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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.
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.
NAGATA COMPACTIFICATION FOR ALGEBRAIC SPACES
"... Abstract. We prove the Nagata compactification theorem for any separated map of finite type between quasicompact and quasiseparated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian approximation for such algebraic spaces, generaliz ..."
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Cited by 20 (3 self)
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Abstract. We prove the Nagata compactification theorem for any separated map of finite type between quasicompact and quasiseparated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian approximation for such algebraic spaces, generalizing earlier results in the case of schemes. To the memory of Masayoshi Nagata 1.
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.
SUBMERSIONS AND EFFECTIVE DESCENT OF ÉTALE MORPHISMS
, 2007
"... Using the flatification by blowup result of Raynaud and Gruson, we obtain new results for submersive and subtrusive morphisms. We show that universally subtrusive morphisms, and in particular universally open morphisms, are morphisms of effective descent for the fibered category of étale morphism ..."
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Cited by 5 (1 self)
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Using the flatification by blowup result of Raynaud and Gruson, we obtain new results for submersive and subtrusive morphisms. We show that universally subtrusive morphisms, and in particular universally open morphisms, are morphisms of effective descent for the fibered category of étale morphisms. Our results extend and supplement previous treatments on submersive morphisms by Grothendieck, Picavet and Voevodsky. Applications include the universality of geometric quotients and the elimination of noetherian hypotheses in many instances.
The Hilbert stack
, 2010
"... We show that for any locally finitely presented morphism of algebraic stacks X → S with quasicompact and separated diagonal, there is an algebraic stack HSX/S, the Hilbert stack, parameterizing proper algebraic stacks with finite diagonal mapping quasifinitely to X. The technical heart of this is ..."
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Cited by 3 (1 self)
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We show that for any locally finitely presented morphism of algebraic stacks X → S with quasicompact and separated diagonal, there is an algebraic stack HSX/S, the Hilbert stack, parameterizing proper algebraic stacks with finite diagonal mapping quasifinitely to X. The technical heart of this is a generalization of formal GAGA to a nonseparated morphism of algebraic stacks, something that was previously unknown for a morphism of schemes. We also employ derived algebraic geometry, in an essential way, to prove the algebraicity of the stack HSX/S. The Hilbert stack, for a morphism of algebraic spaces, was claimed to exist in [Art74, Appendix §1], but was left unproved due to a lack of foundational results for nonseparated algebraic spaces.
LOCAL PROPERTIES OF GOOD MODULI SPACES
, 2009
"... We study the local properties of Artin stacks and their good moduli spaces, if they exist. We show that near closed points with linearly reductive stabilizer, Artin stacks formally locally admit good moduli spaces. We also give conditions for when the existence of good moduli spaces can be deduced ..."
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We study the local properties of Artin stacks and their good moduli spaces, if they exist. We show that near closed points with linearly reductive stabilizer, Artin stacks formally locally admit good moduli spaces. We also give conditions for when the existence of good moduli spaces can be deduced from the existence of étale charts admitting good moduli spaces.