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47
Alternate compactifications of moduli spaces of curves
"... We give an informal survey, emphasizing examples and open problems, of two interconnected research programs in moduli of curves: the systematic classification of modular compactifications of Mg,n, and the study of Mori chamber decompositions of M g,n. ..."
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Cited by 23 (6 self)
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We give an informal survey, emphasizing examples and open problems, of two interconnected research programs in moduli of curves: the systematic classification of modular compactifications of Mg,n, and the study of Mori chamber decompositions of M g,n.
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.
Singularities with Gmaction and the log minimal model program for M g
, 2010
"... We give a precise formulation of the modularity principle for the log canonical models ..."
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Cited by 19 (11 self)
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We give a precise formulation of the modularity principle for the log canonical models
Variation of Geometric Invariant Theory quotients and derived categories
, 2014
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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
COMPACTIFIED PICARD STACKS OVER Mg
, 2007
"... We study algebraic (Artin) stacks over Mg giving a functorial way of compactifying the relative degree d Picard variety for families of stable curves. We also describe for every d the locus of genus g stable curves over which we get DeligneMumford stacks strongly representable over Mg. ..."
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Cited by 10 (3 self)
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We study algebraic (Artin) stacks over Mg giving a functorial way of compactifying the relative degree d Picard variety for families of stable curves. We also describe for every d the locus of genus g stable curves over which we get DeligneMumford stacks strongly representable over Mg.
GROTHENDIECK DUALITY FOR DeligneMumford Stacks
, 2009
"... We prove the existence of the dualizing functor for a separated morphism of algebraic stacks with affine diagonal; then we explicitly develop duality for compact DeligneMumford stacks focusing in particular on the morphism from a stack to its coarse moduli space and on representable morphisms. We ..."
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Cited by 9 (0 self)
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We prove the existence of the dualizing functor for a separated morphism of algebraic stacks with affine diagonal; then we explicitly develop duality for compact DeligneMumford stacks focusing in particular on the morphism from a stack to its coarse moduli space and on representable morphisms. We explicitly compute the dualizing complex for a smooth stack over an algebraically closed field and prove that Serre duality holds for smooth compact DeligneMumford stacks in its usual form. We prove also that a proper CohenMacaulay stack has a dualizing sheaf and it is an invertible sheaf when it is Gorenstein. As an application of this general machinery we compute the dualizing sheaf of a tame nodal curve.
The ample cone of the moduli spaces of sheaves on the plane, preprint
"... Abstract. Let ξ be the Chern character of a stable coherent sheaf on P2. We compute the cone of effective divisors on the moduli space M(ξ) of semistable sheaves on P2 with Chern character ξ. The computation hinges on finding a good resolution of the general sheaf in M(ξ). This resolution is determi ..."
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Cited by 9 (7 self)
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Abstract. Let ξ be the Chern character of a stable coherent sheaf on P2. We compute the cone of effective divisors on the moduli space M(ξ) of semistable sheaves on P2 with Chern character ξ. The computation hinges on finding a good resolution of the general sheaf in M(ξ). This resolution is determined by Bridgeland stability and arises from a wellchosen Beilinson spectral sequence. The existence of a good choice of spectral sequence depends on remarkable numbertheoretic properties of the slopes of exceptional bundles.