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36
Restricting linear syzygies: algebra and geometry
, 2005
"... Let X ⊂ P r be a closed scheme in projective space whose homogeneous ideal is generated by quadrics. We say that X (or its ideal IX) satisfies the condition N2,p if the syzygies of IX are linear for p steps. We show that if X satisfies N2,p then a zerodimensional or onedimensional intersection of ..."
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Cited by 35 (4 self)
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Let X ⊂ P r be a closed scheme in projective space whose homogeneous ideal is generated by quadrics. We say that X (or its ideal IX) satisfies the condition N2,p if the syzygies of IX are linear for p steps. We show that if X satisfies N2,p then a zerodimensional or onedimensional intersection of X with a plane of dimension � p is 2regular. This extends a result of Green and Lazarsfeld. We give conditions when the syzygies of X restrict to the syzygies of the intersection. Many of our results also work for ideals generated by forms of higher degree. As applications, we bound the p for which some wellknown projective varieties satisfy N2,p. Another application, carried out by us in a different paper, is a step in the classification of 2regular reduced projective schemes. Extending a result of Fröberg, we determine which monomial ideals satisfy N2,p. We also apply Green’s ‘linear syzygy theorem ’ to deduce a relation between the resolutions of IX and IX∪Γ for a scheme Γ, and apply the result to bound the number of intersection points of certain pairs of varieties such as rational normal scrolls.
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 26 (6 self)
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We develop a multigraded variant of CastelnuovoMumford regularity. Motivated by toric
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.
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 13 (5 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
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 13 (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.
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 11 (4 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.
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 11 (8 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.
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 7 (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
Regularity on abelian varieties III: relationship with Generic Vanishing and applications
 G. Pareschi, M. Popa: Mregularity and the FourierMukai transform
"... 2. GVsheaves and Mregular sheaves on abelian varieties 3 3. Tensor products of GV and Mregular sheaves 6 4. Nefness of GVsheaves 7 ..."
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Cited by 6 (4 self)
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2. GVsheaves and Mregular sheaves on abelian varieties 3 3. Tensor products of GV and Mregular sheaves 6 4. Nefness of GVsheaves 7