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108
Geometric invariant theory and flips
 Jour. AMS
, 1996
"... Ever since the invention of geometric invariant theory, it has been understood that the quotient it constructs is not entirely canonical, but depends on a choice: the choice of a linearization of the group action. However, the founders of the subject never made a systematic study of this dependence. ..."
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Cited by 88 (3 self)
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Ever since the invention of geometric invariant theory, it has been understood that the quotient it constructs is not entirely canonical, but depends on a choice: the choice of a linearization of the group action. However, the founders of the subject never made a systematic study of this dependence. In light of its fundamental and elementary nature, this is a rather surprising gap, and this paper will attempt to fill it. In a way, this neglect is understandable, because the different quotients must be related by birational transformations, whose structure in higher dimensions is poorly understood. However, it has been considerably clarified in the last dozen years with the advent of Mori theory. In particular, the birational transformations that Mori called flips are ubiquitous in geometric invariant theory; indeed, one of our main results (3.3) describes the mild conditions under which the transformation between two quotients is given by a flip. This paper will not use any of the deep results of Mori theory, but the notion of a flip is central to it. The definition of a flip does not describe the birational transformation explicitly, but in the general case there is not much more to say. So to obtain more concrete results, hypotheses
Stable pairs, linear systems and the Verlinde formula
 Invent. Math
, 1994
"... Let X be a smooth projective complex curve of genus g ≥ 2, let Λ→X be a line bundle of degree d> 0, and let (E, φ) be a pair consisting of a vector bundle E →X such that Λ 2 E = Λ and a section φ ∈ H 0 (E) − 0. This paper will study the moduli theory of such pairs. However, it is by no means a r ..."
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Cited by 86 (8 self)
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Let X be a smooth projective complex curve of genus g ≥ 2, let Λ→X be a line bundle of degree d> 0, and let (E, φ) be a pair consisting of a vector bundle E →X such that Λ 2 E = Λ and a section φ ∈ H 0 (E) − 0. This paper will study the moduli theory of such pairs. However, it is by no means a routine generalization of the wellknown theory of stable
Complete moduli in the presence of semiabelian group action
 Ann. of Math
"... Abstract. I prove the existence, and describe the structure, of moduli space of pairs (P, Θ) consisting of a projective variety P with semiabelian group action and an ample Cartier divisor on it satisfying a few simple conditions. Every connected component of this moduli space is proper. A component ..."
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Cited by 57 (5 self)
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Abstract. I prove the existence, and describe the structure, of moduli space of pairs (P, Θ) consisting of a projective variety P with semiabelian group action and an ample Cartier divisor on it satisfying a few simple conditions. Every connected component of this moduli space is proper. A component containing a projective toric variety is described by a configuration of several polytopes the main of which is the secondary polytope. On the other hand, the component containing a principally polarized abelian variety provides a moduli compactification of Ag. The main irreducible component of this compactification is described by an ”infinite periodic ” analog of secondary polytope and coincides with the
Fibrations of Graphs
 DISCRETE MATH
, 1996
"... A fibration of graphs is a morphism that is a local isomorphism of inneighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. This paper develops systematically the theory of graph fibrations, emphasizing in particular those results that recently found ..."
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Cited by 25 (6 self)
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A fibration of graphs is a morphism that is a local isomorphism of inneighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. This paper develops systematically the theory of graph fibrations, emphasizing in particular those results that recently found application in the theory of distributed systems.
Coarse Alexander duality and duality groups
 JOURNAL OF DIFFERENTIAL GEOMETRY
, 1999
"... We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n − 1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplic ..."
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Cited by 19 (5 self)
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We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n − 1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action G � X determines a collection of “peripheral ” subgroups H1,..., Hk ⊂ G so that the group pair (G, {H1,..., Hk}) is an ndimensional Poincare duality pair. In particular, if G is a 2dimensional 1ended group of type F P2, and G � X is a free simplicial action on a coarse P D(3) space X, then G contains surface subgroups; if in addition X is simply connected, then we obtain a partial generalization of the Scott/Shalen compact core theorem to the setting of coarse P D(3) spaces. In the process we develop coarse topological language and a formulation of coarse Alexander duality which is suitable for applications involving quasiisometries and geometric group theory.
Explicit construction of universal deformation rings
 Modular Forms and Fermat’s Last Theorem
, 1995
"... Abstract. Let V be an absolutely irreducible representation of a profinite group G over the residue field k of a noetherian local ring O. For local complete Oalgebras A with residue field k the representations of G over A that reduce to V over k are given by Oalgebra homomorphisms R → A, where R i ..."
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Cited by 17 (0 self)
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Abstract. Let V be an absolutely irreducible representation of a profinite group G over the residue field k of a noetherian local ring O. For local complete Oalgebras A with residue field k the representations of G over A that reduce to V over k are given by Oalgebra homomorphisms R → A, where R is the universal deformation ring of V. We show this with an explicit construction of R. The ring R is noetherian if and only if H 1 (G, Endk(V)) has finite dimension over k. 1.
A Survey of Characteristic Classes of Singular Spaces
"... The theory of characteristic classes of vector bundles and smooth manifolds plays an important role in the theory of smooth manifolds. An investigation of reasonable notions of characteristic classes of singular spaces started with a systematic study of singular spaces such as singular algebraic va ..."
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Cited by 15 (4 self)
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The theory of characteristic classes of vector bundles and smooth manifolds plays an important role in the theory of smooth manifolds. An investigation of reasonable notions of characteristic classes of singular spaces started with a systematic study of singular spaces such as singular algebraic varieties. We give a quick survey of characteristic classes of singular varieties, mainly focusing on the functorial aspects of some important ones such as the singular versions of the Chern class, the Todd class and Thom–Hirzebruch’s Lclass. Further we explain our recent “motivic” characteristic classes, which in a sense unify these three different theories of characteristic classes. We also discuss bivariant versions of them and characteristic classes of proalgebraic varieties, which are related to the motivic measures/integrations. Finally we explain some recent work on “stringy” versions of these theories, together with some references for “equivariant” counterparts.
NÉRON MODELS AND COMPACTIFIED PICARD SCHEMES OVER THE MODULI STACK OF STABLE CURVES
"... Abstract. We construct modular DeligneMumford stacks Pd,g representable over Mg parametrizing Néron models of Jacobians as follows. Let B be a smooth curve and K its function field, let XK be a smooth genusg curve over K admitting stable minimal model over B. The Néron model N(PicdXK) → B is then ..."
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Cited by 11 (6 self)
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Abstract. We construct modular DeligneMumford stacks Pd,g representable over Mg parametrizing Néron models of Jacobians as follows. Let B be a smooth curve and K its function field, let XK be a smooth genusg curve over K admitting stable minimal model over B. The Néron model N(PicdXK) → B is then the base change of Pd,g via the moduli map B − → Mg of f, i.e.: N(PicdXK) ∼ = Pd,g × B. Moreover P Mg d,g is compactified by a DeligneMumford stack over Mg, giving a completion of Néron models naturally stratified in terms of Néron models.