. • For an extended stacky fan S e , let cr G S be a cone, we define link(a) := {r : a + T G S, a n r = 0}. Let {pi,...,/5/} be the rays in link(a) and s := |cr|. Then S e / c r = (N(a) = N/ Na,Y,/a,j3e(a)) is an extended stacky fan, where /?e(cr) : Z m ~ s —> N(a) is given by the images of 6, for pi not in a under N —• N(a), here {b\,... ,b{\ generate the quotient fan S/cr, all others are extra data. From the construction of toric 41 Deligne-Mumford stacks, we have X(T,e/a) := [Ze(a)/Ge(a)], where Ze(a) = (A1 -V ( J S / J ) x ( C x ) m " s - ( = Z(a) x ( C x ) m - S - ' , Ge(a) = Homz(DG{pe(a)),Cx). We have an action of ( C x ) m on Ze(a) induced by the natural action of ( C x ) m _ s on Ze{a) and the projection ( C x ) m —> ( C x ) m _ s . We consider pX(Sya) = [(P x ( C x ) m ( C x ) m " s x ( c x ) m - s Ze(a))/Ge(a)} = [(Px(c,)mZe(a))/Ge(a)}. Then we have: Proposition 2.3.5 Let a be a cone in the extended stacky fan ~Ee, then pX(Y,e/a) defines a closed substack of pX(Ee). P R O O F . Let We(a) be the closed subvariety of Ze defined by J(a) :=< Zi : pi C a > in C[zi, ...,zn, z ± l v z ± ) } , then we see that We(a) = W(a) x ( C x ) m - " , where W(a) is the closed subvariety of Z defined by J(a) : = < Zi : pi C a > in C [ z \ , z n ] . From [BCS], there is a map tpo : W(a) —> Z(a) which is (C x)"-equivariant. We define the map We(a) —> Ze(a) by ipa x 1. From the following diagram: 0 0 -> 0 -> N -» N{a) - -»• 0, applying Gale dual yields 0 0 jm-s i (0eM)v -> 0 (/3e)v -+ DG(B%cr)) -> DG(/3e) -> 0. The dimension of the cone dim(a) is s since the cone cr is simplicial, then iV f f = Z s and DG((3) = 0. Applying # o m z ( - , C x ) functor we get: 1 • 1 • G e Ge(a) > 1 a«(a) (2.7) 1 -* (C> ( C x ) m 42 -> ( C > -> 1. From (2.7), <pi is an isomorphism. From the definition of W(a), we have that W{a) 3 Z{a) x ( C x ) n - l ~ s . So We(a) = Z(a) x ( C x ) " - ' - * x ( C x ) ™ - " = Ze{a) and [We(a)/Ge] =< [Ze(a)/Ge(a)}. Since We(a) is closed subvariety of Z x ( C x ) m " " , so the stack X{Y,e/a) is a closed substack of X(Se). Twisting it by the bundle P , we have a map ipo : P X ( C x j m We(a) — • P X ( C x ) m Z e (cr). So we get a map of groupoids: ipo x <p\ : P X ( C x ) m W e(<7) x Ge —• P X ( C x ) m Ze(a) x Ge(a) which is Morita equivalent. So we have an isomorphism of stacks [(P X ( C x ) m W{a))/Ge] = [(P x ( C x ) m Ze{a))/Ge{a)}. Since W e (or) is a subvariety of Ze, and P x ( C x ) m FP^e(cr) is a subvariety of P x ( C * ) m Z e , so [ ( P X ( C x)m We(a))/Ge] is a substack of [(P X ( C x ) m Z e ) / G e ] = pX(Y,e). So p ^ ( E e / ^ ) is a closed substack of p * ( E e ) . • R e m a r k The proof is similar to Proposition 4.2 in [BCS] where the authors used the notion of quotient stacky fan. We found that the exact sequences they wrote down there are wrong. We write the correct sequences here and found that the quotient stacky fan is naturally an extended stacky fan. Note that the underlying stacky fan in the quotient extended stacky fan E e / c r is the quotient stacky fan in the sense of [BCS]. R e m a r k From [BCS], W(a) = Z < 9 u ' " ' 9 r > for some group elements g\ , •• • ,gr e G. From Proposition 2.2.4, the toric Deligne-Mumford stack [Ze(a)/Ge(a)] is isomor-phic to the stack [Z{a)/G{a)]. Let g-[,--- ,gr still represent the elements in Ge through the map yx in (2.4). Then We{cr) = ( Z e ) < 9 i 9 r > . P r o p o s i t i o n 2.3.6 Let F X(Yie) —> B be a toric stack bundle over a smooth va-riety B whose fibre X(Y,e) is the toric Deligne-Mumford stack associated to the 43 extended stacky fan £ e , then the r-th inertia stack of this toric stack bundle is i r { p x { v * ) ) = Jj p x & y o - ( v u - - - , v r ) ) . ( w i , " , U r ) € B o x ( £ e ) r P R O O F . From (2.5), pX(Yf) = [(P x ( C x ) m ZE)/GE] is a quotient stack. Because GE is an abelian group and the the action has finite, reduced stabilizers, we have the r-th inertia stack: Ir (P*(£e)) = n (p x { c * ) m ze)H ]/G° where H is the subgroup in GE generated by the elements g\, • • • , gr • From Lemma 4.6 in [BCS], there is a map from Box(Ee) to G. So from the map \p\ in (2.4), we have a map p : 73oa:(Ee) —> GE such that p(v) = g(v). For a r-tuple (vx, • • • ,vr) in the - B o x ( £ e ) , from Proposition 3.5 and the above Remark, we have: PX(Sy<T(vlr--,Vr)) - [P X { C x ) m (ZE)H/GE]. Taking the disjoint union over all r-tuples in J3oa;(E e) we get a map: 1>: U PX(Ve/a(vx,--- ,vT)) —>lr{pX(i:e)). , i v ) e f l o x ( £ e ) r The toric stack bundle pX(Y,e) locally is the product of a smooth variety with the toric Deligne-Mumford stack X(Ee). From [BCS], the map ip is an isomorphism locally in the Zariski topology of the base B, so ip is an isomorphism globally. We complete the proof of the Proposition. • R e m a r k For any pair (u],t>2) 6 B o a ; ( S e ) 2 , there exists a unique element v3 G Box(£e) such that V1+V2 + V3 G N. This means that in the local group N/Na(y1>y2y the corresponding group elements g\,g2,gs satisfy <?i<?2<?3 = 1. So this implies that o-{v\,V2,v3) = a(v\,V2)- In fact, Proposition 2.3.6 determines all 3-twisted sectors of the toric stack bundle pX(H,e). See also [Jiangl] for the case of toric varieties. 44 2.4 The Orbifold Cohomology Ring. In this section we describe the ring structure of the orbifold cohomology of toric stack bundles. 2.4.1 The module structure on A* r b( pA'(S e)). Let S e be an extended stacky fan, P -> B a ( C x ) m - b u n d l e and pX(Ee) —> B the associated toric stack bundle. Let M = N* be the dual of TV. For 9 G M, let X° : ( C x ) m — • C x be the map induced by 9 o /3e : Z m —> Z. Let & — • B be the line bundle P xxe C . We give several definitions. D e f i n i t i o n 2.4.1 Let A*(B) denote the Chow ring over Q of the smooth variety B. Define the deformed ring A*(B)[N]1:e as follows: A*(B)[N]^e = A*{B) ® Q Q [ / V ] s e , Q [ / V ] s e = ® c e N Q y c , where y is a formal variable. Multiplication is given by (2.1). The deformed ring A*(B)[N]se has a Q-grading defined as follows: if c = Epica(c) a&i, deg(yc) = £ a* G Q. If 7 € A*(B), then deg(Tyc) = deg(j)+deg(yc). D e f i n i t i o n 2.4.2 Let E e = (TV, S, /9 E ) be an extended stacky fan. Let S E be the ring A*(B)[x\,..., xn]/I^, where the ideal Is is generated by the square-free mono-mials {xix • • • x i s : H h pi3 ^ £}. Note that Ss is a subring of A*(B)[N}T'e given by the map Xi 1—> ybi for 1 < i < n. Let { p i , . . . , p „ } be the rays of S e , then each pi corresponds to a line bundle Li over the toric Deligne-Mumford stack ^ ( S e ) . This line bundle can be defined as follow. The line bundle L j on the toric Deligne-Mumford stack X(Y,) is given by the trivial line bundle C x Z over Z with the G action on C given by the 45 i-th component of a : G —> ( C x ) ™ in (2.3) when S e = S. From (2.4), we have: (2.8) G —>Ge ( C x ) " ——>• (C* Definition 2.4.3 For each pi, define the line bundle L , over ^f(Se) to be the quo-tient of the trivial line bundle Ze x <C over Ze under the action of Ge on C through the component of a e such that the pullback component in a through (2.8) is a , . Twisting it by the principal ( C x ) m - b u n d l e P, we get the line bundle Ci over the toric stack bundle p , Y ( X e ) . Let X ( p E e ) be the ideal in (2.2). We first describe the ordinary Chow ring of the toric stack bundle p X(Ee). Lemma 2.4.4 Let pX(Yie) —> B be a toric stack bundle over a smooth variety B whose fibre X{Y,e) is the toric Deligne-Mumford stack associated to the extended stacky fan £ e , then there is an isomorphism of Q-graded rings: T ^ r ) 2 A* (p*(£e)) given by xt i—> c i ( A ) . P R O O F . From Corollary 2.4, let X ( £ ) be the coarse moduli space of the toric Deligne-Mumford stack ^(S e). Let E —• B be the principal T-bundle induced from the ( C x ) m -bundle P. Then from Proposition 3.3, EX(E) is the coarse moduli space of the toric stack bundle pX(Ee). Let OJ be the first lattice vector in the ray generated by b{, then b{ = ^a; for some positive integer /j . The ideal I ( p X e ) in (2.2) also defines an ideal in Sg. From [SU], we have T ( p £ e ) " 46 which is given by xi E(V(pi)), where E(V(pi)) is the associated bundle over B corresponding to the T-invariant divisor V(pi). From [V], the Chow ring of the stack pX(Yie) is isomorphic to the Chow ring of its coarse moduli space EX(T,) given by Cl(d) >—> l~x • E{V(Pi)). Then we conclude by ci(&) + E L i 8(ai)hybi = Now we discuss the module structure of A*orb ( p A " ( S e ) ) . Because E is a simplicial fan, we have: L e m m a 2.4.5 For any c € N, let a be the minimal cone in E containing c, then there exists a unique expression PiCa where mt € Z>o, and v € Box (a). • L e m m a 2.4.6 Let r is a cone in the complete simplicial fan E and {p-[,..., ps} C link(r). Suppose p\,... ,ps are contained in a cone a C E . Then aUr is contained in a cone of E . P R O O F . Using the following result, see [F]. Let p \ , . . . ,ps be rays in the complete simplicial fan E . If for any pi,pj generate a cone, then pj, • • • ,ps generate a cone. • P r o p o s i t i o n 2.4.7 Let pX(Tie) —> B be a toric stack bundle over a smooth va-riety B whose fibre X(Ee) is the toric Deligne-Mumford stack associated to the extended stacky fan X e , then we have an isomorphism of A* (p^(Xe))-modules: w £ B o x ( S e ) 0 A* (pX(^/a(v))) [deg(y*)} A*(B)[Nf I ( p S e ) 47 P R O O F . From the definition of A*(B)[N]se and Lemma 2.4.5, we see that A*(B)[N]^' © „ g B o a . ( E . ) y° • ST.- Since I ( p S e ) is the ideal in A*(B)[N]^e defined in (2.2). Then @veBo*(V)yv • J ( P s e ) i s t h e i d e a l Z ( P S e ) i n @veBox(v>)yv • = A * ( B ) [ T V ] s e . So we obtain the isomorphism of A*( p ^(S e ) ) -module s : For any v G Box(Ee), let cr(u) be the minimal cone in S containing iJ. Let /0] , . . . , pi G link(a(v)), and /5j be the image of pi under the natural map TV —> N(a(v)) = TV/TV f f ( 1 J ) . Then 5 E / F R ( ^ C A * ( £ ) [ J V ( c r ( y ) ) ] s < > ( « ) i s the subring given by: Xi i—> y b i , for /9j G link(a(v)). Consider the morphism: i : A*(B)[xi,..., x{\ —> A*(B)[x\,... ,xn] given by X{ —> X{. From Lemma 2.4.6, it is easy to check that the ideal IT/O(V) g ° e s to the ideal Is, so we have a morphism S-£/a(v) — • ST,- Since ST is a subring of A*(B)[N]se given by X{ \—> y b i , we use the notations y b i . Let * D : ^ / ^ [ d e ^ g / " ) ] —>• yv • S E be the morphism given by: ybi i—>• • y b i . If E{=i + ci (fj) belongs to the ideal Z ( p £ e / V ( t i ) ) , then = j," - ( f > ( & ^ + c a ( £ 0 ) V where # is determined by the diagram: TV (2.10) N(a(v)) 0 So 6(bi) = #(&j). From the definition of the line bundle we have = We obtain that < M £ J = 1 0(&i)s/6< + c i ( ^ ) ) G y" • Z ( P £ e ) - So induce a morphism * » : Z P ^ T W ' W ) ] - > s u c h t h a t = [y° • J / H 48 Conversely, for such v G £?o:c(Ee) and pi C <r(v), choose #, G Hom(N,Q) such that 6i(bi) = 1 and Oi(bj) — 0 for bj ^ b{ G c(^)- We consider the following morphism p : A* (B) [x\,..., xn] —> A* (B) [ x \ , x { \ , where p is given by: X{ if pi C link(a{v)), ~ E j = i if Pi C cr(u), 0 if pi % a(v) Ulink{a(v)). For any x^ • • • Xis in 1%, also from Lemma 2.4.6 we prove that p(x^ • • • Xis) G I-£/a(v)-We also use the notations ybi to replace Xi. The map p induces a surjective map: 5 S —>• 5 s / f r ( ^ and a surjective map: $ „ : • S s —>• Siyo-o^esCs/'')]. Let • (E"=i 6(°i)ybi +ci(Co)) belong to the ideal j / u • 7 J ( p £ e ) . For 8 G M , we have 9 = 6V + 6'v, where G N(a(v))* = M D criv)-1 and 6(, belongs to the orthogonal complement of the subspace er(tJ)-1- in M. From (2.10), we have: «=1 PiCa(v) \ j=l + c ^ o + E t o A i=l Note that ( E i = i 6v(bi)ybi + ci (&„)) G X ( p £ e / t r ( u ) ) . From the definition of ^ over A"(Se/c7(i7)), ^ = 0. Now let 6>^ = T/pica(v)a^^ where a* G Q, then E p ^ ) to = E P i c ^ ) a A ( M - We have: E P i c ^ ) A ^ ( M ( - E i = i W V J ) + ZPlc*(v) E$=i = 0, so we have £ „ (y« • ( £ ? = 1 + ci(&))) G 7 ; ( p £ e / V ( ^ ) ) -So induces a morphism y« • X ( p £ e ) 7 J ( p £ e / V ( « ) ) 49 Note that ^ v ^ v = 1 is easy to check. For any [yv • ybi] G y „ f ^ ' p | e ^ since yv • ( - E j = i Oi(bj)yb' + E " = i ft(&i)i/6') = y" • we have • ( - £ < = 1 O^y*)] = [yv • y b i ] i w e check that = 1- So $ „ is an isomorphism. From Lemma 2.4.4, for any v G - B o x ( £ e ) , we have an isomorphism of Chow rings: x(pii^jl\v)) ~ A*(pX(Ee/a(v))). Taking into account all the v in Box(Ze) and (2.9) we have the isomorphism: 0 „ S B o x ( £ e ) A* {pX(J:e/a(v))) [deg(yv)} S ^ f f i ^ j f • Note that both sides of (2.9) are 5 s / X ( p S e ) = A*( p A / (S e ) ) -modules , we complete the proof. • R e m a r k In Proposition 5.2 of [BCS], the authors give a proof of Proposition 2.4.7 for toric Deligne-Mumford stacks. We give a more explicit proof of this isomorphism for the toric stack bundle. 2.4.2 The orbifold cup product. In this section we consider the orbifold cup product on A*orb ( p ^ ( S e ) ) . First we determine the 3-twisted sectors of p X(Yie) which are the components of the double inertia stack Z2(pX(Y,e)) of p < Y ( E e ) , see [CR2]. It follows that all 3-twisted sectors of pX(T,e) are: pX(Ve/o-(gl,g2,g3)), (2.11) (.91.92,93)GBox(Ee)3,919293=1 where o'(g1,g2,g3) is the minimal cone in £ containing g\,g2,g3- For any 3-twisted sector p ; t ( £ e ) ( 9 u 9 2 i 9 3 ) = F'X(Eej'a(gx,g2,g3)), we have an inclusion e: p X ( ^ / a ( g „ g 2 , g 3 ) ) ^ p * ( £ e ) because pX(J^e/a(gi,g2,g3)) is a substack of pX(Yie). Let H be the subgroup generated by gi,g2,g3, then the genus zero, degree zero orbifold stable map to pX(Y,e) determines a Galois covering 7r : C —> P 1 branching over three marked 50 points 0 , l ,oo such that the transformation group of this covering is H. We have the definition: D e f i n i t i o n 2.4.8 ( [CM]) The obstruction bundle 0( 9 , , 9 2 , 9 3 ) over pX(Ee/o-(g1,g2,g3)) is defined as the iJ-invariant bundle: (e*T(pX{Ve))®H\C, O c ) ) H • P r o p o s i t i o n 2.4.9 Let p A ' ( S e ) ( 9 ] : S 2 ; S 3 ) = p X(He / a(^i,g2,g3)) be a S-twisted sec-tor of the stack pX(£e). Let gx + g2 + ga = J2Pica(gug2,g3) aibi> ai = 2> t h e n t h e Euler class of the obstruction bundle 0 ( S l , 9 2 , 9 3 ) over P < ; f (S e ) ( g i i f l 2 j f l 3 ) is: II Cl(Ci)\pX(V°/a(g,,g2,g3)), a.i=2 where Ci is the line bundle overpX(Yie) in definition 2-4-3. P R O O F . Let X(Y,e) be the toric Deligne-Mumford stack corresponding to the ex-tended stacky fan S e . Let a (jgx, g2, g3) be the minimal cone in £ containing g3, g2, g3. From (2.11) we have the 3-twisted sector ••f ( E e ) ( 9 l ) 9 2 ) 9 3 ) = X(Ee/cr(g1, g2,g3)) and p ^ ( S e ) ( 9 l , 9 2 , 9 3 ) = pX{Y,°la(gl,g2,g3))- Since e : *(E% l l 9 a , 9 3 ) —• * ( E e ) is an inclusion, we have an exact sequence: 0^TX(Se/a(g„g2,g3)) —> e * T * ( £ e ) — • i V ( ^ ( S e / ( 7 ( 3 l ! 3 2 l f f 3 ) ) / ^ ( S e ) ) - > 0 , where T V ^ ^ / c r ^ , #2, g3))/X(£e)) is the normal bundle of - f ( S e / a ( g 1 , g 2 , g 3 ) ) in A"(S e ) . Since A"(S e ) = [ Z e / G e ] , the tangent bundle T ( ^ ( S e ) ) = [ T ( Z e ) / T ( G e ) ] is a quotient stack. Z e is an open subvariety of A™ x ( C x ) m _ ™ , so T(Ze) = 0^e. 51 Now from the construction of the line bundle Lk over ^f (S e ) , we have a canon-ical map: 0 £ = 1 Lk —>• T(X(Ze)). Since we have a natural map T ( ^ ( S e ) ) —>• N(X('Ee/(7(g-l Jg2,g3))/X('Se)), we obtain a map of vector bundles over X(Ee/a(gl,g2, <p: 0 Lk—>/V(^(Se/a(91,92,53))/A'(Se)). Then from the definition of the line bundle Ck over pX(Ee), we have the map: 0 £k —> N(pX(^e/a(gug2,g3))/pX(^)), PkCo-(g1,g2,g3) where N(pX(-£e/a(g1,g2,g3))/pX(-£e)) is the normal bundle oipX('SB/iT{g1,g2,g3)) in P / Y ( £ e ) . For any point map: x : Spec C X { V e / a ^ g ^ ) ) ^ pX(i:e/a(g^g2,g3)), note that x*<p is an isomorphism, so (p is an isomorphism. We have the exact sequence: 0 _ > T ( p * ( E e M s i , 3 2 > 3 3 ) ) ) — • e*T ( p * ( £ e ) ) —> 0 A - > 0. Now using the result in the proof of Proposition 6.3 in [BCS], we have 0 if ak - 1, dimc{Ck®H\C,Oc))H = I [ l if ak = 2. So from the Definition 2.4.8, we have: e ( ° ( . 