## The Orbifold Chow Ring of Toric Deligne-Mumford Stacks (2004)

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Venue: | JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY |

Citations: | 67 - 5 self |

### BibTeX

@MISC{Borisov04theorbifold,

author = {Lev A. Borisov and Linda Chen and Gregory G. Smith},

title = { The Orbifold Chow Ring of Toric Deligne-Mumford Stacks },

year = {2004}

}

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### Abstract

### Citations

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Citation Context ... conclude that our family is flat on a Zariski open subset of P 1 which contains both 0 and #. # THE ORBIFOLD CHOW RING OF TORIC DELIGNE-MUMFORD STACKS 23 Remark 7.3. In analogy with Theorem 15.17 in =-=[11]-=-, the flat family constructed in the proof of Theorem 7.1 can be interpreted as a pair of Grobner deformations with respect to the appropriate weight orders connecting the ideal I 1 + I 2 with its ini... |

826 |
Introduction to toric varieties
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Citation Context ...ollows that the set {a 1 , . . . , a n } defines a map # # : (Z n ) # # Z n-d # = Pic(X) and the short exact sequence (2.0.1) becomes 0 # N # # # ---# (Z n ) # # # ---# Pic(X) # 0; see Section 3.4 in =-=[15]-=-. Our goal is to extend this theory to a larger class of maps. Let N be a finitely generated abelian group and consider a group homomorphism # : Z n # N . The map # is determined by a finite subset {b... |

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Citation Context ...ly write (-) # for the functor Hom Z (-, Z). The dual map # # : (Z n ) # # DG(#) is defined as follows. Choose projective resolutions E and F of the Z-modules Z n and N respectively. Theorem 2.2.6 in =-=[26]-=- shows that # : Z n # N lifts to a morphism E # F and Subsection 1.5.8 in [26] shows that the mapping cone Cone(#) fits into an exact sequence of cochain complexes 0 # F # Cone(#) # E[1] # 0. Since E ... |

316 | The homogeneous coordinate ring of a toric variety
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Citation Context ...e purpose of this section is to associate a smooth Deligne-Mumford stack to certain combinatorial data. This construction is inspired by the quotient construction for toric varieties; for example see =-=[6]-=-. Let N be a finitely generated abelian group of rank d. We write N for the lattice generated by N in the d-dimensional Q-vector space NQ := N# Z Q. The natural map N # N is denoted by b ## b. Let # b... |

208 | Convex bodies and algebraic geometry, an introduction to the theory of toric varieties, Ergebnisse der Math. 15, Springer-Verlag - Oda - 1988 |

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Citation Context ...orresponding to lattice points in N . Proposition 4.8 in [13] implies that the sequence # n i=1 # j (b i )y b i for 1 # j # d forms a homogeneous system of parameters on S/I 2 , and Theorem 5.1.16 in =-=[5]-=- shows that this sequence is also a homogeneous system of parameters on S/I# # . Thus, # m i=1 # j (b i )y b i for 1 # j # d is a regular sequence on both S/I 2 and S/I# # and the Hilbert functions of... |

190 | Lectures on Polytopes. Graduate Texts in Mathematics 152, Springer-Verlag - Ziegler - 1995 |

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Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A
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Citation Context ...s of K v1 ,v 2 ,v 3 and [Z H /G]. Since both K v1 ,v 2 ,v 3 and [Z H /G] are smooth Deligne-Mumford stacks, their coarse moduli spaces have at worst quotient singularities. Applying Theorem VI.1.5 in =-=[17]-=-, we deduce that, in fact, e produces an isomorphism between the coarse moduli spaces. To prove that e is an isomorphism of stacks, it remains to show that e gives an isomorphism between the isotropy ... |

159 |
Intersection theory on algebraic stacks and on their moduli spaces
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Citation Context ... ## # t i which implies A #,# # is integral over B #,# # and completes the proof. # Proposition 3.2. The quotient X (#) is a Deligne-Mumford stack. Proof. By Corollary 2.2 in [10] (or Example 7.17 in =-=[25]-=-), it is enough to show that the stabilizers of the geometric points of Z are finite and reduced. Lemma 3.1 shows that 8 LEV A. BORISOV, LINDA CHEN, AND GREGORY G. SMITH the map Z G # Z Z defined by (... |

143 |
Moret-Bailly L., Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3
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Citation Context ...LIGNE-MUMFORD STACKS 7 E # S with a G-equivariant map E # Z and the morphisms are isomorphisms which preserve the map to Z. Since Z is smooth, X (#) is a smooth algebraic stack; see Remark 10.13.2 in =-=[19]-=-. The next proposition shows that X (#) is in fact a DeligneMumford stack. We call X (#) the toric Deligne-Mumford stack associated to the stacky fan #. Lemma 3.1. The map Z G # Z Z with (z, g) ## (z,... |

