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Multigraded CastelnuovoMumford Regularity
 J. REINE ANGEW. MATH
, 2003
"... We develop a multigraded variant of CastelnuovoMumford regularity. Motivated by toric ..."
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Cited by 27 (6 self)
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We develop a multigraded variant of CastelnuovoMumford regularity. Motivated by toric
A derived category approach to generic vanishing
 J. Reine Angew. Math
"... Abstract. We prove a Generic Vanishing Theorem for coherent sheaves on an abelian variety over an algebraically closed field k. When k = C this implies a conjecture of Green and Lazarsfeld. 1. ..."
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Cited by 21 (2 self)
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Abstract. We prove a Generic Vanishing Theorem for coherent sheaves on an abelian variety over an algebraically closed field k. When k = C this implies a conjecture of Green and Lazarsfeld. 1.
Regularity on abelian varieties. II. Basic results on linear series and defining equations
 J. Algebraic Geom
"... This paper is mainly concerned with applying the theory of Mukai regularity (or Mregularity) introduced in [PP] to the study of linear series given by multiples of ample line bundles on abelian varieties. We show that this regularity notion allows one to define a new invariant of a line bundle, cal ..."
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Cited by 18 (7 self)
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This paper is mainly concerned with applying the theory of Mukai regularity (or Mregularity) introduced in [PP] to the study of linear series given by multiples of ample line bundles on abelian varieties. We show that this regularity notion allows one to define a new invariant of a line bundle, called Mregularity index, which will be seen to roughly
GVSHEAVES, FOURIERMUKAI TRANSFORM, AND GENERIC VANISHING
"... Abstract. We prove a formal criterion for generic vanishing, in the sense originated by Green and Lazarsfeld and pursued further by Hacon, but in the context of an arbitrary FourierMukai correspondence. For smooth projective varieties we apply this to deduce a Kodairatype generic vanishing theorem ..."
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Cited by 18 (13 self)
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Abstract. We prove a formal criterion for generic vanishing, in the sense originated by Green and Lazarsfeld and pursued further by Hacon, but in the context of an arbitrary FourierMukai correspondence. For smooth projective varieties we apply this to deduce a Kodairatype generic vanishing theorem for adjoint bundles associated to nef line bundles, and in fact a more general generic Nadeltype vanishing theorem for multiplier ideal sheaves. Still in the context of the Picard variety, the same method gives various other generic vanishing results, by reduction to standard vanishing theorems. We further use our criterion in order to address some examples related to generic vanishing on higher rank moduli spaces.
Dimension estimates for Hilbert schemes and effective base point freeness on moduli spaces of vector bundles on curves
 Duke Math Jour
, 2001
"... We give an upper bound on the dimension of the Hilbert scheme of quotients of an arbitrary vector bundle on a smooth projective curve, depending on the minimal degree of such a quotient. The bound is used for deriving effective base point freeness statements for generalized theta linear series on mo ..."
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Cited by 15 (6 self)
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We give an upper bound on the dimension of the Hilbert scheme of quotients of an arbitrary vector bundle on a smooth projective curve, depending on the minimal degree of such a quotient. The bound is used for deriving effective base point freeness statements for generalized theta linear series on moduli spaces of vector bundles.
Syzygies, multigraded regularity and toric varieties
, 2006
"... Using multigraded Castelnuovo–Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B1,...,Bℓ on X and m1,...,mℓ ∈ N, consider the line bundle L: = B m1 1 ⊗···⊗Bmℓ ℓ. We give conditions on the mi which guarantee that the ide ..."
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Cited by 13 (4 self)
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Using multigraded Castelnuovo–Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B1,...,Bℓ on X and m1,...,mℓ ∈ N, consider the line bundle L: = B m1 1 ⊗···⊗Bmℓ ℓ. We give conditions on the mi which guarantee that the ideal of X in P(H0 (X, L) ∗ ) is generated by quadrics and that the first p syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.
GENERIC VANISHING AND MINIMAL COHOMOLOGY CLASSES ON ABELIAN VARIETIES
, 2006
"... This paper is concerned with a relationship between the existence of subvarieties of principally polarized abelian varieties (ppav’s) having minimal cohomology class and the (generic) vanishing of certain sheaf cohomology, based on the Generic Vanishing criterion studied in [PP3]. This is in analogy ..."
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Cited by 10 (3 self)
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This paper is concerned with a relationship between the existence of subvarieties of principally polarized abelian varieties (ppav’s) having minimal cohomology class and the (generic) vanishing of certain sheaf cohomology, based on the Generic Vanishing criterion studied in [PP3]. This is in analogy with the wellknown equivalence between a subvariety in projective space being
CASTELNUOVO THEORY AND THE GEOMETRIC SCHOTTKY PROBLEM
"... The aim of this paper is to show that Castelnuovo theory in projective space (cf. [ACGH] Ch.III §2 and [GH] Ch.4 §3) has a precise analogue for abelian varieties. This can be quite surprisingly related in a concrete way to the geometric Schottky problem, namely the problem of identifying Jacobians a ..."
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Cited by 9 (3 self)
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The aim of this paper is to show that Castelnuovo theory in projective space (cf. [ACGH] Ch.III §2 and [GH] Ch.4 §3) has a precise analogue for abelian varieties. This can be quite surprisingly related in a concrete way to the geometric Schottky problem, namely the problem of identifying Jacobians among all principally polarized abelian varieties (ppav’s) via geometric