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The Orbifold Chow Ring of Toric DeligneMumford Stacks
 JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY
, 2004
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Gromov–Witten theory of Deligne–Mumford stacks
, 2006
"... 2. Chow rings, cohomology and homology of stacks 5 3. The cyclotomic inertia stack and its rigidification 10 4. Twisted curves and their maps 18 ..."
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Cited by 129 (10 self)
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2. Chow rings, cohomology and homology of stacks 5 3. The cyclotomic inertia stack and its rigidification 10 4. Twisted curves and their maps 18
Using stacks to impose tangency conditions on curves
 math.AG/0210398. [Ch1] [Ch2] [Co87] [DM69] [FSZ] [Gr68] [Ha83
"... From a scheme Y, an effective Cartier divisor D ⊂ Y, and a positive integer r, we define a stack YD,r and work out some of its basic properties. The most important of these relates morphisms from a curve C into YD,r to morphisms from C into Y such that the order of contact of C with D is a multiple ..."
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Cited by 65 (5 self)
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From a scheme Y, an effective Cartier divisor D ⊂ Y, and a positive integer r, we define a stack YD,r and work out some of its basic properties. The most important of these relates morphisms from a curve C into YD,r to morphisms from C into Y such that the order of contact of C with D is a multiple of r at each point. This is a foundational paper whose results will be applied to the enumerative geometry of curves with tangency conditions in a future paper. 1
The Quantum Orbifold Cohomology of Weighted Projective Spaces
, 2007
"... We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S 1equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit for ..."
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Cited by 56 (20 self)
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We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S 1equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small Jfunction, a generating function for certain genuszero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first nontrivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small Jfunctions of weighted projective complete intersections satisfying a combinatorial condition; this condition
Tame stacks in positive characteristic
"... Since their introduction in [8, 4], algebraic stacks have been a key tool in the algebraic theory of moduli. In characteristic 0, one often is able to work with Deligne–Mumford stacks, which, especially in characteristic 0, enjoy a number of nice properties making them almost as easy to handle as al ..."
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Cited by 56 (14 self)
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Since their introduction in [8, 4], algebraic stacks have been a key tool in the algebraic theory of moduli. In characteristic 0, one often is able to work with Deligne–Mumford stacks, which, especially in characteristic 0, enjoy a number of nice properties making them almost as easy to handle as algebraic
Group actions on stacks and applications
 Michigan Math. J
"... Abstract: We provide the correct framework for the treatment of group actions on algebraic stacks (including fixed points and quotients). It is then used to exploit some natural actions on moduli stacks of maps of curves. This leads to the construction of a nice desingularization of the normal stack ..."
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Cited by 49 (1 self)
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Abstract: We provide the correct framework for the treatment of group actions on algebraic stacks (including fixed points and quotients). It is then used to exploit some natural actions on moduli stacks of maps of curves. This leads to the construction of a nice desingularization of the normal stack of curves with level structures considered by Deligne and Mumford ([DM]), and to a presentation of stacks of Galois covers of curves as quotients of a scheme by a finite group. The motivation at the origin of this article is to investigate some ways in which one can construct moduli for curves and covers above them, using tools from stack theory. This idea arose from reading BertinMézard [BM] (especially its §5) and AbramovichCortiVistoli [ACV]. Our approach is in the spirit of most recent works where one uses the flexibility of the language of algebraic stacks. This language has two (twin) aspects, categorytheoretic on one side, and geometric on the other side. Some of our arguments, especially in section 8, are formal arguments involving general constructions concerning group actions on algebraic stacks (this is more on the categoric side). They are, intrinsically, natural enough so they preserve the ”modular ” aspect. In trying to isolate these arguments, we were led to write results of independent interest. It seemed therefore more adequate to present them in a separated, selfcontained part. Thus the article is split into two parts, of comparable size. More specifically, groups are ubiquitous in Algebraic Geometry (when one focuses on curves and
Moduli of Twisted Sheaves
, 2004
"... Abstract. We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of su ..."
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Cited by 48 (9 self)
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Abstract. We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of surfaces, we show (under a mild hypothesis on the twisting class) that the spaces are asympotically geometrically irreducible, normal, generically smooth, and l.c.i. over the base. We also develop general tools necessary for these results: the theory of associated points and purity of sheaves on Artin stacks, twisted Bogomolov inequalities,
Classical and minimal models of the moduli space of curves of genus two
, 2004
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ORBIFOLD QUANTUM RIEMANNROCH, LEFSCHETZ AND SERRE
, 2009
"... Given a vector bundle F on a smooth DeligneMumford stack X and an invertible multiplicative characteristic class c, we define orbifold GromovWitten invariants of X twisted by F and c. We prove a “quantum RiemannRoch theorem” (Theorem 4.2.1) which expresses the generating function of the twisted i ..."
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Cited by 30 (9 self)
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Given a vector bundle F on a smooth DeligneMumford stack X and an invertible multiplicative characteristic class c, we define orbifold GromovWitten invariants of X twisted by F and c. We prove a “quantum RiemannRoch theorem” (Theorem 4.2.1) which expresses the generating function of the twisted invariants in terms of the generating function of the untwisted invariants. A quantum Lefschetz hyperplane theorem is derived from this by specializing to genus zero. As an application, we determine the relationship between genus0 orbifold GromovWitten invariants of X and that of a complete intersection, under additional assumptions. This provides a way to verify mirror symmetry predictions for some complete intersection orbifolds.
SMOOTH TORIC DELIGNEMUMFORD STACKS
, 2009
"... We give a geometric definition of smooth toric DeligneMumford stacks using the action of a “torus”. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a sta ..."
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Cited by 27 (0 self)
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We give a geometric definition of smooth toric DeligneMumford stacks using the action of a “torus”. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.