138 |
Les schémas de modules de courbes elliptiques
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Citation Context ... [ 6 4 ]. Furthermore, Z := A 2 - {(0, 0)} and # # G # = C # acts by (z 1 , z 2 ) ## (# 6 z 1 , # 4 z 2 ). In this case, X (#) is precisely the moduli stack of elliptic curves M 1,1 ; see Page 126 in =-=[9]-=-. To illustrate that a toric Deligne-Mumford stack depends on the set {b i }, we include the following: Example 3.6. Let # be the complete fan in Q, which implies Z := A 2 - {(0, 0)}, and let N := Z #... |

134 |
A new cohomology theory for orbifold
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Citation Context ... Introduction The orbifold Chow ring of a Deligne-Mumford stack, defined by Abramovich, Graber and Vistoli [2], is the algebraic version of the orbifold cohomology ring introduced by W. Chen and Ruan =-=[7]-=- [8]. By design, this ring incorporates numerical invariants, such as the orbifold Euler characteristic and the orbifold Hodge numbers, of the underlying variety. The product structure is induced by t... |

80 |
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Citation Context ... Moreover, it follows from the decomposition of I 2 in Proposition 4.8 in [13] that the sequence # n i=1 # j (b i )y b i for 1 # j # d forms a homogeneous system of parameters on S/I 2 . Lemma 4.6 in =-=[23]-=- shows that S/I 2 is a Cohen-Macaulay ring, so we deduce that # n i=1 # j (b i )y b i for 1 # j # d is a regular sequence. Being a regular sequence is an open condition on the set of d-tuples of degre... |

78 | Non-Archimedian integrals and stringy Euler numbers of log terminal pairs - Batyrev - 1999 |

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Citation Context ... 1 # i # n, let a i denote the unique minimal lattice generator of # i in # and let # i be the positive integer satisfying the relation b i = # i a i . The Jurkiewicz-Danilov Theorem (see Page 134 in =-=[21]-=-) states that there is a surjective homomorphism of graded rings from Q[x 1 , . . . , x n ] to A # # X(#) # given by x i ## D i where D i is the torus invariant Weil divisor on X(#) associated with # ... |

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Twisted bundles and admissible covers
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Citation Context ...rem 2.2.6 in [26] shows that # : Z n # N lifts to a morphism E # F and Subsection 1.5.8 in [26] shows that the mapping cone Cone(#) fits into an exact sequence of cochain complexes 0 # F # Cone(#) # E=-=[1]-=- # 0. Since E is projective, we have the exact sequence of cochain complexes (2.0.2) 0 -# E[1] # -# Cone(#) # -# F # -# 0 and the associated long exact sequence in cohomology contains the exact sequen... |

58 | Orbifold Gromov–Witten theory, in: Orbifolds in mathematics and physics - Chen, Ruan - 2002 |

55 | Orbifold cohomology for global quotients
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Citation Context ...ng theory and results in Batyrev [3] and Yasuda [27], one expects that, in nice situations, the orbifold Chow ring coincides with the Chow ring of a resolution of singularities. Fantechi and Gottsche =-=[14]-=- and Uribe [24] verify this conjecture when the orbifold is Sym n (S) where S is a smooth projective surface with K S = 0 and the resolution is Hilb n (S). The initial motivation for this project was ... |

45 | Algebraic orbifold quantum products, Orbifolds in mathematics and physics - Abramovich, Graber, et al. - 2002 |

35 |
The geometry of schemes, Graduate Texts
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Citation Context ...s: S[t 1 , t -1 1 ] I 1 + I 2 # = S I 1 + I 2 # = S[t 2 , t -1 2 ] I 1 + I 2 . Since a family of a#ne cones is a flat family if and only if the Hilbert function is constant (see Proposition III-56 in =-=[12]-=-), we conclude that our family is flat on a Zariski open subset of P 1 which contains both 0 and #. # THE ORBIFOLD CHOW RING OF TORIC DELIGNE-MUMFORD STACKS 23 Remark 7.3. In analogy with Theorem 15.1... |

35 | Orbifolds as groupoids: an introduction, in: Orbifolds in mathematics and physics - Moerdijk - 2002 |

33 | Binomials ideals - Eisenbud, Sturmfels - 1996 |

31 | Algebraic orbifold quantum products
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Citation Context ...formation of (but is not necessarily isomorphic to) the Chow ring of a crepant resolution. 1. Introduction The orbifold Chow ring of a Deligne-Mumford stack, defined by Abramovich, Graber and Vistoli =-=[2]-=-, is the algebraic version of the orbifold cohomology ring introduced by W. Chen and Ruan [7] [8]. By design, this ring incorporates numerical invariants, such as the orbifold Euler characteristic and... |

28 | Orbifold cohomology of the symmetric product
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Citation Context ...esults in Batyrev [3] and Yasuda [27], one expects that, in nice situations, the orbifold Chow ring coincides with the Chow ring of a resolution of singularities. Fantechi and Gottsche [14] and Uribe =-=[24]-=- verify this conjecture when the orbifold is Sym n (S) where S is a smooth projective surface with K S = 0 and the resolution is Hilb n (S). The initial motivation for this project was to compare the ... |

