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182
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
The witten equation, mirror symmetry and quantum singularity theory
, 2009
"... For any quasihomogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to GromovWitten theory and generalizes the theory of rspin curves, which corresponds ..."
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Cited by 53 (2 self)
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For any quasihomogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to GromovWitten theory and generalizes the theory of rspin curves, which corresponds to the simple singularity Ar−1. The main results are that we resolve two outstanding conjectures of Witten. The first conjecture is that ADEsingularities are selfdual; and the second conjecture is that the total potential functions of ADEsingularities satisfy corresponding ADEintegrable hierarchies. Other cases of integrable hierarchies are also discussed.
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,
Quot functors for DeligneMumford stacks
 Comm. Algebra
"... Abstract. Given a separated and locally finitelypresented DeligneMumford stack X over an algebraic space S, and a locally finitelypresented OXmodule F, we prove that the Quot functor Quot(F/X/S) is represented by a separated and locally finitelypresented algebraic space over S. Under additional ..."
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Cited by 42 (5 self)
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Abstract. Given a separated and locally finitelypresented DeligneMumford stack X over an algebraic space S, and a locally finitelypresented OXmodule F, we prove that the Quot functor Quot(F/X/S) is represented by a separated and locally finitelypresented algebraic space over S. Under additional hypotheses, we prove that the connected components of Quot(F/X/S) are quasiprojective over S. Contents 1. Statement of results 1 2. Representability by an algebraic space 3
The crepant resolution conjecture
, 2006
"... Abstract. For orbifolds admitting a crepant resolution and satisfying a hard Lefschetz condition, we formulate a conjectural equivalence between the GromovWitten theories of the orbifold and the resolution. We prove the conjecture for the equivariant GromovWitten theories of Sym n C 2 and Hilb n C ..."
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Cited by 42 (8 self)
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Abstract. For orbifolds admitting a crepant resolution and satisfying a hard Lefschetz condition, we formulate a conjectural equivalence between the GromovWitten theories of the orbifold and the resolution. We prove the conjecture for the equivariant GromovWitten theories of Sym n C 2 and Hilb n C 2. 1.