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56
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
Good moduli spaces for Artin stacks
"... Abstract. We develop the theory of associating moduli spaces with nice geometric properties to arbitrary Artin stacks generalizing Mumford's geometric invariant theory and tame stacks. ..."
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Cited by 47 (8 self)
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Abstract. We develop the theory of associating moduli spaces with nice geometric properties to arbitrary Artin stacks generalizing Mumford's geometric invariant theory and tame stacks.
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.
Noetherian approximation of algebraic spaces and stacks
, 2008
"... Abstract. We show that every scheme (resp. algebraic space, resp. algebraic stack) that is quasicompact with quasifinite diagonal can be approximated by a noetherian scheme (resp. algebraic space, resp. stack). Examples of applications are generalizations of Chevalley’s, Serre’s and Zariski’s theo ..."
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Cited by 21 (4 self)
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Abstract. We show that every scheme (resp. algebraic space, resp. algebraic stack) that is quasicompact with quasifinite diagonal can be approximated by a noetherian scheme (resp. algebraic space, resp. stack). Examples of applications are generalizations of Chevalley’s, Serre’s and Zariski’s theorems and Chow’s lemma.
Moduli of weighted stable maps and their gravitational descendants
, 2006
"... We study the intersection theory on the moduli spaces of maps of npointed curves f: (C,s1,... sn) → V which are stable with respect to the weight data (a1,..., an), 0 ≤ ai ≤ 1. After describing the structure of these moduli spaces, we prove a formula describing the way each descendant changes und ..."
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Cited by 19 (1 self)
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We study the intersection theory on the moduli spaces of maps of npointed curves f: (C,s1,... sn) → V which are stable with respect to the weight data (a1,..., an), 0 ≤ ai ≤ 1. After describing the structure of these moduli spaces, we prove a formula describing the way each descendant changes under a wall crossing. As a corollary, we compute the weighted descendants in terms of the usual ones, i.e. for the weight data (1,..., 1), and vice versa.
TWISTED STABLE MAPS TO TAME ARTIN STACKS
, 801
"... 2. Twisted curves 3 3. Interlude: Relative moduli spaces 10 4. Twisted stable maps 14 ..."
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Cited by 18 (4 self)
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2. Twisted curves 3 3. Interlude: Relative moduli spaces 10 4. Twisted stable maps 14
Existence of quotients by finite groups and coarse moduli spaces
 Aug 2007, arXiv:0708.3333v1. D. RYDH
"... Abstract. In this paper we prove the existence of several quotients in a very general setting. We consider finite group actions and more generally groupoid actions with finite stabilizers generalizing the results of Keel and Mori. In particular we show that any algebraic stack with finite inertia st ..."
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Cited by 13 (4 self)
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Abstract. In this paper we prove the existence of several quotients in a very general setting. We consider finite group actions and more generally groupoid actions with finite stabilizers generalizing the results of Keel and Mori. In particular we show that any algebraic stack with finite inertia stack has a coarse moduli space. We also show that any algebraic stack with quasifinite diagonal has a locally quasifinite flat cover. The proofs do not use noetherian methods and are valid for general algebraic spaces and algebraic stacks.
MODULI SPACES OF SEMISTABLE SHEAVES ON PROJECTIVE DELIGNEMUMFORD STACKS
, 811
"... Abstract. In this paper we introduce a notion of Gieseker stability for coherent sheaves on tame DeligneMumford stacks with projective moduli scheme and some chosen generating sheaf on the stack in the sense of Olsson and Starr [OS03]. We prove that this stability condition is open, and pure dimens ..."
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Cited by 12 (1 self)
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Abstract. In this paper we introduce a notion of Gieseker stability for coherent sheaves on tame DeligneMumford stacks with projective moduli scheme and some chosen generating sheaf on the stack in the sense of Olsson and Starr [OS03]. We prove that this stability condition is open, and pure dimensional semistable sheaves form a bounded family. We explicitly construct the moduli stack of semistable sheaves as a finite type global quotient, and study the moduli scheme of stable sheaves and its natural compactification in the same spirit as the seminal paper of Simpson [Sim94]. With this general machinery we are able to retrieve, as special cases, results of Lieblich [Lie07] and Yoshioka [Yos06] about moduli of twisted sheaves and results of MaruyamaYokogawa [MY92] about moduli of parabolic bundles. Overview We define a notion of stability for coherent sheaves on stacks, and construct a moduli stack of semistable sheaves. The class of stacks that is suitable to approach this problem is the class of projective stacks: tame stacks (for instance DeligneMumford stacks in characteristic zero) with projective moduli scheme and a locally free sheaf that is “very ample ” with respect to
ON THE LOCAL QUOTIENT STRUCTURE OF ARTIN STACKS
, 2009
"... We show that near closed points with linearly reductive stabilizer, Artin stacks are formally locally quotient stacks by the stabilizer and conjecture that the statement holds étale locally. In particular, we prove that if the stabilizer of a point is linearly reductive, the stabilizer acts algebra ..."
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Cited by 11 (6 self)
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We show that near closed points with linearly reductive stabilizer, Artin stacks are formally locally quotient stacks by the stabilizer and conjecture that the statement holds étale locally. In particular, we prove that if the stabilizer of a point is linearly reductive, the stabilizer acts algebraically on a miniversal deformation space generalizing results of Pinkham and Rim.
GromovWitten theory of étale gerbes I: root gerbes
, 2009
"... Let X be a smooth complex projective algebraic variety. Given a line bundle L over X and an integer r> 1 we study the GromovWitten theory of the stack rp L/X of rth root of L. We prove an exact formula expressing genus 0 GromovWitten invariants of rp L/X in terms of those of X. Assuming that ..."
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Cited by 9 (3 self)
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Let X be a smooth complex projective algebraic variety. Given a line bundle L over X and an integer r> 1 we study the GromovWitten theory of the stack rp L/X of rth root of L. We prove an exact formula expressing genus 0 GromovWitten invariants of rp L/X in terms of those of X. Assuming that either rp L/X or X has semisimple quantum cohomology, we prove an exact formula between higher genus invariants. We also present constructions of moduli stacks of twisted stable maps to rp L/X starting from moduli stack of stable maps to X.