26 | schémas de modules de courbes elliptiques. In: Modular functions of one variable - Deligne, Rapoport, et al. - 1973 |

20 | Twisted jet, motivic measure and orbifold cohomology
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Citation Context .... The product structure is induced by the degree zero part of the quantum product; in particular, it involves Gromov-Witten invariants. Inspired by string theory and results in Batyrev [3] and Yasuda =-=[27]-=-, one expects that, in nice situations, the orbifold Chow ring coincides with the Chow ring of a resolution of singularities. Fantechi and Gottsche [14] and Uribe [24] verify this conjecture when the ... |

14 | Pavages des simplexes, schémas de graphes recollés et compactification des PGLn+1r /PGLr - Lafforgue - 1999 |

10 | The Chen-Ruan cohomology of weighted projective spaces
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Citation Context ...#) is a complete toric Deligne-Mumford stack, then there is an isomorphism of Q-graded rings: A # orb # X (#) # # = Q[N ] # # # n i=1 #(b i )y b i : # # Hom(N,Z) # . Using di#erential geometry, Jiang =-=[16]-=- establishes this result for the weighted projective space P(1, 2, 2, 3, 3, 3). Our proof of this theorem involves two steps. By definition, the orbifold Chow ring A # orb # X (#) # is isomorphic as a... |

8 | Notes on the construction of the moduli space of curves
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Citation Context ...#((z # b # # ) -1 ) # # i ## # t i which implies A #,# # is integral over B #,# # and completes the proof. # Proposition 3.2. The quotient X (#) is a Deligne-Mumford stack. Proof. By Corollary 2.2 in =-=[10]-=- (or Example 7.17 in [25]), it is enough to show that the stabilizers of the geometric points of Z are finite and reduced. Lemma 3.1 shows that 8 LEV A. BORISOV, LINDA CHEN, AND GREGORY G. SMITH the m... |

7 | Notes on the construction of the moduli space of curves, Recent progress in intersection theory - Edidin - 1997 |

6 |
Lectures on polytopes, Graduate Texts in Math. 152
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Citation Context ... b n which span Q d , there is a dual configuration [a 1 a n ] # Q (n-d)n such that (2.0.1) 0 -# Q d [b 1 b n ] T -----# Q n [a 1 a n ] ----# Q n-d -# 0 is a short exact sequence; see Theorem 6.14 in =-=[28]-=-. The set of vectors {a 1 , . . . , a n } is uniquely determined up to a linear coordinates transformation in Q n-d . This duality plays a role in study of smooth toric varieties. Specifically, let # ... |

4 | Borisov, String cohomology of a toroidal singularity - A |

4 |
Orbifold cohomology group of toric varieties, Orbifolds in mathematics and physics
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Citation Context ...ee that the coarse moduli space of I # X (#) # is isomorphic to the disjoint union of X # #/#(v) # for all v # Box(#). In particular, we recover the description of the twisted sectors in Section 6 of =-=[22]-=-. 5. Module Structure on A # orb # X (#) # The goal of this section to describe the orbifold Chow ring of a complete toric DeligneMumford stack as an abelian group. Throughout this section, we assume ... |

2 | String cohomology of a toroidal singularity - Borisov |

1 |
Orbifold Gromov-Witten theory" in Orbifolds in mathematics and physics
- Chen, Ruan
- 2002
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Citation Context ...roduction The orbifold Chow ring of a Deligne-Mumford stack, defined by Abramovich, Graber and Vistoli [2], is the algebraic version of the orbifold cohomology ring introduced by W. Chen and Ruan [7] =-=[8]-=-. By design, this ring incorporates numerical invariants, such as the orbifold Euler characteristic and the orbifold Hodge numbers, of the underlying variety. The product structure is induced by the d... |

1 |
Pavages des simplexes, schemas de graphes recolles et compactification des
- Laorgue
- 1999
(Show Context)
Citation Context ... (z, g) ## (z, z g) is a finite morphism. It follows that each stabilizer is a finite group scheme. Since we are working in characteristic zero, all finite group schemes are reduced. # Remark 3.3. In =-=[18], a "toric stac-=-k" is defined to be the quotient of a toric variety by its torus. Since such a quotient is never a Deligne-Mumford stack, X (#) is not a "toric stack". Remark 3.4. The definition of X (... |

1 |
Orbifolds as groupoids: an introduction" in Orbifolds in mathematics and physics
- Moerdijk
- 2002
(Show Context)
Citation Context ...k I # X (#) # as a disjoint union of certain closed substacks. To describe the connection between the combinatorics of the stacky fan # and the substacks of X (#), we use the theory of groupoids; see =-=[20]-=- for an introduction. Recall that a homomorphism of groupoids # : (R # # U # ) -# (R # U) is called a Morita equivalence if (1) the square R # (s,t) ---# U # U # # # # # # # ### R (s,t) ---# U U is Ca... |

1 | Ruan,Cohomology Ring of Crepant Resolutions of Orbifolds - unknown authors |