9 i ,92,9s)) - II C l ( ^ ) l p * ( E - M 3 i . 3 a . 3 3 ) ) ' a,i=2 • 52 2.4.3 Proof of Theorem 2.1.1 From the definition of the orbifold cohomology in [CR1], we have that A*orb (PX(^)) = ® g e B o x { v e ) A* {pX^/a{g))) [deg(y9)]. From Proposition 2.4.7, we have an isomorphism between A*( p ^ ; f(S e ))-modules: 0 A* {pX{^/a(g))) [deg{y>)] = ^ f f f l ' • gEBox{Y,e) So we have an isomorphism of J 4*( p A'(S e ))-modules: A*orb (pXCEe)) S A * $ [ £ } * ' • Next we show that the orbifold cup product defined in [CR1] coincides with the product in ring A*(B)[N]se/Z(pEe). From the above isomorphisms, it suffices to consider the canonical generators ybi, y9 where g G 2 ? o a ; ( £ e ) and 7 G A*(B). Since hi G N, the twisted sector determined by b{ is the whole toric stack bundle p A ' ( £ e ) , ybi U o r(, 7 is the usual product ybi • 7 in the deformed ring because ybi and 7 belong to the ordinary Chow ring of pX(Ee). For y9 Uorb ybi and y9 U o r(, 7 , where g G Box(Ee). g determines a twisted sector pX(Ee/cr(g)). The corresponding twisted sectors to bi and 7 are the whole toric stack bundle pX(Ee). It is easy to see that the 3-twisted sector corresponding to (g, b{) and ( 3 ,7 ) are p X(Tie)^g^g-\^ = pX(Ee/a(g)), where g~l is the inverse of g in the local group. From the dimension formula in [CR1], the obstruction bundle over p X ( £ e ) ( 9 i l ; f f - i ) has rank zero. So from the definition of orbifold cup product in [CR1] it is easy to check that y9 U o r(, ybi = y9 • ybi, y9 Uorb 7 = y9 • 7 . For the orbifold product y91 Uorby92, where g\,g2 G 5 o x ( S e ) . From (2.11), we see that if there is no cone in £ containing g-\,g2, then there is no 3-twisted sector corresponding to the elements gi,g2, so the orbifold cup product is zero from the definition. O n the other hand from the definition of the group ring A*(B)[N]se, y9i .y92 - Q) s 0 ygi \jorby92 = y 9 1 -y92. If there is a cone a G £ such that gx,g2 G a, let 53 g3 S Box(£e) such that g3 € a(g1,g2) and g\g2g3 = 1 in the local group. Using the same method in the proof of main Theorem in [BCS], we get: y9x DorbV92 = V91 'V92-The theorem is proved. • 2.5 The //-Gerbe. In this section we study the degenerate case of toric Deligne-Mumford stacks. In this case TV is a finite abelian group, the simplicial fan £ is 0. The toric stack bundle is a ju-gerbe X over B for a finite abelian group p. Let TV = Z nj ©• • -®Z ns be a finite abelian group, where p i , •• • ,ps are prime numbers and n\, • • • , ns > 1. Let 0e : Z —> TV be given by the vector (1,1, • • • , 1). TVQ = 0 implies that £ = 0, then £ E = (TV, £ , / ? e ) is an extended stacky fan from Section 2.1. Let n = lcm(p™}, • • • ,p™s), then n = p" M • • -p™ H , where pix, • • • ,pit are the distinct prime numbers which have the highest powers , • • • , riit. Note that the vector (1,1, • • • , 1) generates an order n cyclic subgroup of TV. We calculate the Gale dual ( ^ ) v : Z —> Z © 0 W L , . . . , I ( } Z £ , where £ G ( / 3 e ) = Z © 0 ^ { I L , . . , I T } Z £ . We have the following exact sequence: o — • z —-> z A TV —> 0 Z £ — • 0, 0 — • 0 —»• z z © 0 z * — • z n e 0 Z £ —> 0. So we obtain l _ , / i _ , C x x T J M « i ^ ! > C x _ ^ 1 ) ( 2 . 1 2 ) 54 where the map ae in (2.12) is given by the matrix n 0 and n = f i n x T \ m i u . . . -t} * L0 J AT. The toric Deligne-Mumford stack is X(Y,e) = [ C x / C x x Tii<t{iu-,it} Pp'] = Bp,, the classifying stack of the group p. Let L be a line bundle over a smooth va-riety B and L x the principal C x - b u n d l e induced from L removing the zero sec-tion. From our twist we have LXX(T,e) = L x x c x [ C x / C x x Yl^{iu-,it} Ppf] = [ L x / C x x riigiz! ••• it} which is exactly a p-gevbe X over B. The structure of this gerbe is a ^ n -gerbe coming from the line bundle L plus a trivial n ^ f i j ... it} P'p]' gerbe over B. For this toric stack bundle, Box{Y,e) = N, so we have the following Proposition for the inertia stack. P r o p o s i t i o n 2.5.1 The inertia stack of this toric stack bundle X is p™1 p" s copies of the p-gerbe X. From our main Theorem, we have: P r o p o s i t i o n 2.5.2 The orbifold cohomology ring of the toric stack bundle X is given by: H*orb(X,Q) - H*(B,Q)®HU(BIM,Q), where H*orb(Bpi;®) = ®[tu • • • ,ts}/(ff - 1, • • • X ' ' ~ !)• Let N = Z r , and 0 : Z —> Z r be the natural projection. The toric Deligne-Mumford stack X(£e) = Bpr. Let L —> B be a line bundle, then the toric stack bundle X = i? (£ , r ) is the pr-gevbe over B determined by the line bundle L. We have: 55 C o r o l l a r y 2.5.3 The orbifold cohomology ring of B^Lr^ is isomorphic to H*(B)[t]/(tr— !)• If the variety B is not a toric variety, then the toric stack bundle over B is not a toric Deligne-Mumford stack. But suppose B is a smooth toric variety, then a /i-gerbe X can give a toric Deligne-Mumford stack in the sense of [BCS]. E x a m p l e Let B — FD be the d-dimensional projective space. We give stacky fan S = (TV, £ , / ? ) as follows. Let TV = ZD © Z r and /3 : Z d + 1 —>• TV be the map deter-mined by the vectors: { ( 1 , 0 , 0 , 0 ) , ( 0 ,1 , . . . , 0,0),. . . . ,(0,0, . . . , 1 , 0 ) , ( - 1 , - 1 , . . . , - 1 , Then DG(fi) = Z , and the Gale dual / ? v is given by the matrix [r, r , . . . ,r]. So we have the following exact sequences: 0 —> Z — • Z d + 1 A Z r f © Z r — • 0 — • 0, 0 —> ZD — » Z d + 1 Z —> Z r —> 0. Then we obtain the exact sequence: 1 —>• M r —>• C x - A ( C x ) r f + 1 — * ( C x ) d —>• 1. The toric Deligne-Mumford stack X(Y\) :— [ C d + 1 - { 0 } / C x ] is the canonical p r-gerbe over the projective space Td coming from the canonical line bundle, where the C x action is given by A- ( z \ , z a + i ) = (A r -z\,..., A r -Zd+i). Denote this toric Deligne-Mumford stack by QT = P ( r , . . . , r) . If the homomorphism : Z d + 1 — • TV is deter-mined by the vectors: {(1,0, . . . , 0,0), (0 ,1 , . . . , 0 , 0 ) , . . . , (0 ,0 , . . . , 1,0), ( -1 , - 1 , . . . , - 1 , then DG(f3) = Z@ZR. Comparing to the former exact sequence, we have the exact sequence: 1 —» Hr —> <CX X M r ^ ( C X ) d + 1 —> ( C X ) d — • 1. 56 The corresponding toric Deligne-Mumford stack is the trivial /i r-gerbe ¥ d x Bfir coming from the trivial line bundle over ¥ d . The coarse moduli spaces of these two stacks are both projective space fd. From the Theorem of this paper or the main Theorem in [BCS], the orbifold cohomology rings of these two stacks are isomorphic, although as stacks they are different. R e m a r k Let H represent the hyperplane class of P d , then H*rb(gr, Q) £ Q[H]/(Hd+1) Q[t]/(tr — 1). We conjecture that the orbifold quantum cohomology ring of Qr de-fined in [CR2] is isomorphic to Q[H}/(Hd+1 - f(H,q)) (g> Q[t]/(tr - 1 - g(t,q)), where / , g are two relations and q is the quantum parameter. The orbifold quan-tum cohomology of trivial gerbe case is easy to compute, where f(H, q) = q and R e m a r k We conjecture that the small orbifold quantum cohomology ring of the nontrivial /i r-gerbe and trivial /ur-gerbe over the projective space FD should be dif-ferent. This means that the orbifold quantum cohomology can classify these two different stacks. 2.6 Application. In this section we generalize a result of Borisov, Chen and Smith [BCS] to the toric stack bundle case. Let X ( S ) be a simplicial toric variety, and let be the associated toric Deligne-Mumford stack, where £ = (N, £ , / ? ) is the stacky fan associated to S. Let S' be a subdivision of S such that X(Y,') is a crepant resolution of X(E). Suppose there are m rays in S', let i : ( C x ) n —> ( C x ) m be the inclusion. From the following g(t,q) = 0. 57 commutative diagram: 0 • Z n " o > zm-d > Z m taking Gale dual we get: -»• N- > 0 id -> N • 0, 0 -> N* - i ^ A DG{0 ->• 0 0 • iV* -)• (Z n )* •+ DG(0) >• 0. So applying the i?om functor we have the following diagram: r -id • T. Let P —> B be a principal (C x )"-bundle, we still use P to represent the principal ( C x ) m - b u n d l e induced by i, then they induce the same principal T bundle E over B. So EX(T!) •—>E X(E) is a crepant resolution. A n d EX(E) is the coarse moduli space of the toric stack bundle pX(Y1) from Proposition 3.3. We have the following result. P r o p o s i t i o n 2.6.1 If the Chow ring of the smooth variety B is a Cohen-Macaulay ring. Then there is a flat family S —> P 1 of schemes such thatSo = Spec(A*rb(pX(T:))) and Soc 3 Spec(A*{EX(Z'))). P R O O F . We also construct a family of algebras over P 1 such that the fiber over 0 and oo are A*orb(pXC£)) and A * ( E X ( S ' ) ) respectively. X ( S ' ) is a smooth variety, and {bi, • • • , bn, bn+-[, • • • bm] generate the whole lattice N, then A*(B)[N]S is the quotient ring of the ring S := A*(B)[ybl , ybm] by the binomial ideal determined 58 by (2.1). Let I2 denote this ideal. Let I\ denote the ideal generated by ci(ffl.) + Y^i=\ @j{bi)ybi for 1 < j < d, where 6\, • • • , 6d is a basis of N*. Since S' is a regular subdivision of E , then there is a £ ' - l i n e a r support function h : N — • Z such that h(bi) = 0 for 1 < j < n, h(bi) > 0 for n + 1 < i < m. For any lattice points ci ,C2 lying in the same cone of E , h{c\ + c2) > h{c\) + h(c2), and the inequality is strict unless C] , c2 lies in the same cone of S'. We describe the family over P 1 - {oo}. Let h be the ideal in S[t{\ generated by C ! ( ^ ) 4 ( 6 i ) + YULi Qj{1bi)yhit'{{hi) for 1 < j < d. So the choice of h implies that S ^ ~ £ ~ A* ( p y i y \ \ h + I 2 + < U > ~ <c1(teJ) + Z ? ^ 8 j ( k ) y b ' : l < j < d > + I 2 - ° r b [ 1 ) y The sequence C\(£Q.) + Y^=i^j(bi)ybi for 1 < j < d is also a homogeneous system of parameters on S/I2. The Chow ring A* (B) is a Cohen-Macaulay ring, so 5/72 is also Cohen-Macaulay. So the sequence is a regular sequence. Therefore, the Hilbert function of the family 5[ i i / (7] + I2)} is constant outside a finite set in Q*. O n the other hand, for the family over fx — {0}, let I2 be the binomial ideal in S[*2] given by Ln+ctMci+cJ-hM-hte) i f 3 a G s s u c h t h a t S ] e CTj c 2 G a , I 0 otherwise. From the property of the function h. This product becomes (2.1) for the fan S' over t2 = 0. Hence S[t2]/(h + h+ < t2 >) = A * ( E X ( S ' ) ) . The sequence c i ( £ 0 j . ) + X)r=i @j(bi)ybi for 1 < j < d is a regular sequence on 5 / I2 and 5 / / £ / , and we have the same Hilbert function for 5/(7] + I2) and 5/(7] + 7 S ' ) -There exists an automorphism </? between these two families so that we con-struct such a family over P 1 . The rest of the proof is the same as in [BCS]. We omit the details. • 59 R e m a r k Ruan [R] conjectured that the cohomology ring of crepant resolution is isomorphic to the orbifold Chow ring of the orbifold if we add some quantum corrections on the ordinary cohomology ring of the crepant resolution coming from the exceptional divisors. Let P ( l , l , 2 ) be the weighted projective plane with one orbifold point whose local group is Z 2 . The Hirzburch surface F 2 is the crepant resolution of F ( l , 1,2). 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Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math., 97 (1989) 613-670. [Zagier] G . M . Ziegler, Lectures on polytopes, Graduate Texts in Math. 152, Springer-Verlag, New York, 1995. 63 Chapter 3 Hypertoric Deligne-Mumford Stacks i 3.1 Introduction Hypertoric varieties (cf. [BD], [P]) are the hyperkahler analogue of Kahler toric varieties. The algebraic construction of hypertoric varieties was given by Hausel and Sturmfels [HS]. Modelling on their construction, in this chapter we construct hypertoric Deligne-Mumford stacks and study their orbifold Chow rings. According to [BD], the topology of hypertoric varieties is determined by hyperplane arrangements. In this chapter we define stacky hyperplane arrangements from which we define hypertoric Deligne-Mumford stacks. Let N be a finitely generated abelian group of rank d and N —> N the natural projection modulo torsion. Let /3 : Z m -> N be a homomorphism determined by 1 The content of this chapter has been accepted by Journal fur die reine und angewandte Mathematik for publication. 64 a collection of nontorsion integral vectors - ,bm} C JV. We require that /? has finite cokernel. The Gale dual of j3 is denoted by / 3 V : (%m)* -> DG(/3). A generic element 6 in DG(f3) and the vectors {61, • • • ,bm} determine a hyperplane arrangement H = (Hi, - • • , Hm) in JVj*,. We call A := (JV,/3,6) a stacky hyperplane arrangement. For /3 : Z m —> JV in .4, we consider the Lawrence lifting / 3 L : Z m © Z m - ^ JV L of /? where JV/; is a finitely generated abelian group with rank m + d. The map fit is given by vectors {&L,I, • • • > &L,m ; b'ri,--- , b'L m } C JV/,. The generic element # deter-mines a Lawrence simplicial fan in Ni. We call So = (JV^,So,/?/,) a Lawrence stacky fan and ^(S^) the Lawrence toric Deligne-Mumford stack. The hypertoric Deligne-Mumford stack M (A) associated to A is defined as a quotient stack which is a closed substack of the Lawrence toric Deligne-Mumford stack ^ ( £ # ) , generalizing the construction of [HS]. The stacky hyperplane arrangement A also determines an extended stacky fan S = (JV, £ , / ? ) introduced in [Jiang2]. Here £ is the normal fan of the bounded polytope T of the hyperplane arrangement H. The toric Deligne-Mumford stack ( £ ) defined in [Jiang2] is the associated toric Deligne-Mumford stack of M(A). To the map /3 we associate a multi-fan Ap in the sense of [HM], which consists of cones generated by linearly independent subsets {b^, • • • ,b~ik} in JV for {^ 1, • • • ,h} C {1, • • • ,m}, see Section 3.4. We assume that the supp(Ap) = JV. We prove that each top dimensional cone in Ap gives a local chart for the hypertoric Deligne-Mumford stack M(A). We define a set Box(Ap) consisting of all pairs (v,a), where cr is a cone in the multi-fan Ap, v G JV such that v — YlPicaa^>i ^ o r 0 < cti < 1. For (v,a) G Box(Ap) we consider a closed substack of M(A) given by the quotient stacky hyperplane arrangement A(a). The inertia stack of M(A) 65 is the disjoint union of all such closed substacks, see Section 3.4. We now describe the orbifold Chow ring of M(A). The multi-fan A ^ nat-urally gives a "matroid" Mp. The vertex set is {1, • • • , m}, and the faces are the subsets ,ik} Q {!>••• >m} s u c h that (b^, • • • ,b~ik} are linearly independent in N. Note that the faces of Mp are the cones in Ap. According to [HS], the or-dinary cohomology ring of the hypertoric variety corresponding to the hyperplane arrangement 7i is isomorphic to the "Stanley-Reisner" ring of the matroid Mp. Our result shows that the orbifold Chow ring of hypertoric Deligne-Mumford stacks is a generalization of the Stanley-Reisner ring of the matroid Mp to the multi-fan Ap. Let A^ A ^ denote all the pairs (c, a), where c G N, a is a cone in A ^ such that c = Y^piCa aib~i a n d ai > 0 a r e rational numbers. Then NA& gives rise to a group ring ®[Ap]= © Q-yM, (c,a)eNA0 where y is a formal variable. For any (c, a) G NAff, there exists a unique element (v, r) G Box(Ap) such that T C u and c = v + E p j C u m i ^ i where mj are non-negative integers. We call (V,T) the fractional part of (c,a). For (c,a) we define the ceiling function \c\a by \c\a = E ^ C T ^ * + E f t C a m J ^ - Note that if v = 0, \c~\o- = £ f t c o - m A - For two pairs ( c i , c i ) , (02,172)) if o"i UCT2 is a cone in A ^ , define e (c 1 ; c 2 ) := [calo-j + \c2]„2 - fa + c2] <TiU<T2- Let c r 6 C ( T ] U ( T 2 b e the minimal cone in Ap containing e{c-\,c2) so that (e(c],c2),ae) G NA0. We define the grading on QfA^] as follows. For any (c, a), write c = v + Y2PiCo- m i ° i , then deg(y^) := | r | + £ ™i . where | r | is the dimension of r . By abuse of notation, we write y(bi>p>) as y b i . The 66 multiplication is defined by (_l)kly(ci+c2+e(c,,c2),.7ju<72) if a i U cr2 is a cone in A0 , 0 otherwise. (3.1) Using the property of ceiling functions we check that the multiplication is com-mutative and associative. So QfA^] is a unital associative commutative ring. Let Cir(Ap) be the ideal in QfA/?] generated by the elements: m Y,<bi)yb\ eeN*. (3.2) i=i Let A*rb(M(A)) be the orbifold Chow ring of the hypertoric Deligne-Mumford stack M(A). We have the following Theorem: Theorem 3.1.1 Let M.(A) be the hypertoric Deligne-Mumford stack associated to the stacky hyperplane arrangement A. Then there is an isomorphism of graded Q-algebras: A*orb(M(A)) = Q [ A / 3 ] Cir(Ap) The orbifold Chow ring of the hypertoric Deligne-Mumford stack M.(A) is indepen-dent of the generic element 9. It only depends on the map /?. Theorem 3.1.1 is proven by a direct approach. The inertia stack of a hy-pertoric Deligne-Mumford stack M(A) is the disjoint union of closed substacks M(A(a)) for all (v, a) € Box(Ap). To determine the ring structure, we identify the 3-twisted sectors as closed substacks of M (A) indexed by triples ({v\, o\), {y2, a2), (v3,03)) in Box(Ap)3 such that V] + v2 + W3 € N is a integral linear combination of bi's. We then determine the obstruction bundle over any 3-twisted sector and prove that the orbifold cup product is the same as the product of the ring QfA^] described above. 67 The multi-fan Ap is equal to the simplicial fan £ in £ induced from the stacky hyperplane arrangement A if and only if H has n hyperplanes {H\, • • • , Hn} whose normal polytope is a product of simplices. So in this case £ is a stacky fan and the simplicial fan £ is a product of normal fans of simplices, the toric variety X(T,) is a product of weighted projective spaces. Then by [BD] the associated hypertoric variety is the cotangent bundle of the toric variety X(E). So M(A) ~ T*X(£), the cotangent bundle of the toric Deligne-Mumford stack A'(S) . The ring Q[A^] coincides (as vector spaces) with the deformed ring Q [ i V ] s as defined in [BCS]. Corollary 3.1.2 Let £ be as above. Then there is an isomorphism of Q-vector spaces A*orb(M(A))~A*orb(XCZ)). Here is an example which shows that the orbifold Chow ring of M(A) is not isomorphic as a ring to the orbifold Chow ring of the associated toric Deligne-Mumford stack ^ ( S ) . Consider the weighted projective stack 1P(1,2) which is a toric Deligne-Mumford stack with stacky fan £ = (AT, £ , / ? ) , where N = Z, 0 : Z2 -> N is given by the vectors b\ = (1),&2 = (—2) and £ is the simplicial fan in the lattice N consisting cones p\ and p2 generated by b\ = (1) and b2 = (-2) respectively. The Gale dual map 0y : Z 2 —> Z is given by the matrix (2). Choosing generic element 8 — (1) G Z , we get a stacky hyperplane arrangement A = (N, 0,8). The hypertoric Deligne-Mumford stack M(A) is the cotangent bundle T*P(1,2) whose core is the toric Deligne-Mumford stack P( l ,2 ) . Both Q[Ap] and Q[AT] E are generated by y b l , yb2, and y(^b2<P2), According to Theorem 1.1 and the main theorem in [BCS], their orbifold Chow rings are given as follows: Q[xux2,v] ^Q[v] 4 U ( * ( E ) ; Q ) £ (Xi - 2X2,V2 - X2,VXl,XlX2) (v3)' 68 A*orb(M(A);Q) ^ Q[xi,x2,v] (x\ - 2X2,X\X2,VX\,V <2) (xl,VX2,V2)' It is easy to see that these two rings are not isomorphic. Thus the orbifold Chow ring of a hypertoric Deligne-Mumford stack is not necessarily isomorphic to the orbifold Chow ring of its core. (However, their Chow rings are isomorphic, see Theorem 1.1 of [HS].) This also proves that the orbifold Chow ring has no homotopy invariance property. We remark that the core of a general hypertoric Deligne-Mumford stack can be singular, it is not clear how to define orbifold Chow ring. But in the case of a cotangent bundle over weighted projective space, the core is the weighted projective space and the orbifold Chow ring is well-defined. O n the other hand, the orbifold Chow ring of a Lawrence toric Deligne-Mumford stack is isomorphic to its associated hypertoric Deligne-Mumford stack, see [JT2]. Computations of orbifold cohomology rings of hypertoric orbifolds in sym-plectic geometry have been pursued in [GH]. They used the method in [GHK] and symplectic method to compute the orbifold cohomology. Our method is purely algebraic. It would be interesting to compare these two cohomology rings combina-torially. This chapter is organized as follows. In Section 3.2 we discuss the rela-tion between stacky hyperplane arrangements and extended stacky fans. We de-fine hypertoric Deligne-Mumford stack M(A) associated to the stacky hyperplane arrangement A. In Section 3.3 we discuss the properties of hypertoric Deligne-Mumford stacks. In Section 3.4 we determine closed substacks of a hypertoric Deligne-Mumford stack. This yields a description of its inertia stacks. We prove Theorem 3.1.1 in Section 3.5, and in Section 3.6 we give some examples. 69 Conventions For cones oi,o2 in E D , we use o\ U a2 to represent the set of union of the generators of and a2-3.2 The Hypertoric Deligne-Mumford Stacks In this section we define hypertoric Deligne-Mumford stacks, mimicking the construction of hypertoric varieties in [HS]. 3.2.1 Stacky hyperplane arrangements We introduce stacky hyperplane arrangements. We explain how a stacky hyperplane arrangement gives extended stacky fans. Let TV be a finitely generated abelian group and /? : Z M —r N a map given by nontorsion integral vectors {bi, ...,bm}. We have the following exact sequences: 0 — • DG{B)* A * Z M A JV — • Coker(p) — • 0, (3.3) 0 —> N* — • Z M A L>G(/3) — • Coker(/3Y) — * 0, (3.4) where /3 V is the Gale dual of /? (see [BCS]). The map / ? v is given by the integral vectors {01, • • • , a m } C DG(0). Choose a generic element 8 G DG(fi) which lies in the image of /3 V and let ip := (r i , • • • , r m ) be a lifting of 6 in Z M such that 9 = —{5yip. Note that # is generic if and only if it is not in any hyperplane of the configuration determined by /3 V in £>G(/3)R. Let M = N* be the dual of JV and M M = M ® Z E , then M R is a d-dimensional E-vector space. Associated to 9 there is a hyperplane arrangement H = {Hi, • • • , Hm} in M R defined by Hi the hyperplane H i := {v E AfRI < 6j, v > + n = 0} C M R . (3.5) 70 This determines hyperplane arrangements in MJJ, up to translation. D e f i n i t i o n 3.2.1 We call A := (N,0,9) a stacky hyperplane arrangement. It is well-known that hyperplane arrangements determine the topology of hypertoric varieties [BD]. Let m T = p| Fi, where Ft = {v £ A f H | < buv > +n > 0}. 2 = 1 Let £ be the normal fan of T in MJJ = E D with one dimensional rays generated by b~i, • • • ,bn. By reordering, we may assume that Hi,-- - ,Hn are the hyperplanes that bound the polytope T, and Hn+i, • • • , Hm are the other hyperplanes. Then we have an extended stacky fan £ = (N, £ , / ? ) defined in [Jiang2], where 0 : Z m ->• N is given by {bi, • • • , b n , b n + i , • • • ,bm} C N, and {bn+i, • • • ,bm} are the extra data. B y [Jiang2], the extended stacky fan £ determines a toric Deligne-Mumford stack <f ( £ ) • It is the same stack as in [BCS]. Its coarse moduli space is the toric variety corresponding to the normal fan of T . According to [BD], a hyperplane arrangement H is simple if the codimension of the nonempty intersection of any I hyperplanes is I. A hypertoric variety is the coarse moduli space of an orbifold if the corresponding hyperplane arrangement is simple. E x a m p l e Let V. = {Hi, H2, H3, H4}, see Figure 1. The polytope T of the hyper-plane arrangement is the shaded triangle whose toric variety is the projective plane. The extended stacky fan is given by the fan of the projective plane P 2 and an extra ray (0,1). 71 double 2 1 N 4 Figure 1: The correspondence of the hyperplane arrangement and an extended stacky fan 3 R e m a r k If for a generic element 8 6 DG{8) the hyperplane arrangement Ti bounds a polytope whose normal fan is S, then £ = (N,T,,8) is a stacky fan defined in [BCS]. 3.2.2 Lawrence toric Deligne-Mumford stacks Consider the Gale dual map 8V : Z m -> DG{8) in (3.4). We denote the Gale dual map of where Ni is a lattice of dimension 2m — (m — d). The map Br, is given by the integral vectors • • • , & L , m , b ' L 1 , • • • ,b'Lm} and is called the Lawrence lifting of 8. Given the generic element 8, let 8 be the natural image of 8 under the projection DG{8) -> DG((3). Then the map 8 : Z m -> DGG0) is given by /3V = (ai,--.- ,am). For any basis of DG(0) of the form C = { a ^ , , a i m _ d } , Z m © Z m ( / 3 V l - f V ) £ )G( |9 ) by ^ : Z 2 m -»• JVjr, (3.6) 72 there exist unique A i , • • • , \ m - d such that Ai + • • • + ~a~im_d\m-d = 9. Let C[zi, • • • , zm, wi, • • • , wm] be the coordinate ring of C 2 m . Let a(C,9) = {bLtij \Xj > 0}U{b'Ltij| Xj < 0} and C(9) = {z{j \\j > 0}U{wtj| A,- < 0}. We put 1 0 := <JfJ C(0)\ C is a basis of DG{B)} , (3.7) and £ 0 := {a(C, 9): C is a basis of L>G(/?)}, (3.8) where a(C,9) = {bj^x,--- , & £ , m j i > •"" , & / J ] m } \ c ( G , #) is the complement of <J(C, 9) and corresponds to a maximal cone in So- From [HS], So is the fan of a Lawrence toric variety X(Hg) corresponding to 9 in the lattice NL, and Ig is the irrelevant ideal. The construction above establishes the following: Proposition 3.2.2 A stacky hyperplane arrangement A = (N,f3,9) also gives a stacky fan "Eg = (NL, SO, /?L) which is called a Lawrence stacky fan. P R O O F . From Proposition 4.3 in [HS], So is a simplicial fan in NL- The rays PL,i, PLI a r e generated by b~L,i, b L i . The map PL is the map (3.6) given by • • • , bL,m, b'Lii, ••• , b'L,m}- So by [BCS], S 0 = (NL, So, PL) is a stacky fan. • Definition 3.2.3 The toric Deligne-Mumford stack ^ ( E o ) is called the Lawrence toric Deligne-Mumford stack. For the map ^ : Z m ® Z r a - > DG(fi) given by ( £ v , - / ? v ) , there is an exact sequence 0 —»• N*L —> 1 2 M -A DG{B) — • Coker{fil) —> 0. (3.9) 73 Applying / J o m z ( - , C x ) to (3.9) yields 1 —> n —> G -^-» ( C x ) 2 m —> Ti —> 1, (3.10) where fi := Homz{Coker(6fJ,Cx) and Ti is the torus of dimension m + d. From [BCS] and Proposition 3.2.2, the toric Deligne-Mumford stack X(Eg) is the quotient stack [ ( C 2 m \ V{l0))/G], where G acts on C 2 m \ V{10) through the map o A 3.2.3 H y p e r t o r i c D e l i g n e - M u m f o r d s t a c k s Define an ideal in C[z, w] by: Ipy ••= (^(8y(x)iZiWi\ x G DG(0)*^J , (3.11) where (/?v)* is the map in (3.3) and {By)*{x)i is the «-th component of the vector According to Section 6 in [HS], i^v is a prime ideal. Let Y be the closed subvariety of C 2 m \ V(l0) determined by the ideal (3.11). Since ( C x ) 2 m acts on Y naturally and the group G acts on Y through the map a L , we have the quotient stack [Y/G]. Since Y C C 2 m \ V(lo) is a closed subvariety, the quotient stack [Y/G] is a closed substack of X(Eo), and is Deligne-Mumford. D e f i n i t i o n 3.2.4 The hypertoric Deligne-Mumford stack M(A) associated to the stacky hyperplane arrangement A is defined to be the quotient stack [Y/G]. E x a m p l e Let N — Z ® Z 2 , £ the fan of projective line P 1 , and 6 : Z3 ^ N given by {6] = (1,0), b2 = ( - 1 , 1 ) , h = (1,0)}. Then the Gale dual /? v : Z 3 -> Z 2 is given by the matrix 1 0 1 . Choose a generic element 6 = (1,1) in Z 2 2 2 0 ' which determines the fan £ . The stacky hyperplane arrangement is A = (N,0,6), 74 G = ( C x ) 2 and Y is the subvariety of Spec(C[2i, z 2 , z 3 , wi, w2,103]) determined by the ideal Ipv = {z\W\ + z3w3,2z\W\ + 2z2w2). Then by [HS], the coarse moduli space is the crepant resolution of the Gorenstein orbifold [C 2 /Z3] , see Figure 3. The corresponding hyperplane arrangement H consists of three distinct points on the real line K 1 , and the bounded polyhedron is two segments intersecting at one point. So the core of the hypertoric variety is two P 1 intersecting at one point. The hypertoric Deligne-Mumford stack M{A) is a nontrivial /i2-gerbe over the crepant resolution according to the action given by the inverse of the above matrix. If we change b2 to (—1,0), we will see an example in Section 3.4 that the hypertoric Deligne-Mumford stack is a trivial p2-gerbe over the crepant resolution. 3.3 Properties of Hypertoric Deligne-Mumford Stacks The coarse moduli space Each Deligne-Mumford stack has an underlying coarse moduli space. In this section we prove that the coarse moduli space of M(A) is the underlying hypertoric variety. Consider again the map By : Z m ->• DG(B) in (3.4), which is given by the vectors (a\,--- ,am). As in section 2, let 8 be the natural image of 8 under the projection DG(B) ->• DG(B). Then the map /? V : Z m ->• DG(B) is given by /?v = (ai,--- ,am). From the map /? V we get the simplicial fan in (3.8). By [BCS], the toric variety X(Eo), which is the geometric quotient ( C 2 m — V(lo))/G, is the coarse moduli space of the Lawrence toric Deligne-Mumford stack X(Eg). The toric variety X(Eg) is semi-projective, but not projective. In [HS], from /? v and 8, the authors define the hypertoric variety Y(By ,8) as the complete intersection of 75 the toric variety X(Eo) by the ideal (3.11), which is the geometric quotient Y/G. We have the following Proposition. P r o p o s i t i o n 3.3.1 The coarse moduli space of Ai(A) is Y(0V,9). P R O O F . Let X = ( C 2 m — V(To)). By the universal property of geometric quotients ([KM]), we have that X x.X(xe) Y(0W,9) = Y. From Lemma 3.3 in [JT2], the stabilizers of points in X are the same as the stabilizers of the points in Y, which are determined by the box elements in the Lawrence simplicial fan and extended stacky fan. So we have the following diagram M(AY X(ZB) a Y(0\9Y - X ( E f l ) , which is cartesian. The Lawrence toric variety X(T,g) is the coarse moduli space of the Lawrence toric Deligne-Mumford stack X(Y,o). So M(A) has coarse moduli space Y(0V,O). • R e m a r k In [HS], the authors began with the map 0y, and assumed that iV and DG{0) are free. In our case DG{0) is a finitely generated abelian group, the toric variety X(Eo) is again semi-projective since So is a semi-projective fan. The hyper-toric variety Y(0y ,6) is the complete intersection determined by the ideal (3.11). This reduces to the case in [HS] when N and DG{0) are free. Independence of coorientations of hyperplanes From (3.5), a hyperplane Hi is naturally oriented. Changing the orientation of Hi means changing the map 0 by replacing hi by — 76 P r o p o s i t i o n 3.3.2 M.(A) is independent to the coorientations of the hyperplanes in the hyperplane arrangement % — (Hi,--- ,Hm) corresponding to the stacky hy-perplane arrangement A. R e m a r k Note that changing coorientations does change the corresponding normal fan of the weighted polytope T. P R O O F . It suffices to prove the Proposition when we change the coorientation of one hyperplane, say Hj for some j . Let H' = (Hi, • • • , Hj, • • • , Hm). Then we have a new stacky hyperplane arrangement A! = (N,(3',8), where B' : Z m —> N is given by {bw , bra}. Using the technique of Gale dual in [BCS], it is easy to check that if the Gale dual f3v is given by the integral vectors / 3 V = (ai, • • • , a m ) , then the Gale dual ( /3 ' ) v is given by the integral vectors ( /3 ' ) v = (oi, • • • , -aj, • • • , am). Let i\) : Z m -> Z m be the map given by e; ->• e\ if i ^ j and ej -> -ej, then we have the following commutative diagrams: N • id •N, DG(B') ( Z m ) * •DG(P). Consider the diagram (Z 2m\* ( Z 2 m ) * [/9V,-/3V] [/3'v,-/3'v] DG{B') >DG(B). Applying Homz(-,Cx) yields the following diagram of abelian groups G • •G' ( C x ) X \2m ( C x ) 2 m . (3.12) 77 Recall that Y is a subvariety of C2m\V(le) defined by the ideal Ipv in (3.11). When we change the coorientation of Hj, the ideals do not change, so Y' = Y. By (3.12), the following diagram is Cartesian: Y x G ^ Y ' x G ' (s,t) (s,t) (3.13) Y x Y ^ Y ' x Y ' , where tpo is determined by the map ip. So the groupoid Y x G :=} Y is Morita equivalent to the groupoid Y' x G' =4 Y'. The stack [Y/G] is isomorphic to the stack [Y'/G'], and M{A) S M(A'). • R e m a r k Let £ = (TV, £ , / ? ) be the extended stacky fan induced by A. The toric Deligne-Mumford stack # ( £ ) is the quotient stack [ Z / G ] , where Z = ( C " \ V r ( J s ) ) x ( C x ) m _ ™ as in [Jiang2], and is the square-free ideal of the fan £ . So every hypertoric Deligne-Mumford stack M(A) has an associated toric Deligne-Mumford stack <-f ( £ ) whose simplicial fan is the normal fan of the bounded polytope T in the hyperplane arrangement H. determined by the stacky hyperplane arrangement A. But by Proposition 3.3.2, M(A) does not determine E x a m p l e Consider Figure 1 again. The corresponding toric variety is P 2 . If we change the coorientation of the hyperplane 2, then the corresponding normal fan S of T changes. The resulting toric variety is a Hirzebruch surface. So the associated toric Deligne-Mumford stacks are different. But the hypertoric Deligne-Mumford stacks are the same. 78 3.4 Substacks of Hypertoric Deligne-Mumford Stacks In this section we consider substacks of hypertoric Deligne-Mumford stacks. In particular, we determine the inertia stack of a hypertoric Deligne-Mumford stack. Let A = (N, 8,8) be a stacky hyperplane arrangement and £ = (N, £ , 8) the extended stacky fan induced from A. Let M.(A) denote the corresponding hypertoric Deligne-Mumford stack. Consider the map 8 : 7Lm -> N given by {b\, • • • , bm}. Let Cone(8) be a partially ordered finite set of cones generated by t>i,--- , b m . The partial ordering is defined by requiring that a -< r if a is a face of T . We have the minimum element 0 which is the cone consisting of the origin. Let Cone(N) be the set of all convex polyhedral cones in the lattice N. Then we have a map C :Cone(B) — • Cone(N), such that for any a 6 Cone(B), C(a) is the cone in N. Then Ap := (C, Cone(8)) is a simplicial multi-fan in the sense of [HM]. Closed substacks Recall that in Section 3.2 we have the fan £ # for the Lawrence toric variety corresponding to ±8^. Let A(B) = {6 L , i ,-- - , b L t m , V L A , - • • ,b'Ltm} C 77 L be the Lawrence lifting of 5 = {&],•• • , bm} C N. We have the following lemma. L e m m a 3.4.1 If ag = (b^,--- , ~b~L,ik, b'L^, • • • ,b'L>ik) forms a cone in T,g, then a = (Tjjj, • • • ,bik) forms a cone in Ap. P R O O F . This can be easily proved from the definition of fan T,Q in (3.8). • For a cone a in the multi-fan Ap, let link{a) = {bi : p, + a is a cone in A^}. Then we have a quotient extended stacky fan £/CT = (N(a), £/a, 8(a)), where 79 8(a) : Z ' -> N(a) is given by the images of {6,}'s in link(a). Let s :-dim(Na) = s since a is simplicial. Consider the commutative diagrams 0 > Z l + S > 1 M — > Z m - l ~ s y 0 \a\, then and -+ N —> Zs -> JV l+s ->• 0 -»• z< ->• 0 0 -> JV(cr) >• 0. Applying the Gale dual yields 0 >• zm~l-s -> Z * z ' + s -> 0 (3.14) Z">- ' -» > DG(8) — ^ DG(8) • 0, and 1} Jl + S -> Z s -»• 0 (3.15) 0 • DG(8(a)) DG(8) • DG(Ba) -> 0. Since Z s S JV f f, the Gale dual DG(8A) - 0. A n d again applying the # o m z ( - , C x ) functor to the above two diagrams (3.14), (3.15) yields 1 —->• G • G > (Cx)m~l-S > 1 + (G a X N / + . S (3.16) -> ( C x ) r -> (C L and ->• 1 1 ( C x ) * • (C -+ G(a) -a(a) X N i •+ 1 (3.17) -> (<cx) -»• 1. Since 9 G DG(B), from the map </>i in (3.14) 6 induces (9 in DG(8). From the isomorphism (fo in (3.15), we get 9(a) G DG(B(a)). Then 4^ = (N,8,d) gives ^4(cr) = (N(a), 8(a), 9(a)) whose induced extended stacky fan is S/cr. 80 From (3.14),(3.15) we have the following diagrams Z 2 m ^ Z2(l+s) Z2l. ( / 3 V , - / 3 V ) ( / 3 V 0 DG{B) ^DG(B), ( £ v , - £ v ) DG(J3(a)) *DG(j3). (3.18) Taking Hom%( — , C x ) gives G G G •G(tr) I <*{o-)L (3.19) ( C x ) 2 ( / + 5 ) >- ( C x ) 2 m , ( C x ) 2 ( / + S ) » ( C x ) 2 ' . Let X(a) := ( C 2 ' \ V(lo(<r))) a n d Y(a) the closed subvariety of X(a) defined by the ideal W := { X > W T ( z ) ^ = e D G 0 J ( « 7 ) ) * } , (3.20) where ( £ ( a ) v ) * : DG(B{a))* -> Z ( is the dual map of /i(cr) v and {B(aY)*(x)i the i-th component of the vector (B(a)v)*(x). Then from the definition of hypertoric Deligne-Mumford stacks, we have A^(^4(cr)) = [Y(a)/G(a)]. We have the following result: Proposition 3.4.2 If a is a cone in the multi-fan Ap, then M(A((T)) is a closed substack of M(A). P R O O F . Let TQ be the irrelevant ideal in (3.7). The hypertoric stack M(A) is the quotient stack [Y/G], where Y C X :— ( C 2 m \ V(To)) is the subvariety determined by the ideal Ipv in (3.11). Taking duals to (3.14), (3.15) we get: 0 • DG{BY • DG(B)* • Z m ~ l - S > 0 GSV)* ( / 3 V ) * (3.21) -> Zr -»• o, 81 and 0 • 0 • DG{B)* —=->• DG{B{a))* • 0 ^ ) * JGS(*)V)* (3-22) 0 • Z " -> Z ' + s — A - Z ' • 0, Let W(a) be the subvariety of X defined by the ideal J(a) := (zi,Wi : ^ C cr). Then W(CT) contains the C-points (z,w) G C 2 m such that the cone spanned by {pi : Zi = Wi = 0} containing a belongs to A ^ . From Lemma 3.4.1, the C-point (z,w) in W(a) such that pi <jt a U link(cr) implies that Z{ ^ 0 or u>, ^ 0. It is clear that W(<T) is invariant under the G-action defined by (3.10). Let Via) := YnW(a). Then from (3.11),(3.20) and (3.21),(3.22), V(a) S F(o-) x ( C x ) m - S - ' x ( 0 ) M " S - ' and the components 0 are determined by the choice of the generic element 8. Let </?o : Y(a) —> V(a) be the inclusion given by (z, w) \—> (z, w, 1,0). From the map (pi in (3.19), we have a morphism of groupoids ipo x <P\ '• Y(&) x G(a) =t V(a) x G which induces a morphism of stacks ip : [Y(a)/G(a)] —> [V(a)/G]. To prove that it is an isomorphism, we first prove that the following diagram is cartesian: Y ( a ) x G ( a ) ^ U v ( a ) x G (s,t) (8,t) Y ( a ) x Y ( a ) ^ V ( a ) x V ( a ) . This is easy to prove. Given an element ((z\,wi), (z2,w2)) € Y(a) x Y(a), under the map ipo x ip0, we get ((z\, w\, 1,0), (z2,W2,1,0)) G V(a) x V(a). If there is an element g G G such that g(z\,w\, 1,0) = (z 2 ,u;2,1,0), then from the exact sequence in the first row of (3.16), there is an element g(a) G G(a) such that g(a)(z\,w\) = (z2,w2). Thus we have an element ((z\,wi),g(a)) G Y(a) x G(a). So the morphism ip : \Y(a)/G(a)} —> [V(a)/G] is injective. Let (z,w,s,0) be an element in V(a), then there exists an element g G (Cx)m~l~s such that g(z, w, s, 0) = 82 (z,w, 1,0). From (3.16), g determines an element in G, so ip is surjective and <p is an isomorphism. Clearly the stack [V(a)/G] is a closed substack of M(A), so the stack M(A(a)) = [Y(a)/G(a)} is also a closed substack of M(A). • Open substacks We now study open substacks of M(A). Let a be a top dimensional cone in Ap. Then a = (Zd,a,8a) is a stacky fan, where 8a '• Zd —> N is given by b{ for /9j C a. Since TV has rank d, we find that DG(0a) is a finite abelian group. So in this case the generic element 9 induces zero in DG(Ba). This is the degenerate case, which means that the corresponding ideal (3.11) is zero. Thus Ya = C 2 d . Note that Ga — Homz(DG(8a),Cx) is a finite abelian group. According to the construction of hypertoric Deligne-Mumford stack in Section 3.2.3, the hypertoric Deligne-Mumford stack M(o-) associated to a is the quotient stack [Ya/Ga] which can be regarded as a local chart of the hypertoric orbifold [Y/G]. P r o p o s i t i o n 3.4.3 If a is a top-dimensional cone in the multi-fan Ap, then M(a) is an open substack of M. (A). P R O O F . Since a is a top dimensional cone in Ap, from (3.4) we get a basis C of DG(8). Let Ug- be the open subvariety of C2m\V(Xo) defined by the monomials ]T C{9) in (3.7). Let Vc = Uan Y, i.e. the points in Ua staisfying (3.11). Then we have the groupoid Va x G Va associated to the action of G on Va. It is clear that this groupoid defines an open substack of M(A). Next we show that this substack is isomorphic to M(cr). 83 Consider the following commutative diagram: 0 > zd >• z m >• z m ~ f l ->• 0 0 -+ N id -> 0. Applying Gale dual and i ? o m z ( - , C x ) , we obtain 1 -» ( C x ) x \d G -> ( C x ) r -4 ( C x ) -» 1 -»• (C id X \m—d (3.23) -> 1. We construct a morphism ipo : Y a - ^ V a . For pj a, we set Zj = 1 if Zj is component of a monomial of C(6) in (3.7) or Wj = 1 if IOJ is component of a monomial of C(0) in (3.7), then from (3.11), the corresponding Wj or ZJ can be represented as linear component of {ztWi} for pi C er. Let <£>o : C 2 < i —>• £/,j be the morphism given by Zi, Wi i—> Z{, Wi for pi C cr, and Zj,Wj to the corresponding 1 or linear combination of {ziWi} for pi C cr in the above analysis. Then £>o induces a morphism </50 : Ya —y Va. Hence we have a morphism of groupoids $ := (ip0 x ipo,ipa x i^i) : [Ya x G . z j Yg-] — > [ K x G ^ Va], where ip\ is the morphism in (3.23). This morphism determines a morphism of the associated stacks. The isomorphism of these two stacks comes from the following Cartesian diagram: (s,t) (3.24) Yo- x K > V„ x K • Inertia stacks Let Na be the sublattice generated by a, and N(a) := N/Na. Note that when a is a top dimensional cone, N(a) is the local orbifold group in the local chart of 84 the coarse moduli space of the hypertoric toric Deligne-Mumford stack. Namely: L e m m a 3.4.4 Let a be a top-dimensional cone in the multi-fan Ap. Then Ga — N(a). P R O O F . The proof is the same as the proof for a top dimensional cone in a simplicial fan in Proposition 4.3 in [BCS]. • Recall that G acts on ( C x ) 2 m via the map a L : G ->• ( C x ) 2 m in (3.10). We write <xL{9) = (<*i(9), • • • . a&(ff)> a i + m ( s ) , ' • • . a2m(9))-L e m m a 3.4.5 Let (z,w) £ Y be a point fixed by g £ G. If af'(g) ^ 1, then Zi = Wi = 0. P R O O F . Since G acts on C 2 m through the matrix ByL = [Bv,-By] in (3.9), we have that af+m(g) = a\(g)-1- The Lemma follows immediately. • Given the multi-fan Ap, we consider the pairs (v,a), where a is a cone in Ap, v 6 TV such that v — £ p . C ( T aibi for 0 < on < 1. Note that a is the minimal cone in Ap satisfying the above condition. Let Box(Ap) be the set of all such pairs (v,a). P r o p o s i t i o n 3.4.6 There is an one-to-one correspondence between g £ G with nonempty fixed point set and (v, a) £ Box(Ap). Moreover, for such g and (v, a) we have [Y3/G] = M{A(a)). P R O O F . Let (v,a) £ Box(Ap). Since a is contained in a top dimensional cone r in Ap, we have v £ TV(r). By Lemma 3.4.4, JV(r) = G T . Hence v determines an element in GT. Using the morphism ip\ in (3.23), we see that g fixes a point in Y. 85 Conversely, suppose g G G fixes a point (z,w) in Y, where (z,w) G C 2 m . By Lemma 3.4.5, the point (z,w) satisfies the condition that if (<?) ^ 1 then zi — u>i= 0. From the definition of C2m\V(Ig), there is a cone in containing the rays for which Zi = u>i = 0. B y Lemma 3.4.1, the rays pi for which Z{ = 0 is a cone in A/3 which we call a. So # stabilizes YT — C2d in VT through <po in (3.24) for any top dimensional cone r containing a, and # corresponds to an element (v, a) G Box(Ap). From the definition of W(a) and V(cr) in Proposition 3.4.2, we have W(a) = Y9 and [V(a) /G] = [Y9/G] which is 7W(^l(cr)). • We determine the inertia stack of a hypertoric Deligne-Mumford stack. P r o p o s i t i o n 3.4.7 The inertia stack of M(A) is given by I(M(A)) = M(A(a)). (v,a)eBox(A0) P R O O F . The hypertoric Deligne-Mumford stack M(A) — [Y/G] is a quotient stack. Its inertia stack is determined as I(M{A)) = By Proposition 3.4.6, the stack [Y9/G] is isomorphic to the stack A^(^4(cr)) for some (v,a) G Box(Ap). • E x a m p l e Let S = (TV, X,/?) be an extended stacky fan, where TV = Z 2 , the simplicial 86 extra data Figure 2: The correspondence of the hyperplane arrangement and an extended stacky fan fan £ is the fan of weighted projective plane P ( l , 2,2), and 0 : Z4 N is given by the vectors {bi = (1,0), b2 = (0,1), 63 = (—2, —2), 64 = (0, —1)}, where 61,62, &3 are the generators of the rays in £ . Choose generic element 8 = (1,1) e DG(0) = Z 2 . Then A = (N, 0,8) is the stacky hyperplane arrangement whose induced extended stacky fan is £ . A lifting of 8 in Z 4 through the Gale dual map 0y is r = (1,1, — 3,0). The corresponding hyperplane arrangement H = (Hi, i72, #3,7/4) consists of 4 lines, see Figure 2. Take v = 563, then (v,a) G Box(A.p), where a is the ray generated by 63. Consider the following diagram 0 0 -+ Z z 3 -+ 0 p -¥ N -> z © z 2 - -> 0. We have the quotient extended stacky fan S/cr = (N(a), S/cr, 0(a)), where 0(a) : Z 3 -»• AT(cr) is given by the vectors {(1,0), ( -1,0) , (1,0)}, and (1,0) is the extra data in the quotient extended stacky fan. Taking Gale dual, we get 0 Z 3 ->• 0 -> z 2 © z 2 -»• z 2 -+ 0 -> 0, where 0y is given by the matrix 2 2 1 0 0 1 0 1 and 0(a)v is given by 2 2 1 0 1 0 The associated generic element 8(a) = (1,1,0) and the lifting of 8(a) in Z 3 is r(a) = 87 (1 ,1 , -3) . So the quotient hyperplane arrangement A(a) = (N(a),0(a),8'(cr)) is a line with three distinct points { — 1,1,3}. The bounded polyhedron of this hyper-plane arrangement is two segments intersecting at one point, see Figure 3. i i i -1 1 3 Figure 3: The bounded polyhedron The core of M(A(a)) corresponds to these two segments, hence is two ff^'s meeting at one point. Adding the stacky structure the twisted sector M(A(a)) corresponding to the element v is the trivial /f2-gerbe over the crepant resolution of the stack [ C 2 / Z 3 ] . 3.5 Orbifold Chow Ring of M { A ) In this section we discuss the orbifold Chow ring of hypertoric Deligne-Mumford stacks. We determine its module structure, then compute the orbifold cup product. 3.5.1 T h e m o d u l e s t r u c t u r e We first consider the ordinary Chow ring for hypertoric Deligne-Mumford stacks. According to [K], the cohomology ring of M (A) is generated by the Chern classes of some line bundles defined as follows. Applying Homz(—,Cx) to (3.4), we have 1 —> p —> G -^4 ( C x ) m —> T —> 1. D e f i n i t i o n 3.5.1 For every bi in the stacky hyperplane arrangement, define the line bundle Li over M (A) to be the trivial line bundle Y x C with the G-action on C defined via the «-th component of the morphism a : G —> ( C x ) m in the above exact sequence. 88 For any c G N, there is a cone a €. Ap such that c = £ p . C ( 7 a A where a, > 0 are rational numbers. Let NA& denote all the pairs (c,a). Then NA& gives rise a group ring Q[&p] - © Q - y ( C l < r ) , (c,ff)6WA/3 where ?/ is a formal variable. By abuse of notation, we write j / 6 *^) as ybi. The multiplication is given in terms of the ceiling function for fans which we define below. Since the multi-fan is simplicial, we have the following Lemma. L e m m a 3.5.2 For any c G N, there exists a unique cone a G Ap and (V,T) G Box(Ap) such that T C a and c = v + ^2 mibi PiCa where mi G Z>o'. • D e f i n i t i o n 3.5.3 (v,r) is called the fractional part of (c, a). Now for (c,a) G NA&, from Lemma 3.5.2, we write c = v + J ^ c c r m A > where m '^s are nonnegative integers. We define the ceiling function \c\a by = Z ~ l h i + / ~ l m i h i -P;CT pj Co-Note that if v = 0, \c\a = YlPiCamibi- For two pairs (c\,a\), (02,0-2), if o\ U 02 is a cone in Ap, define e(ci,c 2) := f c i l^ i + \ci\<n ~ \ci + C21<T1U<T2- L E T °~e Q °~\ u °"2 be the minimal cone in A ^ containing t(c\,C2) so that (e(c\,c2),ae) G i V A 0 . The ceiling function \c\a is an integral linear combination of b^s for pi C cr. We define the grading on QfA^] as follows. For any (c, a), write c = v + Y^PiCami°i-> then d e f f ( s / ( c , f l r ) ) : = | r | + £ " » i > p;C<r 89 where | r | is the dimension of r . Let Cir(Ap) be the ideal in Q[A^] generated by the elements in (3.2). The multiplication y^w) • y(c^) i s defined by (3.1). L e m m a 3.5.4 The multiplication (3.1) is associative. P R O O F . For any three pairs (ci, o i ) , (C2, 02), (C3,03), if a\ U 02 U 03 is a cone in Ap, let 0" C o\ U o"2 U 03 be the minimal cone in Ap containing e (c i ,C 2 ,C 3 ) := \c{\ai + \c2}a2 + \c3]a3 ~ [~cl + C2 + C3I Ua2U<73, such that (e(ci,C2,C3),a) G i V A 0 . Then we check from the properties of ceiling function that (y(cu<n). yfaw)). y(c3,<T3) a n c q y{cx,^). (y(c2,*2). y(c3,*3)j a r e both equal to It is easy to check that the product preserves the grading, and the proof is left to readers. • Consider the map 8 : Z m —• N which is given by {b\,--- ,bm}. We take {1, • • • , m) as the vertex set. The matroid complex Mp is defined using 8 by requiring that F G Mp iff the normal vectors {b~i}i£F are linearly independent in N. The Stanley-Reisner ring of the matroid Mp is (_l)kly(ci+c2-i-c3+€(ci,c2,c3),<7iu<72U<T3) if cr•] u o~2 U CT3 is a cone in Ap otherwise. Q[Mp] = where IM0 is the matroid ideal generated by the set of square-free monomials {y»n . . . ybik , • • • ,bik linearly dependent in N}. It is clear that Q[M^] is a subring of Q[A/3] under the injection y' y •(k,Pi) 90 L e m m a 3.5.5 Let A = (N,0,8) be a stacky hyperplane arrangement and M.(A) the corresponding hypertoric Deligne-Mumford stack, then we have an isomorphism of graded rings A*(M{A))^ <*MA given by c\(Li) —> y b i , where Cir(Ap) is the ideal generated by elements in (3.2). P R O O F . Let Y(0y ,8) be the coarse moduli space of the hypertoric Deligne-Mumford stack M(A). By [HS], we have A*(Y(0v ,8)) = given by Di -» ybi, where Di is the T-equivariant Weil divisor on Y{0y ,8). Let etj be the first lattice vector in the ray generated by bi, then bi = ken for some positive integer l{. B y [V], the Chow ring of the stack M(A) is isomorphic to the Chow ring of its coarse moduli space Y(A,8) via c\{Li) —>• IT1 • Di, and YllLi e(ai)hybi = £ ™ i e(^)2/6i for e £ N*. • Let A*orb{M{A)) denote the orbifold Chow ring of M(A), which by definition is A*(I(M(A))) as a group. By Proposition 3.4.7, we have A*(I(M(A)))* © A*(M(A(a))). (v,a)eBox(Ag) For (v,a) G Box(Ap), there is an exact sequence of vector bundles, 0 -»• TM(A(a)) -> TM{A)\M{A[a)) -»• Nv -> 0, where Nv denotes the normal bundle of M(A(cr)) in M(A). O n the other hand, there is a surjective morphism m Q)(Li@L.l)^TM{A). i=i 91 Restricting this to M(A(a)) yields a surjective map 0 (Li®Lr*)->NV. Pi<Zo{v) Moreover, the element in the local group represented by v acts trivially on the kernel. Let v act on Li with eigenvalue e 2 ' K y / ^ W i , where Wi G [0,1) D Q. It follows that the age function on the component M(A(a)) assumes the value ] T (wi + (1 - Wi)) = \a\. PiCo Hence A*orb(M(A)) as a graded module coincides with 0 A*(M(A(a)))[\a\}. (v,a)eBox{Ai3) Note that A*orb(M(A)). is Z-graded, due to the fact that M(A) is hyperkahler. Again since the multi-fan is simplicial, we have the following result, sim-ilar to Lemma 4.6 in [Jiang2]. L e m m a 3.5.6 Let r be a cone in the multi-fan Ap. If {pi, • • • ,pt} C Zmfc(r), and suppose pi, • • • ,pt are contained in a cone a G A^g. Then a U r is contained in a cone of Ap. • P r o p o s i t i o n 3.5.7 Let M(A) be the hypertoric Deligne-Mumford stack associated to the stacky hyperplane arrangement A, then we have an isomorphism of graded A* (M(A))-modules: J ^ T = © A*(M(A(a)))[deg(y^)}. (v,a)&Box(A0) P R O O F . We use arguments similar to those in Proposition 4.7 of [Jiang2]. From Lemma 3.5.2 it is easy to see that Q[A/3] = © y^-Q[Mp]. (v,a)6Box(A0) 92 Consider © ( „ i f f ) e B o x ( A / ) ) y^-Cir(Ap). It is an ideal of the ring ® ( „ ) ( j ) e B o i ( A / 3 ) y M -Q[Mp], so For an element G Box(Ap), let p i , - - - , P( G link(a). Then we have an induced matroid complex Mp^, where 8(a) is the map in the quotient stacky hyperplane arrangement A(a) and the quotient extended stacky fan S/cr. Similarly from 8(a), we have multi-fan Ap^ in N(a). B y Lemma 3.5.5, A*(M(A(a))) = <Q)[Mp(a)]/Cir(Ap(a)). For any element (v,a) G Boa;(A /g), we construct an isomor-phism as follows. Consider the quotient stacky hyperplane arrangement A(a) = (N(a), 8(a), 6(a)). The hypertoric Deligne-Mumford stack M(A(a)) is a closed substack of M(A). Consider the morphism i : Q>[ybl,..., yb'] —> Q[yb>,..., ybm] given by ybi —> ybi. By Lemma 3.5.6, it is easy to check that the ideal IM0^ is mapped to the ideal IM0, SO we have a morphism Q[Mp^) —> QfM^]. Since QfM^] is a subring of Q[Ap]. Let # „ : q{Mp{a)][deg(y{-v^)] -> y^v^ • Q[Mp] be the morphism given by yk _^ y(v,<r) . yhm Using similar arguments as in Proposition 4.7 of [Jiang2], we find that the ideal Cir(Ap^) goes to the ideal y(v'a) -Cir(Ap), so we have the morphism such that *u([j/^]) = [ y M • ybi]. Conversely, for (v,a) G Box(Ap), since a is simplicial, for pi C a we can choose Oi G Homz(N,Q) such that = 1 and Oi(b-i) = 0 for b-> ^ bi G a. We 93 consider the following morphism p : Q [ y b l y b m ] —> Q[yb* , ybl] given by: ybi if pi C link(a), 0 if pi £ a U Zmfc(cr). Again by Lemma 3.5.6, the ideal IM0 is mapped by p to the ideal IM^^- Then p induces a surjective map Q[Mp] -> QfMg^)] and a surjective map : y(v'a) • Q[Mp] -> Q[M(g(<7)][de5(j/("'<7))]. Using the same computation as in Proposition 4.7 in [Jiang2], the relations j/"'"7) • Cir(Ap) is seen to go to Cir(Ap^). This yields another morphism so that = l,\I>„<fr„ = 1. So \t„ is an isomorphism. We conclude by Lemma 3.5.5. • 3.5.2 The orbifold product In this section we compute the orbifold cup product. First for any (vii&i), (^ 2,02) € Box(Ap), we have the following lemma: L e m m a 3.5.8 If u\ U ui is a cone in the multi-fan Ap, there exists a unique (va> 0-3) € Box(Ap) such that CT1UCT2U03 is a cone in the multi-fan Ap and V1+V2+V3 has no fractional part. PROOF. Let V3 = \vi + t ^ l ^ u ^ — vi — and 03 the minimal cone in o\ U 02 containing V3. Then (^ 3,03) satisfies the conditions of the Lemma. • The notation (ui, a"i)-r-(v2, cr2)+(v3, CT3) = Omeansthat ((vi,a\), (i>2,0"2), (^ 3,03)) satisfies the conditions in Lemma 3.5.8. 94 By [CR2], the 3-twisted sector M(A)(gi,g2tg3) is the moduli space of 3-pointed genus 0 degree 0 orbifold stable maps to M(A). Let P l (0 , l ,oo) be a genus 0 twisted curve with stacky structures possibly at 0,1, oo. Consider a constant map / : P 1 (0 ,1 , oo) —» M(A) with image x G M(A). This induces a morphism p : ^ b ( F \ 0 , l , o o ) ) ^ G x , where Gx is the local group of the point x. Let 7, be generators of 7 r ° r b ( P 1 ( 0 , 1 , 00)) and gi := p(-yi). The gi fixes the point x. By Proposition 3.4.6, <?j corresponds to an element (vi,o~i) G Box(Ap). A n argument similar to that in Proposition 6.1 in [BCS] shows that 3-twisted sectors of the hypertoric Deligne-Mumford stack A4(A) are given by H M(A(a123)), (3.25) {(vj ,aj),(v2,<J2),(,V3,o-3))eBox(A0)3,(vi ,<T\)+{v2,a2)+{va,oa)=Q where 0 1 2 3 is the cone in A ^ satisfying v\ + v2 + v3 - J2Pico-123 a A , cn = 1,2. Let evi : M(.4(0123)) - * M(A{(Ji)) be the evaluation map. We have the obstruction bundle (see [CR1]) O ^ ^ ^ ) over the 3-twisted sector A-f (.4.(0123)), O b i v u V 2 , V 3 ) = {e*T{M{A))®H\C,Oc))H (3.26) where e : M{A{a 123)) ->• M(A) is the embedding, C -»• P 1 is the if-covering branching over three marked points {0, l,oo} C P 1 , and H is the group generated by V\,V2,v$. Let (v,a) G Box(Ap), say v G AT(r) for some top dimensional cone r . Let {v,a) G Box(Ap) be the inverse of v as an element in the group N(T). Equivalently, ifv = YjPica for 0 < a i < 1, then t = Y , P i c S l ~ ai)°~i-L e m m a 3.5.9 Let (i>i, 01), (i>2,02), (^3,03) € Box(Ap) such that v\ + v2 + V3 = 0. Then ifvi+v2+v3 = E P i c < r 1 2 3 a ^ > ^1+^2+^3 = E f t c < r 1 2 3 C A > ^ e r e a*> Cj = 1 or 2, i/ien aj + Cj = 2 or 3. 95 P R O O F . Let Ui = I ^ - c ^ aj&j> w i t n 0 < aj < 1 and i = 1,2,3. Then Vi — YlpjCaiQ ~ a ) ) b r From the condition we have a j + aj'• + a | = aj = 1 or 2 and (1 — aj) + (1 — a 2 ) + (1 — otj) — Cj = 2 or 1. So if /9j belongs to u i , a2 and 0-3, then a j , aj, a | exist and if a3• = 1 or 2, then Cj = 2 or 1. If pj belongs to <7], 02, but not (T3, then a^ doesn't exist and a j + a 2 = aj = 1, (1 — aj) + (1 — a 2 ) = Cj = 1. The cases that pj belongs to o~\, 0-3 but not 02> to 172> 03 but not <7i are similar. We omit them. • The stack A4(A) is an abelian Deligne-Mumford stack, i.e. the local groups are all abelian groups. For any 3-twisted sector M(A(a-[23)), the normal bundle N(M(A(a\2z))IM{A)) can be split into the direct sum of some line bundles under the group action. It follows from the definition that if v — Ep ;Co - i 2 3 a i b i ' then the action of v on the normal bundle N (M(A(a\23)) / M(A)) is given by the diagonal matrix diag(cti,l — aj). A general result in [CH] and [JKK] about the obstruction bundle and Lemma 3.5.9 imply the following Proposition. P r o p o s i t i o n 3.5.10 Let M.(A)(Vl,V2,v3) = -M(.4(ci23)) be a 3-twisted sector of the stack A4(A) such that vi,v2,V3 ^ 0. Then the Euler class of the obstruction bundle 0h{vx,v2,v3) onM{A{axiz)) is Qi=2 a.i=l, aj,Q?,a| exist where Li is the line bundle over M(A) defined in Definition 3.5.1. To prove the main theorem, we introduce two Lemmas. For any two pairs {c\,a\), (02,0-2) G N A 0 , there exist two unique elements (VI,TI), (v2,T2) 6 Box(Ap) such that n C ( 7 i , r 2 C a2 and c\=v\ + ^ 2 P i C a i m i b i , c2 = v2 + J2Pica2 U i b i > w h e r e rrti, Hi are nonnegative integers. Let (V3, (73) be the unique element in Box(Ap) such 96 that v\ + v2 + v3 = 0 in the local group given by o\ Uer 2. Let Vi = ^ 2 P j C a . ctljbj, with 0 < a7- < 1 and i — 1,2,3. The existence of ctj,a2,a^ means that pj belongs to 0-1,0-2,0-3. Let a\23 be the cone in A ^ such that v-\ + v2 + V3 = Ep;Co-i 23 aJ>i, with ^ = 1 or 2. Let / be the set of i such that a, = 1 and a j , a 2 , a:- exist, J the set of j such that pj belongs to 0-123 but not to 03. We have the following Lemma for the ceiling functions: L e m m a 3.5.11 \c{]ai + \c2]a2 - \cX + C2]AIU(72 - \v{\-rx + \v2]T2 - \v\ + t>2]TiUT2-P R O O F . By the definition of ceiling functions, we have \c{\ax = [wi] T ] + YlPiccr1 m"^>i and \c2]a2 - \v2]T2 + Y,Pica2 n i b i - T n e Lemma follows. • L e m m a 3.5.12 If o\ U <r2 is a cone in Ap for the two pairs (c\, a\), (02,0-2), then the product y^Cl'"^ .yto^) 4 - N (S.l) can be given by ( _ l ) | / | + M y ( c i + < a + £ i 6 / f c + E i 6 J f c . < n u « ) l f ^ , v 2 ^ 0 and vx ^ V 2 , (_ l ) |J | y (c i+c2+£i € l / 6i .<nuo2) ifvi,v2 # 0 andv, = V2, (3-27) yCd+cs^uo-s) ifv1orv2 = 0. P R O O F . First for a fixed ray pi and 0 < 0:1,0:2 < 1, by the definition of ceiling functions, we find that ( 0 if a i + a 2 > 1, (3.28) bi if a i + a 2 < 1. Since e(ci ,c 2 ) = \c{\ai + [~c2"]o-2 — \c\ + C21OIU<T2) D v Lemma 3.5.11, we need to check that ( E i e / & i + E i e J ^ if «i ,W2 # 0 and ui # v 2 , T,ieJbi if Ui ,U2 7^ 0 a n d U i = t 2 , 0 if v\ or U 2 = 0. 97 fuilr, + \v2~\T2 - \vr + v 2 ] T l U T 2 = < This can be easily proven using (3.28) and Lemma 3.5.9. • 3.5.3 Proof of Theorem 3.1.1 B y Proposition 3.5.7, it remains to prove that the orbifold cup product is the same as the product in the ring QfA^]. By Lemma 3.5.12, we need to prove that the orbifold cup product is the same as the product in (3.27). It suffices to consider the canonical generators ybi, y^v'a) for (v,a) £ Box(Ap). Consider y(v>a) Uorbybi with (v,a) £ Box(Ap). The element (w,cr) determines a twisted sector M(A(a)). The corresponding twisted sector to 6, is the whole hypertoric stack M{A). It is easy to see that the 3-twisted sector relevant to this product is M{A)^v^tV-\^ = M(A(a)), where v~l denotes the inverse of v in the local group. It follows from the dimension formula in [CR1] that the obstruction bundle over M(A)(Vji^v-i} has rank zero. It is immediate from definition that y^v^Uorbybi = y(v+bi,aupi) jf t h e r e i s a c o n e m ^ containing v, bi. This is the third case in (3.27). Now consider y^^) \jorb y(V2<°2\ where {vx,CTI), (V2, a2) € Box(Ap). By (3.25), we see that if o\ U a2 is not a cone in A ^ , then there is no 3-twisted sector corresponding to the elements vi,v2. Thus the product is zero by definition. O n the other hand, by definition of the ring Q[Ap], we have y ^ ^ ) • y(«2,<T2) = Q. SO y^'^Uorby^2'"^ =y(vuo-i).y(v2,a2)_ If is a cone in A ^ , let {v3,a3) e Box{Ap) such that v3 £ a-\23 and vjv2v3 = 1 in the local group. Then we have the 3-twisted sector M{A(o-[23)). Let evi : M{A{o-\23)) —> M(A(ai)) be the evaluation maps. The element y(v<a) is the class 1 in the cohomology of the twisted sector M{A(a)). From the definition of orbifold cup product [CR1], [AGV2], we have: y { v i ' a i ) U o r b y ^ = (evMevty^^ • ev*2y^ • e ( 0 6 ( l ) 1 ) W 2 i „ 3 ) ) ) , where ev3 = 1 o ev3 : -M(.4(<7i23)) ->• M(A)^3) is the composite of ev3 and the 98 natural involution X : M(A)(V3) ->• M(A)^3y Let Vi = J2Pjc<Ti afij' 0 < alj < 1 and i = 1,2,3. Let I denote the set of i such that aj = 1 and a}-, a?, a3 exist, J the set of j such that belongs to (T123, but not belong to 173. If some vi = 0, for example, v\ — 0, then i>i is a torsion element in N which means that the action of v\ is trivial on the hypertoric Deligne-Mumford stack. Then the 3-twisted sector corresponding to v\, v2 is isomorphic to the twisted sector M{A{u2)) and the obstruction bundle over M(A(cr2)) is zero by [CR1]. In this case the set I and J are all empty. So y(v<^) u o r b - y(«i+«2,<xiUa 2)_ T h i s i s a g a i n the third case in (3.27). Now we assume that v\,v2 ^ 0. If v\ = v2, then U3 = 0, 17123 = o\ and vi + v2 — YlpjCai bj- So the 3-twisted sector corresponding to v\,v2 is isomorphic to the twisted sector A4(A(ai)) and the obstruction bundle over A4(A(ai)) is zero by [CR1]. The set J is the set j such that pj C ^ , So we have i / ^ i W " ^ = y ° - l [ y b i - l [ ( - y b ' ) ieJ ieJ = (-1)I JI • + , ;2+Ei6J hi,' TlU f f2) j which is the second case in (3.27). If Uj ^ v2, then v 3 ^ 0 and the obstruction bundle over the 3-twisted sector M{A{o-\2z)) is given by Proposition 3.5.10. So we have: ! / ( u , , < n ) U o r 6 yl»»°>) = j/(fl3.'3) . J T ybi Y[ybi. T J ( _ y 6 i ) . J J ( _ ^ ) . Since £3 4- £ a i = 2 + 12ieJ h = + v2, we have y { v u a ' ] ^ o r b y { V 2 , c r 2 ) = ( - i ) m + | J | .y(«i+"a.^u^). Y[ybi • Y[ybi iel i€J which is the first case in (3.27). • 99 3.6 Applications In this section we compute some examples of the orbifold Chow rings of hypertoric Deligne-Mumford stacks. In particular, we relate the hypertoric stack to crepant resolutions. Let N = Z and S the fan of projective line P 1 generated by {(1),(—1)}. Let 0 : Zn N be the map given by bi = (1),&2 = (-1) and &j = (1) for i > 2. Consider the following exact sequences 0 —•» Z —>• Zn — N —> 0 —> 0, 0 — • Z —> Zn Z " _ 1 —> 0 — • 0, where the Gale dual 0y is given by the column vectors of the matrix 1 1 0 0 ••• 0 1 0 - 1 0 ••• 0 A = 1 0 0 - 1 ••• 0 1 0 0 0 ••• - 1 Note that A is unimodular in the sense of [HS]. Taking Homz(—,CX) yields ( C x ) \ n - l a , (m* \n ( C x ) " —). So G = ( C x ) " " 1 . Choose 6> = (1,1, • • • , 1) in Z n _ 1 , then it is a generic element. The extended stacky fan £ = (N, T,,0) is induced from the stacky hyperplane arrange-ment A = (N,0,9), where H is the hyperplane arrangement whose normal fan is £ . It is easy to see that the toric Deligne-Mumford stack is the projective line P 1 . The hypertoric Deligne-Mumford stack is the crepant resolution of the Gorenstein 100 orbifold [ C 2 / Z „ ] . To see this, from the construction of hypertoric Deligne-Mumford stack, we have: 1 (CX) X\n-1 a . [ir\X.\2n ( C x ) \n+l 1, (3.29) where a L is given by the matrix [8y, — 8V]. Let <C[zi, ...,zn,w\, ...,wn] be the coordi-nate ring of C 2 " . So the ideal Ipv in (3.11) is generated by the following equations: ZiW\ + Z2W2 = 0 , Z\W\ - ZzWz = 0, z\w\ - znwn = 0. Hence Y is the subvariety of C 2 n — V(Ig) determined by the above ideal. The action of G on Y is through the map a L in (3.29). The hypertoric Deligne-Mumford stack associated to A is M(A) — [Y/G]. From Proposition 3.3.2, the hypertoric Deligne-Mumford stack is independent to the coorientations of the hyperplanes. This means that we can give the stacky hyperplane arrangement A as follows. Let 6j = 1 for 1 < i < n. Then the Gale dual map By : Z n —> Zn~l is given by the matrix 1 - 1 0 0 ••• 0 0 1 - 1 0 ••• 0 0 0 1 - 1 ••• 0 0 0 0 ••• 1 - 1 which is exactly the matrix in Lemma 10.2 in [HS], from which it follows that the coarse moduli space Y(8y,6) of M(A) = [Y/G] is the crepant resolution of the Gorenstein orbifold [ C 2 / Z „ ] . The core of the hypertoric Deligne-Mumford stack 101 M (A) is a chain of n — 1 copies of P 1 with normal crossing divisors corresponding to the multi-fan Ap. R e m a r k This is an example of [Kro], in which it is shown that the minimal res-olution of a surface singularity of A D E type can be constructed as a hyperkahler quotient. The Z „ - a c t i o n defining the Gorenstein orbifold [ C 2 / Z „ ] is given by \(x,y) = (Xx,\-^y) for A G Z n . There are n — 1 twisted sectors each of which is isomorphic to BZn with age 1. There are only dimension zero cohomology on the untwisted sector and twisted sectors. So we prove the following Proposition: P r o p o s i t i o n 3.6.1 The orbifold Chow ring A*orb([C2/Zn]) o / [ C 2 / Z „ ] is isomorphic to the ring C[xi,- • • ,xn-i] {xiXj : 1 < i,j < n — 1} Since the crepant resolution is a manifold, the orbifold Chow ring is the ordinary Chow ring. B y Theorem 3.1.1, we have P r o p o s i t i o n 3.6.2 The Chow ring of M(A) is isomorphic to the ring ,Vn-i] {ViVj • 1 < i,j < n - 1}' which is isomorphic to the orbifold cohomology ring of the Gorenstein orbifold [ C 2 / Z „ ] . P R O O F . By Theorem 3.1.1, the Chow ring of M(A) is isomorphic to the ring: C[yi ,Vn] {y\ - yn + 2/3 H \-yn-\mvj: i < j < n - 1} which we can easily check that this ring is isomorphic to the orbifold cohomology ring of [ C 2 / Z „ ] in Proposition 3.6.1. • 102 Y . Ruan [R] conjectured that, among other things, the orbifold cohomology ring of a hyperkahler orbifold is isomorphic to the ordinary cohomology ring of a hyperkahler resolution (which is crepant). For the orbifold [ C 2 / Z „ ] , the crepant resolution Y(8V,9) is smooth, we have that M(A) = Y(0y,9). From Proposition 3.6.2, the conjecture is true. A conjecture equating Gromov-Witten theories of an orbifold and its crepant resolutions, as proposed in [BG], is recently proven in genus 0 for [C 2/Z3], see [BGP]. The comparison of two Gromov-Witten theories requires certain change of variables. For [C 2 /Zs ] case, see [BGP]. For [C2/Z4] case the following change of variables is found in [BJ]: 2/1 = \{\/2x\ + 2ix2 - V2x3), * 2/2 = l(y/2ix\ - 2ix2 + V2ix3), 2/3 = T(—y/2xi + 2ix2 + V2x3). v Under this change of variables, the genus zero Gromov-Witten potential of the crepant resolution is seen to coincide with the genus zero orbifold Gromov-Witten potential of the orbifold [ C 2 / Z 4 ] , see [BJ]. For a toric orbifold, it is known that adding rays in the simplicial fan can give a crepant resolution. In the end of the paper we compute an example and explain that adding rays in the stacky hyperplane arrangement doesn't give a smooth hypertoric variety which means that it is not easy in general to give a crepant resolution in hyperkahler geometry. E x a m p l e Let S = (JV, £ , / ? ) be an extended stacky fan, where iV = Z 2 , the simplicial fan S is the fan of weighted projective plane IP(1,1,2), and 3 : Z 3 —> N is given by the vectors {61 = (1,0), 62 = (0,1), 63 = ( -1 , -2 ) , } , where 61,02,03 103 are the generators of the rays in £ . The generic element 8 = (1) G DG(0) = Z determines the fan E . The stacky hyperplane arrangement A = (N,0,8) induces E . The hypertoric D M stack is M{A) = T*(IP(1,1,2)). From Theorem 3.1.1, Q[xux2,x3,X4] ^ Q[x3,x4] AU(M(A)) = (x\ ~X3,X2 ~ 2x3,X%XlX2X3,X4X2,X4X\X3) (xj, X%, X3X4) ' Let 64 = (0, —1) and consider the new map 0' : Z 4 —> JV which is given by the vectors {61, b2, b3,64}. Choose generic element 8' = (1,1) G Z 2 = DG(0') and we get a new stacky hyperplane arrangement A' = (JV,/?',#') which induces the extended stacky fan £ ' = (JV, £ , / ? ' ) . The hypertoric D M stack M(A') is the stack corresponding to A'. From the definition of Box, {^b\ + 563,p\ + pz) is again a box element which determines a twisted sector. We compute that A*orb{M(A')) is isomorphic to Q[x-i,x2,x3,x4,v] ^ Q[x3,X4,v] (Xi - X3,X2 - 2x3 - X4, X2X4, X\X2X3, X-[X3X4, V2, VX2, VX4) (X3X4 + x\,x\,x1x4,V2,VX3,VX4)' We check that Alrb(M(A)) is not isomorphic to the ring Alrb(M(A')). We give two comments here. First, if the crepant resolution conjecture is true, then M(A') is not a hyperkahler resolution. O n the other hand, the map 0 is given by the matrix B — 1 0 - 1 0 1 - 2 and the map 0 given by B 1 0 - 1 0 0 1 - 2 - 1 It is easy to see that B' is not unimodular which means that Ai(A') is not smooth, see [HS]. 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Ziegler, Lectures on polytopes, Graduate Texts in Math. 152, Springer-Verlag, New York, 1995. 109 C h a p t e r 4 Semi-projective Toric Deligne-Mumford Stacks 4.1 Introduction The main goal of this chapter is to generalize the orbifold Chow ring formula of Borisov-Chen-Smith for projective toric Deligne-Mumford stacks to the case of semi-projective toric Deligne-Mumford stacks. In Chapter 1 we reviewed the construction of toric Deligne-Mumford stacks. The construction of toric Deligne-Mumford stacks was slightly generalized in Chap-ter 2, in which the notion of extended stacky fans was introduced. This new notion is based on that of stacky fans plus some extra data. Extended stacky fans yield toric Deligne-Mumford stacks in the same way as stacky fans do. The main point is that extended stacky fans provide presentations of toric Deligne-Mumford stacks 1 The content of this chapter has been submitted for publication. 110 not available from stacky fans. When is projective, it is found in [BCS] that the orbifold Chow ring (or Chen-Ruan cohomology ring) of ( £ ) is isomorphic to a deformed ring of the group ring of N. We call a toric Deligne-Mumford stack A'(S) semi-projective if its coarse moduli space X(T,) is semi-projective. Hausel and Sturmfels [HS] computed the Chow ring of semi-projective toric varieties. Their answer is also known as the "Stanley-Reisner" ring of a fan. Using their result, we prove a formula of the orbifold Chow ring of semi-projective toric Deligne-Mumford stacks. Consider an extended stacky fan £ = (N, £ , 0), where S is the simplicial fan of the semi-projective toric variety X(E). Let Ntor be the torsion subgroup of N, then N = N © N t o r . Let i V s := |S | © N t o r . Note that |S | is convex, so |S | © Ntor is a subgroup of N. Define the deformed ring QfiVs] := 0 c e / v E Qyc with the product structure given by ( yCi+c2 jf there is a cone a 6 S such that c\ e a,c2 £ cr; (4.1) 0 otherwise. Note that if ^ ( S ) is projective, then NY, = N and Q[-/Vs] is the deformed ring Q[A^] E in [BCS]. Let A*orb{X{Y,)) denote the orbifold Chow ring of the toric Deligne-Mumford stack X(H). Theorem 4.1.1 Assume that X(E) is semi-projective. There is an isomorphism of rings { E f a i * ) ! / » < :eeN*Y The strategy of proving Theorem 4.1.1 is as follows. We use a formula in [HS] for the ordinary Chow ring of semi-projective toric varieties. We prove that each twisted sector is also a semi-projective toric Deligne-Mumford stack. Wi th this, we 111 use a method similar to that in [BCS] and [Jiang2] to prove the isomorphism as modules. The argument to show the isomorphism as rings is the same as that in [BCS], except that we only take elements in the support of the fan. A n interesting class of examples of semi-projective toric Deligne-Mumford stack is the Lawrence toric Deligne-Mumford stacks. We discuss the properties of such stacks. We prove that each 3-twisted sector or twisted sector is again a Lawrence toric Deligne-Mumford stack. This allows us to draw connections to hy-pertoric Deligne-Mumford stacks studied in [JT1]. We prove that the orbifold Chow ring of a Lawrence toric Deligne-Mumford stack is isomorphic to the orbifold Chow ring of its associated hypertoric Deligne-Mumford stack. This is an analog of The-orem 1.1 in [HS] for orbifold Chow rings. The rest of this text is organized as follows. In Section 4.2 we define semi-projective toric Deligne-Mumford stacks and prove Theorem 4.1.1. Results on Lawrence toric Deligne-Mumford stacks are discussed in Section 4.3. Convention In this chapter we use N* to represent the dual of N and C * the multiplicative group C - {0}. 4.2 Semi-projective Toric Deligne-Mumford Stacks and Their Orbifold Chow Rings In this section we define semi-projective toric Deligne-Mumford stacks and discuss their properties. 112 4.2.1 Semi-projective toric Deligne-Mumford stacks Definition 4.2.1 ([HS]) A toric variety X is called semi-projective if the natural map Tr : X ->• X0 = Spec(H°(X, Ox)), is projective and X has at least one torus-fixed point. Definition 4.2.2 ([Jiang2]) A n extended stacky fan £ is a triple (N, X , 8), where TV is a finitely generated abelian group, X is a simplicial fan in 7% and 8 : Z m —> N is the map determined by the elements {b\,--- ,bm} in N such that {&],••• ,bn} generate the simplicial fan X (here m >n). Given an extended stacky fan £ = (N, X , 0), we have the following exact sequences: 0 —> DG(3)* —• Z m - A N —> Coker{0) —>• 0, (4.2) 0 —> N* —> Z m - A I>G(/3) —)• Coker{0y) —> 0, ' (4.3) where /3V is the Gale dual of 8 (see Chapter 2). Applying Homz(-,C*) to (4.3) yields 1 —> p, —> G ( C X —>• (C*) r f —»• 1. (4.4) The toric Deligne-Mumford stack A"(£ ) is the quotient stack [Z/G], where Z := ( C n \ V ( J E ) ) x ( C * ) m - n , J s is the irrelevant ideal of the fan X and G acts on Z through the map a in (4.4). The coarse moduli space of ^ ( S ) is the simplicial toric variety X{T) corresponding to the simplicial fan X , see [BCS] and [Jiang2]. Definition 4.2.3 A toric Deligne-Mumford stack is semi-projective if the coarse moduli space X ( S ) is semi-projective. Theorem 4.2.4 The following notions are equivalent: 113 1. A semi-projective toric Deligne-Mumford stack <f ( £ ) ; 2. A toric Deligne-Mumford stack A'(S) such that the simplicial fan £ is a regular triangulation of B = {61, • • • ,bn} which spans the lattice N. P R O O F . Since the toric Deligne-Mumford stack is semi-projective if its coarse moduli space is semi-projective, the theorem follows from results in [HS]. • 4.2.2 The inertia stack Let £ be an extended stacky fan and a G £ a cone. Define link{o~) := {r : <7 + T G £ , c r n T = 0}. Let {pi,... ,pi} be the rays in link(a). Consider the quotient extended stacky fan £ / c r = (N(a), £ /<r, 6(a)), with 0(a) : Z l + m ~ n ->• N(a) given by the images of 6 1 , . . . , 6/ and 6 n + i , . . . ,bm under N —• N(a). B y the construction of toric Deligne-Mumford stacks, if a is contained in a top dimensional cone in £ , we have X(^/a) := [Z(a)/G(a)], where Z(a) = (A1 \ V ( J s / ( 7 ) ) x ( C * ) ™ - " and G(a) =Homz(DG(0(a)),C*). Lemma 4.2.5 If X(Yi) is semi-projective, so is X(Yl/a). P R O O F . Semi-projectivity of the stack X(E) means the simplicial fan £ is a fan coming from a regular triangulation of B = {b\, • • • ,bn} which spans the lattice N. Let pos(B) be the convex polyhedral cone generated by B. Then from [HS], the triangulation is supported on pos(B) and is dermined by a simple polyhedron whose normal fan is £ . So a is contained in a top-dimensional cone r in £ . The image r of r under quotient by a is a top-dimensional cone in the quotient fan £ / o \ So the toric variety X(T,/a) is semi-projective by Theorem 4.2.4, and the stack X(E/a) is semi-projective by definition. • 114 Recall in [BCS] that for each top-dimensional cone a in S, define Box(a) to be the set of elements v 6 TV such that v = E ^ o ^ i f ° r s o m e 0 < a* < 1. Elements in Box(a) are in one-to-one correspondence with elements in the finite group iV(cr) = N/N„, where N(a) is a local group of the stack X(T,). In fact, we write v = Y^Pica{v) f ° r some 0 < < 1, where <r(u) is the minimal cone containing v. Denoted by Box(H) the union of Box(a) for all top-dimensional cones a. P r o p o s i t i o n 4.2.6 The r-inertia stack is given by Z r ( * ( £ ) ) = JJ X(V/a(vlr-- ,vr)), (4.5) •where cr(vi, • • • ,vr) is the minimal cone in £ containing v\, • • • ,vT. P R O O F . Since G is an abelian group, we have 2r (*(£)) = [( II Z^'"^)/G], (vu-,vr)e(GY where z(Vu'"'Vr*) C Z is the subvariety fixed by v\,--- ,vr. Since a(v\,--- ,vr) is contained in a top-dimensional cone in £ . We use the same method as in Lemma 4.6 and Proposition 4.7 of [BCS] to prove that [Z^'-'^/G] = X(E/a(vi, • • • ,vr)). • Note that in (4.5) each component is semi-projective. 4.2.3 The orbifold Chow ring In this section we compute the orbifold Chow ring of semi-projective toric Deligne-Mumford stacks and prove Theorem 4.1.1. 115 The module structure Let £ = (iV, £ , / ? ) be an extended stacky fan such that the toric Deligne-Mumford stack X(S) is semi-projective. Since the fan £ is convex, | £ | is an abelian subgroup of N. We put TVs := | £ | © N t o r , where N t o r is the torsion subgroup of N. Define the deformed ring Q[-/VE] := 0 c € W e Qyc with the product structure given by (4.1). Let {pi,... ,pn] be the rays of £ , then each pi corresponds to a line bundle Li over the toric Deligne-Mumford stack A f ( £ ) given by the trivial line bundle C x Z over Z with the G action on C given by the i-th. component « j of a : G -> ( C * ) m in (4.4). The first Chern classes of the line bundles Li, which we identify with ybi, generate the cohomology ring of the simplicial toric variety X(E). Let 5 s be the quotient ring !2[lLL^l>2L2l) where J E is the square-free ideal of the fan £ generated by the monomials {ybij • • • ybik '• hi, •' • ,bik do not generate a cone in £ } . It is clear that 5 s is a subring of the deformed ring QpVs] . L e m m a 4.2.7 Let A * ( ^ ( £ ) ) be the ordinary Chow ring of a semi-projective toric Deligne-Mumford stack Then there is a ring isomorphism: A * W £ ) ) - { E r = i e ( ^ : e G ^ } . P R O O F . The Lemma is easily proven from the fact that the Chow ring of a Deligne-Mumford stack is isomorphic to the Chow ring of its coarse moduli space ([V]) and Proposition 2.11 in [HS]. • Now we study the module structure on A*orb ( # ( £ ) ) . Because £ is a simplicial fan, we have: 116 L e m m a 4.2.8 For any c G N^,, let a be the minimal cone in X containing c. Then there is a unique expression c = v + ^2PiCcT mibi where m{ G Z>o, and v G Box(a). P r o p o s i t i o n 4.2.9 Let X(E) be a semi-projective toric Deligne-Mumford stack as-sociated to an extended stacky fan S . We have an isomorphism of A*(X(S))-modules: P R O O F . From the definition of Q[JV S ] and Lemma 4.2.8, we see that Q[7VS] = © u € B o x ( £ ) VV ' "SE- The rest is similar to the proof of Proposition 4.7 in [Jiang2], we leave it to the readers. • T h e C h e n - R u a n p r o d u c t s t r u c t u r e The orbifold cup product on a Deligne-Mumford stack X is defined using genus zero, degree zero 3-pointed orbifold Gromov-Witten invariants on X. The relevant moduli space is the disjoint union of all 3-twisted sectors (i.e. the double inertia stack). B y (4.5), the 3-twisted sectors of a semi-projective toric Deligne-Mumford stack X(E) are JJ ^ ( E / a ( i 7 i , U 2 , U 3 ) ) . (4.6) (V1,V2,V3)€B0X(E)3,V\V2V3 = 1 Let evi : X (T, / a(vi,V2,v3)) -¥ X(E/a(vi)) be the evaluation maps. The ob-struction bundle (see [CR2]) Ob(VltV2tV3) over the 3-twisted sector X(E/a(vi,V2,V3)) are defined by O b i v u V 2 t V 3 ) := {e*T{X{V))®H\C,Oc))H, (4.7) where e : X(E/a(v-\,v2,vs)) -> X(E) is the embedding, C -> P 1 is the i7-covering branched over three marked points {0, l,oo} C P 1 , and H is the group generated 117 by v i , v 2 , v 3 . A general result in [CH] and [JKK] about the obstruction bundle implies the following. P r o p o s i t i o n 4.2.10 Let X(S/<r(wi, v2, v3)) be a 3-twisted sector of the stack X(£). Suppose v\ + v2 +1>3 = Y^Pica(vuv2,v3) a^i, &i = 1 or 2. Then the Euler class of the obstruction bundle Ob(VuV2iV3} on X(E/a(vi,v2,V3)) is IJ Ci(Li)\x{-zio{m,v2,vz)), where Li is the line bundle over <-f (£) corresponding to the ray pi. Let v G Box^E), say v G N(a) for some top-dimensional cone a. Let v G Box(E) be the inverse of v as an element in the group N(a). Equivalently, if v = YpiCaiv) ai°i for 0 < a* < 1, then V = E ^ O r ^ 1 ~ ai)bi- T h e n for a i > A 2 € A*rb(XCS)), the orbifold cup product is defined by a\ Uorfc a2 = ev3*(ev{ai U ev*2a2 U e(Ob(Vl>V2>V3-))), (4.8) where ev3 = I o eu 3 , and 7 : IA"(S) -> X , f ( £ ) is the natural map given by (x, 5) -» P r o o f o f T h e o r e m 4.1.1 By Proposition 4.2.9, it remains to consider the cup product. In this case, for any v\,v2 G Box(E), we also have vi + v2 = v3 + ^ h + ^2 b i ' a,i=2 i£j where J represents the set of j such that pj belongs to a(vi,v2), but not belong to o-(v3). Then the proof is the same as the proof in [BCS]. We omit the details. 118 4.3 Lawrence Toric Deligne-Mumford Stacks In this section we study a special type of semi-projective toric Deligne-Mumford stacks called the Lawrence toric Deligne-Mumford stacks. Their orbifold Chow rings are shown to be isomorphic to the orbifold Chow rings of their associated hypertoric Deligne-Mumford stacks studied in Chapter 3. We refer to Section 3.2 about the construction of Lawrence toric Deligne-Mumford stacks and hypertoric Deligne-Mumford stacks. 4.3.1 Comparison of inertia stacks Next we compare the orbifold Chow ring of the hypertoric Deligne-Mumford stack and the orbifold Chow ring of the Lawrence toric Deligne-Mumford stack. First we compare the inertia stacks. From the map 8 : Z m —> N which is given by vectors {b\, • • • , bm}. Let Cone(ft) be a partially ordered finite set of cones generated by b\, • • • , b m . The partial order is defined by: a -< r if a is a face of r , and we have the minimum element 0 which is the cone consisting of the origin. Let Cone(N) be the set of all convex polyhedral cones in the lattice N. Then we have a map C : Cone(B) —> Cone(N), such that for any a 6 Cone(B), C{a) is the cone in N. Then A ^ := (C, Cone(B)) is a simplicial multi-fan in the sense of [HM]. For the multi-fan A 0 , let Box(Ap) be the set of pairs (v,a), where cr is a cone in A ^ , v 6 N such that v = ^2Picaaih f ° r 0 < Qj < 1. (Note that a is the minimal cone in Ap satisfying the above condition.) From Chapter 3, an element (v,a) € Box(Ap) gives a component of the inertia stack 1(M(A)). Also consider the set BoxCEo) associated to the stacky fan see Section 4.2.2 for its definition. 119 A n element v £ Box(T,g) gives a component of the inertia stack l(X(Y;g)). By the Lawrence lifting property, a vector b{ in TV lifts to two vectors b~L,i,b'Li in NL. Let {bL>ii, • • • , 6 L ) I ) , , b'Lil , b'Lik} be the Lawrence lifting of {b^, • • • , bik}. Lemma 4.3.1 {b^,--- ,bik} generate a cone a in Ap if and only if (BL,^,- •• ,t>Ltik, &£,i i>' ' ' >^£,i*} <? e 7 jera£e a cone ag in Tg. P R O O F . Suppose a is a cone in Ap generated by {b^, • • • , b~ik}, it is contained in a top-dimensional cone r . Assume that r is generated by {b^, • • • , b~ik, h k + i , • • • , t>id}-Let C be the complement {61, • • • , 6 M } \ r . Then C corresponds to a column ba-sis of /3V in the map if : l m -» DG{B). By the definition of T0 in (3.8), C corresponds to a maximal cone TO in To which contains the rays generated by {b~L,ii, • • • ib~L,ikJ>L,i\>''' -^L,ik}- Thus these rays generate a cone ao in To-Conversely, suppose ao is a cone in Tg generated by , • • • , bh,ik, >''' > Using the similar method above we prove that {b^, • • • ,t>ik} must be contained in a top-dimensional cone of A ^ . So {6^, - • • , b~ik} generate a cone a in A ^ . • Lemma 4.3.2 There is an one-to-one correspondence between the elements in Box(T,o) and the elements in Box(Ap). Moreover, their degree shifting numbers coincide. P R O O F . First we rewrite two exact sequences in Chapter 3 here: 1 —-» p —> G ( C * ) m —-> T —> 1, and 1 —> p —> G ( C * ) 2 m — • TL — • 1. The torsion elements in BOX(YJQ) and Box(Ap) are both isomorphic to p = A;er(a) = ker(aL) in the above two exact sequences. Let (u, a) £ Boa;(A jg) with u = Y^Pica a ^ i -120 Then v may be identified with an element (which we ambiguously denote by) v E G := Homz(DG{8),C*). Certainly v fixes a point in C m . Consider the map a in (3.9), put a(v) = ( a 1 ( « ) , • • • ,am{v)). Then a'(u) # 1 if pi C cr, and a'(w) = 1 otherwise. By Lemma 4.3.1, let \pL,iJ>Li : i = 1, - • • , \a\} be the Lawrence lifting of {b~i}PiC(T- Since the action of v on C 2 m is given by (i>, i> - 1), u fixes a point in C 2 m and yields an element VQ in I?02;(Eo). From the map a L , let a ^ N ) = ((*i(vo),--- , a m W . a m + i W i " ' , a L W ) ' (4-9) Then a f ^ e ) ^ 1 and a^+m(v0) ^ 1 if Pi C cr; o f (uo) = a f + m ( u 0 ) = 1 otherwise. So ao{vo) — {bL,i,bLi : i — 1, • • • , is the minimal cone in So containing «o-Furthermore, v0 = E ^ c ^ a i S L , i + E ^ O r C 1 ~ "O^L. i -Conservely, given an element vo £ Box(Eo), let <Jg(vo) be the minimal cone in So containing VQ- Then from the action of G on C 2 m and (4.9), we have OL\[VO) = {af+m(v0))~l• If a f (w0) # 1, then a f + m ( w 0 ) # 1, which means that b L , i , b L , i + m € ao(vg). The cone cro(vo) is the one in So containing b~L,i,b~L,i+m,s satisfying this condition. Then vg = E i ( a ^ £ , j + (1 — ai)bL,i)- B y Lemma 4.3.1, crg(vo) is the Lawrence lifting of a cone CT generated by the {6j}'s in A ^ . Let v = E P i c < r S o it also determines an element (v,a) 6 73o2;(A/g). • For (vi, cr]), (u 2 , (T2), (i>3, 03) G 5oa;(A /g), let a(v-\,U2,^3) be the miniaml cone containing Ui , tJ 2 , v3 in A ^ such that v-[ + v2 + v3 = E f t c«r(iJi ,v 2 ,v 3 ) a ^ a n d a* = 2" Let vo,i,wo ) 2,vo, 3 be the corresponding elements in Box(Eg) and cr(wo,i,1*0,2,^0,3) the minimal cone containing «o,1,^0,2, ^0,3 in So- Then by Lemmas 4.3.1 and 4.3.2, °~(vo,iiVo,2,vo,3) is the Lawrence lifting of a(v^ ,T>2,v3). Suppose that a is generated b y l A , - - " ,bis}, then a(v0,i,v0,2,ve,3) is generated by {bL>il ,•• • , b L > i s , b L M , • • • A ) i s } , the Lawrence lifting of {h^,--- ,t>is}- Let - ,&j m _;_ s } be the rays not in 121 a U link(a), we have the Lawrence lifting { b L J l , • • • , bLjm_,_a, bLJl , b L J m _ l s } . Then from the definition of Lawrence fan T,g in (3.8), we have the following lemma: _ _ _/ _/ Lemma 4.3.3 There exist m—l—s vectors in {OLJI , • • • , bL,jm_t_s, b L j l , • • • , bLjm_t_s} such that the rays they generate plus the rays in c(^0,i 1^0,2,^0,3) generate a cone ao in T,g. • Proposition 4.3.4 The stack X(Eg/ao) is also a Lawrence toric DM stack. P R O O F . For simplicity, put a := a(U\ ,^2,^3). Suppose there are I rays in the link(a). Then by Lemma 4.3.1 there are 21 rays in link(ao), the Lawrence lifting of link(a). Let s := \a\, then 2s + m — I — s = \ao\- Applying Gale dual to the diagrams 0 and -> Z L + S -> N ->• Z m -> Z' -> 0 /3(<T) > Af(cr) >• 0, • J/n-l-s 0 -> AT -> AT • 0 yields 0 > Zl+s -> Z s -)• 0 —4 0 / 3 ( " ) v ft > DG{3{a)) - ^ - > 7JG(/3) • /JG( /J a ) • 0, and 0 jm—l—s -l-s z '+ s -> 0 —> L>G(/i) — £ > G ( / 3 ) > 0. Since Z s = Na, we have that DG(Ba) = 0. We add two exact sequences (4.10) (4.11) 0 — > Z L —> z m —> z m _ ( —-> 0, 122 and 0 —• 0 —>z m — > Z m —• 0, on the rows of the diagrams (4.10),(4.11) and make suitable maps to the Gale duals we get 0 • ^ yl+s+m >. Z s+m-/ y g , - / 3 ( a ) v ) WV , - / 8 v ) (4.12) 0 • DG(0(a)) DG(0) > 0 -> 0, and 0 • Z m—l—s -> z 2m ( / 3 V , - / 3 V ) Z ' + s + m • 0 (4.13) 0 • Z m " ' - 5 > DG(0) • DG(0) • 0. Applying Gale dual to (4.12), (4.13) we get o > z s + m - 1 > z / + s + m > Z 2 Z > 0 PU<re) and 0 Zs+m-l > jyL -> Z / + ' 5 + m >• z 2 m Nr i V L ( c r 0 ) 4 0, -> 0 0 ->• 0. For the generic element 8, from them map </J2 in (4.11), 8 induces 8 £ DG(0), and from the isomorphism tpi in (4.10), # = 8(a) G DG(0(a)). So we a quotient stacky hyperplane arrangement A(a) = (N(a),0(a),0(a)). From the above diagrams we see that the quotient fan To/ao in Nh(ao) also comes from a Lawrence construction of the map 0(a)y : Zl -> DG(0(a)). Let X(a) = C21 \ V(Z0{a)), where X0{(7) is the irrelevant ideal of the quotient fan To/ao- Let G(a) = Homi(DG(0(a)), C ) . The stack ^(Efl /cf l) = [X(o")/G(<r)] is a Lawrence toric Deligne-Mumford stack. • C o r o l l a r y 4.3.5 M.(A(a(v\,V2,v3))) is the hypertoric DM stack associated to the quotient Lawrence toric DM stack X^o/ve)-123 P R O O F . M(A{CF(VI,V2,VZ))) is constructed in [JT1] as a quotient stack [Y(a)/G(a)], where Y(a) C X(a) is defined by Ip^y, which is the ideal in (3.11) correspond-ing to the map /3(cr)v in (4.10). So the stack M{A{a(y\,V2,vz))) is the associated hypertoric Deligne-Mumford stack in the Lawrence toric Deligne-Mumford stack X{E6I(JO). • R e m a r k For any vo G Box(Hig), let VQ1 be its inverse. We have the quotient Lawrence toric stack X(Eg/ag). Let (v,a) be the corresponding element in Box(Ap), then M(A(a(v,v-\l))) ^ M(A{a)). By Proposition 4.3.4 and Corollary 4.3.5, the twisted sector M(A(a)) is the asso-ciated hypertoric Deligne-Mumford stack of the Lawrence toric Deligne-Mumford stack X (Eg/erg). R e m a r k From Lemma 4.3.3, the cone ag is not the minimal cone (j(vgyi,vg^,vg^) containing ^0,1,1^2,^0,3 in Eg. So X(Eg/a(vg^,vgt2,vg^)) is n ° t a Lawrence toric Deligne-Mumford stack. But from the construction of Lawrence toric Deligne-Mumford stack, the quotient stack X(Eg/a(vgti,vgt2,^0,3)) is homotopy equivalent to the quotient stack X(Eg/ag). Since we do not need this to compare the orbifold Chow ring, we omit the details. 4.3.2 Comparison of orbifold Chow rings Recall that NL = NL © NLJOT, where NL,tor is the torsion subgroup of NL-Let N-£G = NLJOT © |£fl | . By Theorem 4.1.1, we have 124 P r o p o s i t i o n 4.3.6 The orbifold Chow ring A*rb(X(Y,o)) of the Lawrence toric Deligne-Mumford stack X(Eo) is isomorphic to the ring {£2=1 e f e ^ + £ ™ i e(b'L.)yb^ : e £ NfJ • (4.14) Recall in Chapter 4 that for any c E N, there is a cone <7 £ A/j such that c = £ f t C ( T where on > 0 are rational numbers. Let i V ^ denote all the pairs (c, cr). Then NA^ gives rise a group ring Q [ A / 3 ] = 0 Q . y M , (c,<r)e7VA/3 where y is a formal variable. For any (c, a) £ J V A 0 , there exists a unique element (v, r) £ Box(Ap) such that T C a and c = u + £ P i c f f Triibi, where mi are nonnegative integers. We call (V,T) the fractional part of (v,a). We define the ceiling function for fans. For (c, cr) define the ceiling function \c]a by \c]a = £ P i C r °* + £ p t c , r m A -Note that if v = 0, [c]CT = £ p . C ( T m,^ £>j. For two pairs (ci, a\), (c2,0-2), if <TI U 02 is a cone in Ap, define e(ci,c 2 ) := [ c i ]^ + \c2\a2 - \c\ + C2]a-1uo-2- Let &e Q °~i u °~i be the minimal cone in A ^ containing e(ci,C2) so that (e(ci,C2), cre) £ NA<3. We define the grading on QJA^] as follows. For any (c,a), write c = v + £ P i c < r m A ' then deg(y(c'a^) = | r | + £ p . C o . mi, where | r | is the dimension of r . By abuse of notation, we write y ( b i >«) as y b i . The multiplication is defined by { ( _ l ) k e | 2 / ( c i + c 2 + £ ( c ] , c 2 ) , o - 1 u 0 - 2 ) i f a i u i s a cone m A * ; 0 otherwise. (4.15) From the property of ceiling function we check that the multiplication is commuta-tive and associative. So QfA^] is a unital associative commutative ring. In Chapter 125 3, it is shown that Consider the map B : Z m —>• TV which is given by the vectors {b\, • • • , bm}. We take {1, • • • , m} as the vertex set of the matroid complex Mp, defined from B by requiring that F G Mp iff the vectors {b~i}i£F are linearly independent in N. A face F G Afy corresponds to a cone in A ^ generated by {bi}i^F- B y [S], the "Stanley-Reisner" ring of the matroid Mp is where is the matroid ideal generated by the set of square-free monomials {z/6'1 • • • ybik \b~n, • '" ,^ i/b linearly dependent in N}. It is proved in Chapter 3 that , Q[Ap] - 0 y ^ • Q[Mp}. (v,a)eBox(Ag) For any (ui,eri), (i>2,0"2) G 5ox(A / g), let (v3,a3) be the unique element in Box(Ap) such that ui + v2 + v3 = 0 in the local group given by o\ U 02, where = 0 means that there exists a cone a(vi, v2, v3) in A ^ such that v\ +v2 + v3 - £ \ _2_a) a ^ > where a{ = 1 or 2. Let ^ = E f t c f f l a j % ^2 = E ^ c ^ a j ^ > "3 = E ^ c f f 3 w i t h 0 < a ] , a 2 , Qj < 1. Let 7 be the set of i such that a; = 1 and a ] , a 2 , a? exist, J the set of j such that pj belongs to a(v-[,v2,v3) but not a 3 . If (v, a) G Box(Ap), let (z), cr) be the inverse of (v, a). Except torsion elements, equivalently, if v = E ^ c u Q A f ° r 0 < a{ < 1, then U = E p ^ o U — Q i )&i- By abuse of notation, we write y(h>>ft) as y h i . We have that v\ + v2 = i>3 + E a I = 2 ^ + YljeJ^j- From (4.15), Lemma 3.5.11 and 126 Lemma 3.5.12 in Chapter 3, if vi,V2 r1 0, we have t T,juhi if ui = #2 -So it is easy to check that the multiplication y(v*>ai) • y^"2^) can be written as (_l)|/|+|J|y(*3,<T3) . Yla_2ybi . Ylierybi • Y[jejy2bi if « i , U 2 6 o- for ( r e ^ and ^ # #2 , (-1)I JI Y l j e J y 2 b j if tJi,I>2 G a for a £ Ap and Ui = #2 , 0 otherwise. (4.17) The following is the main result of this Section. T h e o r e m 4.3.7 There is an isomorphism of orbifold Chow rings A*orb(X(Ylo)) = P R O O F . The ring Q f i V s J is generated by {yb'-\yb'L'' : i = 1, - • • ,m} and yv° for vo £ Box(T>o) by the definition. By Lemma 4.3.2, define a morphism 0 : Q[JV E F L] -> ®[Ap] by ybL'i -> ybi,yb'I"i ->• - y b i and y"* ->• y ^ ) . B y [HS], the ideal I 0 goes to the ideal / M / 3 and the relation {EZie(bL,i)yb^ + £ ™ 1 e ^ ) / ^ : e £ N*} goes to the relation { £ ™ T e(bi)ybi : e G i V * } . Thus the two rings are isomorphic as modules. It remains to check the multiplications. For any yve and ybL<i or ybh>i, let yb>,o-) ^ e corresponding element in QfA^]. By the property of VQ and Lemma 4.3.2, the minimal cone in £ # containing veJ>L,i must contains HLi. By Lemma 4.3.1, there is a cone in A ^ containing v,b{. In this way, yve • ybL<i goes to y(v<a) -ybi and yve • ybh>{ goes to -y(v'a) • ybi. If there is no cone in £ # containing vg,b~Lti,b'Li, 127 then by Lemma 4.3.1 there is no cone in containing v, hi. So yve • y = 0 goes to 2/(';,<T) • ybi = 0 and yVe • yb'^ = 0 goes to - y ^ ) • ybi = 0. For any yVe^, yVe<2, let y(Vl,tri ), yfa^) be the corresponding elements in QfA^g]. If there is no cone in £<? containing voi,vo,2, then by Lemmas 4.3.1 and 4.3.2, there is no cone in A ^ containing Vx,v2- So yVe^ • yV6<2 — 0 goes to y(Vl<ai) • y(v2,(T2) — o Suppose there is a cone containing wo,i,W0,2, let vg^ S Box(Eo) such that voti + vg>2 + ^0,3 = 0. Let cr(voti,vot2,vo,3,) be the minimal cone con-taining 1*0,1,1)0,2,^0,3 in So- Then by Lemmas 4.3.1 and 4.3.2, cr(vgti,vo,2,vgt3) is the Lawrence lifting of a(v\,V2,vz) for (v\,ci), (v2,02), ( « 3 , 0 3 ) e 7?ox(A /g). So we may write v0,i + + v0,3 = EpiCafr&Vs) a ^ + ^ / > , c , ( m « , » 3 ) A$L,i- T h e cor-responding Vi +V2+V3 - Y,PiCa(vuV2,V3)aibi- L < 3 t b e t h e i n v e r S e o f (V>a) in Box(A.p), i.e. if v is nontorsion and v = YlPicaaib~i for 0 < OJJ < 1, then v = E f t C f f ( l — CYi)bi- The £># is defined similarly in Box{Eg). The notation J repre-sents the set of j such that pj belongs to a-(vi,V2,v3) but not 03, the corresponding PL,J,P'LJ belong to o-(v0,t,v0,2,vgj) but not a(uo i 3). If some vo,i = 0 which means that vo,i is a torsion. Then from Lemma (4.3.2) the corresponding v is also a torsion element. In this case we know that the orbifold cup product yV6^ • yve'2 is the usual product, and under the map <j>, is equal to y(vU<ri) . y(v2,<T2)_ If vot\ = vo,2, then vo,3 = 0 and the obstruction bundle over the corresponding 3-twisted sector is zero. The set J is the set j such that pj belongs to cr(vgt\). So from [BCS], we have vva>1 V ' 2 = l[ybL'j Under the map <f> we see that j/"1'*7') • y(v*<<ri) is equal to the second line in the product (4.17). 128 If vo,\ ^ vot2, then vo$ ^ 0 and the obstruction bundle is given by Proposition 3.5.10. If all ctj,a2,a^ exist, the coefficients aj and a\ satisfy that if aj = 1 then a\ = 2, and if aj = 2 then a\ = 1. So from [BCS], Under the map c£ we see that y(Vl<tTl'> .^("2,0-2) j s equal to the first line in the product (4.17). By Lemma 4.3.2, the box elements have the same orbifold degrees. By Corollary 4.3.5 and the definition of orbifold cup product in (4.8), the products y«8,i . yve,2 ar i (} y(vi,oi) . y{v2,02) n a v e game degrees in both Chow rings. So d> induces a ring isomorphism A*orb(X(E0)) = A*orb(M(A)). • R e m a r k The presentation (4.16) of orbifold Chow ring only depends on the ma-troid complex corresponding to the map 0 : Z m -> A'', not 8. Note that the presenta-tion (4.14) depends on the fan £ # . We couldn't see explicitly from this presentation that the ring is independent to the choice of generic elements 8. 129 Bibliography [ A G V l ] D . Abramovich, T . Graber and A . Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, math.AG/0603151. [AGV2] D . Abramovich, T . Graber and A . Vistoli, Algebraic orbifold quantum product, in Orbifolds in mathematics and physics (Madison, WI,2001), 1-24, Contem. Math. 310, Amer. Math. Soc , 2002. math.AG/0112004. [BD] R. Bielawski and A . Dancer, The geometry and topology of toric hy-perkahler manifolds, Comm. Anal. Geom. 8 (2000), 727-760. [BCS] L . Borisov, L . Chen and G . Smith, The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc. 18 (2005), no . l , 193-215, math.AG/0309229. [CR1] W . Chen and Y . Ruan, A new cohomology theory for orbifolds, Comm. Math. Phys. 248 (2004), no. 1, 1-31, math.AG/0004129. [CR2] W . Chen and Y . Ruan, Orbifold Gromov-Witten theory, in Orbifolds in mathematics and physics (Madison, WI, 2001), 25-85, Contem. Math. 310, Amer. Math. S o c , 2002. math.AG/0103156. [Cox] D . Cox, The homogeneous coordinate ring of a toric variety, J. of Alge-braic Geometry, 4 (1995), 17-50. 130 [Cox2] D . Cox, Recent developments in toric geometry, in Algebraic Geometry (Santa Cruz, 1995), 389-436, Proc. Symp. Pure Math . 62.2, Amer. Math . Soc , 1997. [F] W . Fulton, Introduction to toric varieties, Annals of Mathematics Studies 131, Princeton University Press, Princeton, N J , 1993. [HM] A . Hattori and M . Masuda, Theory of multi-fans, Osaka J. Math. 40 (2003), 1-68. [HS] T . Hausel and B . Sturmfels, Toric hyperkahler varieties, Documenta Mathematica, 7 (2002), 495-534. [Jiang2] Y . Jiang, The orbifold cohomology ring of simplicial toric stack bundles, to appear in Illinois J. Math., math.AG/0504563. [JT1] Y . Jiang and H . - H . Tseng, The orbifold Chow ring of hypertoric Deligne-Mumford stacks, to appear in Journal Reine Angew Math., math.AG/0512199. [JT2] Y . Jiang and H . - H . Tseng, Note on orbifold Chow ring of semi-projective toric Deligne-Mumford stacks, math.AG/0606322. [K] H . Konno, Cohomology rings of toric hyperkahler manifolds, Intern. J. of Math.(U) (1997), no.8, 1001-1026. [L-MB] G . Laumon and L . Moret-Bailly, Champs algebriques. (French) [Algebraic stacks], Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 39. Springer-Verlag, Berlin, 2000. 131 [R] Y . Ruan, Cohomology ring of crepant resolutions of orbifolds, in Gromov-Witten theory of spin curves and orbifolds, 117-126, Contem. Math. 403, Amer. Math. S o c , 2006, math.AG/0108195. [S] R. P. Stanley, Combinatorics and commutative algebra, 2nd ed., Birkhauser Boston, 1996. [V] A . Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math., 97 (1989) 613-670. [Zagier] G . M . Ziegler, Lectures on polytopes, Graduate Texts in Math . 152, Springer-Verlag, New York, 1995. 132 Chapter 5 Conclusion 5.1 Relations Among Chapters. Chapter 1 is an introduction chapter. We introduced orbifold Chow ring of smooth Deligne-Mumford stacks. We reviewed the definition of toric Deligne-Mumford stacks by Borisov, Chen and Smith. In Chapter 2 we defined extended stacky fans and constructed toric Deligne-Mumford stacks associated to extended stacky fans. Every extended stacky fan has an underlying stacky fan and they gave isomorphic toric Deligne-Mumford stacks. The main point of extended stacky fan is that it can give different representations of toric Deligne-Mumford stacks. Any extended stacky fan £ can be constructed from a stacky hyperplane arrangement A = (N,8,9) defined in Chapter 3. The hyperplane arrangement determines the topology of hypertoric varieties [BD]. Using stacky hyperplane ar-rangement A we constructed hypertoric Deligne-Mumford stack M (A) so that the toric Deligne-Mumford stack <-f ( £ ) is the associated toric Deligne-Mumford stack of M(A). We computed the orbifold Chow ring of hypertoric Deligne-Mumford stacks. For any stacky hyperplane arrangement A = (N,8,9) , we associate a 133 Lawrence stacky fan £ # constructed from the Lawrence lifting. The hypertoric Deligne-Mumford stack M (A) is defined as a closed substack of the Lawrence toric Deligne-Mumford stack X(Y,o)- The Lawrence toric Deligne-Mumford stack X(Eo) is semi-projective, but not projective. In Chapter 4 we generalized the orbifold Chow ring formula of projective toric Deligne-Mumford stacks to semi-projective toric Deligne-Mumford stacks. We prove that the orbifold Chow ring of a Lawrence toric Deligne-Mumford stack X(Eo) is isomorphic to the orbifold Chow ring of its associated hypertoric Deligne-Mumford stack M (A) which is the orbifold Chow ring analogue of a theorem of Hausel and Sturmfels. 5.2 Importance of the Thesis. 5.2.1 Good reading materials for students. In this thesis we reviewed the basic definition of smooth Deligne-Mumford stacks and the definition of orbifold Chow ring. We gave the construction of toric Deligne-Mumford stacks by Borisov-Chen-Smith and generalized it in Chapter 2. We defined hypertoric Deligne-Mumford stacks in Chapter 3, semi-projective toric Deligne-Mumford stacks in Chapter 4 and computed their orbifold Chow rings. The thesis is a good introductory reading materials for graduate students who want to study orbifold Chow ring of Deligne-Mumford stacks and want to learn about the computations. Through reading students will learn about the techniques for computing the orbifold Chow rings, for example, what's the technical part and how to deal with the obstruction bundles in the definition of orbifold cup product. A n d through reading students will know some nice abelian Deligne-Mumford stacks. 134 5.2.2 Importance of the definitions and results. In the definition of toric Deligne-Mumford stacks by Borisov, Chen and Smith in Chapter 1, the authors used the notion of stacky fans. We introduced a new notion of extended stacky fan and used this notion to define toric Deligne-Mumford stacks. The importance of this new definition mainly lies in two cases: The first case is that it gave more representations of toric Deligne-Mumford stacks as quotients, see the examples in Chapter 2. The second case is that the extended stacky fan £ is natural when we consider closed substacks corresponding to cones in the fan £ . In Proposition 4.2 of [BCS], the authors used quotient stacky fan £ / < r to study the closed substack X(T,/a) of <-f ( £ ) . We found that the exact sequences in that proof are wrong and the quotient stacky fan is naturally an extended stacky fan. We give a new proof in Proposition 2.3.5 of Chapter 2. A n interesting application of the toric stack bundle discussed in Chapter 2 is that any finite abelian gerbes over a smooth scheme is a toric stack bundle. Thus we compute the orbifold Chow ring of any ju-gerbes over smooth scheme B for a finite abelian group fi. We first define the notion of hypertoric Deligne-Mumford stacks in Chapter 3 from stacky hyperplane arrangements and studied their orbifold Chow rings. We generalize the orbifold Chow ring formula of projective toric Deligne-Mumford stacks to semi-projective toric Deligne-Mumford stacks so that we compute more examples of orbifold Chow rings. A l l these results are positive contributions to the research field of stringy orbifold theory. 135 5.2.3 Application of the results. The main results of this thesis are the Chow ring formula for toric stack bun-dles, hypertoric Deligne-Mumford stacks and semi-projective toric Deligne-Mumford stacks. These results will have many applications. One interesting case is the special case of toric stack bundles, i.e. the /x-gerbe Q —> B over a smooth scheme B for a finite abelian group / i . We compute the orbifold cohomology of Q and show that the /i-gerbes over B, no matter they are trivial or not, have the same orbifold cohomology. In a physics paper [PS], the authors predicted that the quantum cohomology of Q should be \/J,\ copies of the quantum cohomology of B. Since quantum cohomology is a deformation of the orbifold cohomology, based on our computation we can study the quantum cohomology of such gerbes. Hypertoric Deligne-Mumford stacks are good examples of hyperkahler Deligne-Mumford stacks. Ruan's C H R C conjecture says that the cohomology of the hyper-kahler resolution of an orbifold is isomorphic to its orbifold cohomology. So our results of the orbifold Chow ring of hypertoric Deligne-Mumford stacks will be used to check the C H R C . Quantum cohomology of Deligne-Mumford stacks will be an interesting topic in the future. We already compute the orbifold cohomology of toric Deligne-Mumford stacks and hypertoric Deligne-Mumford stacks. What's the quantum cohomology of these two stacks? Our results will have applications in studying the quantum cohomology of Deligne-Mumford stacks. 136 5.3 Future Research. 5.3.1 Twisted orbifold Chow ring Ruan [Ruan2] defined twisted orbifold cohomology using inner local systems on the inertia orbifold. Later on people discovered that a C*-gerbe over an orbifold naturally gives an inner local system. I will study the <C*-gerbes over toric Deligne-Mumford stacks. Since a toric Deligne-Mumford stack is a quotient stack [Z/G], a class of discrete torsion a G H2(G,C*) defines a C*-gerbe over the toric Deligne-Mumford stack. But not every C*-gerbe comes in this way. Is there a combinatorial description of the gerbes over toric Deligne-Mumford stacks that comes from discrete torsion elements? I will study this question and compute the twisted orbifold Chow rings of toric Deligne-Mumford stacks. 5.3.2 Quantum cohomology of gerbes Orbifold cohomology of finite abelian gerbes over smooth schemes were computed in [Jiangl]. Let /J, be a finite abelian group and X a smooth scheme. Then trivial and nontrivial /Lt-gerbes over X have the same orbifold cohomology [Jiangl]. For the quantum cohomology, Pantev and Sharpe [PS] predicted k copies of the quantum cohomology of X for the quantum cohomology of a /x^-gerbe over X. For example, for the nontrivial //2-gerbe X over P 1 , they predicted that the quantum cohomology should be QH*rb(X) = Q[x, t, q]/{x2 — qt, t2 - 1). For the trivial gerbe, we have QH*ORB{\FL / p2]) = Q[x,t,q]/(x2 - q,t2 - 1). These two gerbes have the same orbifold cohomology by letting q.be zero, while they have different quantum cohomology which are both 2 copies of the quantum cohomology of P 1 . I will study the quantum cohomology of such gerbes and compute the quantum cohomology of 137 general ju-gerbes. 5.3.3 The Crepant resolution conjecture. One of the motivation to compute orbifold Chow ring is to check the Co-homological Hyperkahler Resolution Conjecture of Ruan [Ruan]. For toric Deligne-Mumford stacks, Borisov, Chen and Smith gave an example [BCS] to show that the C H R C conjecture is not true in general, or maybe true when we add some quantum corrections. I will study this question in the future studies. In Chapter 4 we com-puted the orbifold Chow ring of any hypertoric Deligne-Mumford stack. We don't know if there exists a crepant resolution for a hypertoric orbifold, or suppose it ex-ists, if it is a smooth hypertoric variety. Since hypertoric varieties are hyperkahler, maybe we can check the conjecture if we know the crepant resolution. Bryan and Graber [BG] generalized the C H R C conjecture to the whole Gromov-Witten theory which we call the "crepant resolution conjecture". In [BG], Bryan and Graber solved the genus zero conjecture for the symmetric product orb-ifold [ (C 2 )" /5 n ] . Let G be a finite group and act on C 2 . We take the semi-product Gn x Sn = Gn and the group Gn acts on (C 2 )" . The quotient [(C2)n/Gn] is called the wreath product stack. Let Y be the crepant resolution of C2/G, then the Hilbert scheme Y^ of n-points on Y is the crepant resolution of the wreath product (C2)n/Gn, see [W], [QW]. In [QW], the authors proved that the ordinary cohomol-ogy ring of the Hilbert scheme Y^ is isomorphic to the orbifold cohomology ring of [(C2)n/Gn] when G is a cyclic group. A natural question is to ask if the C R C is true for wreath product stacks. For An resolutions Y, Maulik [Maulik] already computed the Gromov-Witten theory of Y and proved the Gromov-Witten/Hilbert-Scheme correspondence. I will study the orbifold Gromov-Witten theory of the 138 wreath product to see if the C R C is true. 139 Bibliography [BCS] L . Borisov, L . Chen and G . Smith, The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc. 18 (2005), no . l , 193-215, math.AG/0309229. [BG] J . Bryan and T . Graber, The crepant resolution conjecture, math.AG/0610129. [BGP] J . Bryan, T . Graber and R. Pandharipande, The orbifold quantum co-homology of [ C 2 / Z 3 ] and Hurwitz-Hodge integrals, math.AG/0510335. [BJ] J . Bryan and Y . Jiang, The genus zero crepant resolution conjecture for the orbifold [ C 2 / Z 4 ] , preprint. [CR1] W . Chen and Y . Ruan, A new cohomology theory for orbifolds, Comm. Math. Phys. 248 (2004), no. 1, 1-31, math.AG/0004129. [Cox] D . Cox, The homogeneous coordinate ring of a toric variety, J. of Alge-braic Geometry, 4 (1995), 17-50. [Jiangl] Y . Jiang, The orbifold cohomology ring of simplicial toric stack bundles, to appear in Illinois J. Math., math.AG/0504563. [Maulik] D . Maulik, Gromov-Witten theory of An resolutions, preprint. 140 [PS] T . Pantev, E . Sharpe, G L S M ' s for gerbes (and other toric stacks), hep-th/0502053. [QW] Z. Qin, W . Wang, Hilbert schemes of points on the minimal resolution and soliton equations, math.QA/0404540. [Ruan] Y . Ruan, Cohomology ring of crepant resolutions of orbifolds, math.AG/0108195. [Ruan2] Y . Ruan, Discrete torsion and twisted orbifold cohomology, math.AG/0005299. [W] W . Wang, The Farahat-Higman ring of wreath products and Hilbert schemes, Adv. in Math. 187 (2004), 417-446, math.QA/0205071. 141
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Title | Toric orbifold Chow rings |
Creator |
Jiang, Yunfeng |
Publisher | University of British Columbia |
Date Issued | 2007 |
Description | In this thesis we study the orbifold Chow ring of smooth Deligne-Mumford stacks which are related to toric models. We give a new quotient construction of toric Deligne-Mumford stacks defined by Borisov-Chen-Smith such that toric Deligne-Mumford stacks have more representations as quotient stacks. We define toric stack bundles using this new construction and compute their orbifold Chow rings. As an interesting application, we compute the orbifold Chow ring of finite abelian gerbes over smooth schemes. The extended stacky fans we introduced are used to give a new quotient construction of toric Deligne-Mumford stacks. These new combinatorial data have relations to stacky hyperplane arrangements, i.e. every stacky hyperplane arrangement determines an extended stacky fan. The hyperplane arrangement determines the topology of the associated hypertoric varieties. We define hypertoric Deligne- Mumford stacks using stacky hyperplane arrangements, generalizing the construction of Hausel and Sturmfels. Their orbifold Chow rings are computed as well. Borisov, Chen and Smith computed the orbifold Chow ring of projective toric Deligne-Mumford stacks. We generalize their formula to semi-projective toric Deligne-Mumford stacks. The hypertoric Deligne-Mumford stack is a closed substack of the Lawrence toric Deligne-Mumford stack associated to the stacky hyperplane arrangement which is semi-projective, but not projective. We prove that the orbifold Chow ring of a Lawrence toric Deligne-Mumford stack is isomorphic to the orbifold Chow ring of its associated hypertoric Deligne-Mumford stack. This is the orbifold Chow ring analogue of a result of Hausel and Sturmfels. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-01-27 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080418 |
URI | http://hdl.handle.net/2429/30895 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
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UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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