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17
Multigraded regularity: syzygies and fat points
, 2005
"... The CastelnuovoMumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understand ..."
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The CastelnuovoMumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understanding of the multigraded Hilbert function of fat point schemes in P n1 × · · · × P n k.
mBLOCKS COLLECTIONS AND CASTELNUOVOMUMFORD REGULARITY IN MULTIPROJECTIVE SPACES
, 2006
"... The main goal of the paper is to generalize CastelnuovoMumford regularity for coherent sheaves on projective spaces to coherent sheaves on ndimensional smooth projective varieties X with an nblock collection B which generates the bounded derived category D b (OXmod). To this end, we use the the ..."
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The main goal of the paper is to generalize CastelnuovoMumford regularity for coherent sheaves on projective spaces to coherent sheaves on ndimensional smooth projective varieties X with an nblock collection B which generates the bounded derived category D b (OXmod). To this end, we use the theory of nblocks and Beilinson type spectral sequence to define the notion of regularity of a coherent sheaf F on X with respect to the nblock collection B. We show that the basic formal properties of the CastelnuovoMumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we compare our definition of regularity with previous ones. In particular, we show that in case of coherent sheaves on P n and for the nblock collection B = (OP n, OP n(1), · · · , OP n(n)) on Pn CastelnuovoMumford regularity and our new definition of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a multiprojective space P n1 × · · ·×P nr with respect to a suitable n1+ · · ·+nrblock collection and we compare it with the multigraded variant of the CastelnuovoMumford regularity given by Hoffman and Wang in [14].
Equations of parametric surfaces with base points via syzygies
, 2003
"... Let S be a parametrized surface in P 3 given as the image of φ: P 1 × P 1 → P 3. This paper will show that the use of syzygies in the form of a combination of moving planes and moving quadrics provides a valid method for finding the implicit equation of S when certain base points are present. This ..."
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Let S be a parametrized surface in P 3 given as the image of φ: P 1 × P 1 → P 3. This paper will show that the use of syzygies in the form of a combination of moving planes and moving quadrics provides a valid method for finding the implicit equation of S when certain base points are present. This work extends the algorithm provided by Cox [5] for when φ has no base points, and it is analogous to some of the results of Busé, Cox, and D’Andrea [2] for the case when φ: P 2 → P 3 has base points.
Regularity and Cohomological Splitting Conditions for Vector Bundles on Multiprojectives Spaces
, 2008
"... Here we give a definition of regularity on multiprojective spaces which is different from the definitions of HoffmannWang and CostaMiró Roig. By using this notion we prove some splitting criteria for vector bundles. 1 ..."
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Here we give a definition of regularity on multiprojective spaces which is different from the definitions of HoffmannWang and CostaMiró Roig. By using this notion we prove some splitting criteria for vector bundles. 1
nBLOCKS COLLECTIONS ON FANO MANIFOLDS AND SHEAVES WITH REGULARITY −∞
, 2007
"... Let X be a smooth Fano manifold equipped with a “nice” nblocks collection in the sense of [3] and F a coherent sheaf on X. Assume that X is Fano and that all blocks are coherent sheaves. Here we prove that F has regularity −∞ in the sense of [3] if Supp(F) is finite, the converse being true under m ..."
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Let X be a smooth Fano manifold equipped with a “nice” nblocks collection in the sense of [3] and F a coherent sheaf on X. Assume that X is Fano and that all blocks are coherent sheaves. Here we prove that F has regularity −∞ in the sense of [3] if Supp(F) is finite, the converse being true under mild assumptions. The corresponding result is also true when X has a geometric collection in the sense of [2].
MULTIGRADED CASTELNUOVOMUMFORD REGULARITY, a ∗INVARIANTS AND THE MINIMAL FREE RESOLUTION
, 2005
"... Abstract. In recent years, two different multigraded variants of CastelnuovoMumford regularity have been developed, namely multigraded regularity, defined by the vanishing of multigraded pieces of local cohomology modules, and the resolution regularity vector, defined by the multidegrees in a mini ..."
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Abstract. In recent years, two different multigraded variants of CastelnuovoMumford regularity have been developed, namely multigraded regularity, defined by the vanishing of multigraded pieces of local cohomology modules, and the resolution regularity vector, defined by the multidegrees in a minimal free resolution. In this paper, we study the relationship between multigraded regularity and the resolution regularity vector. Our method is to investigate multigraded variants of the usual a ∗invariant. This, in particular, provides an effective approach to examining the vanishing of multigraded pieces of local cohomology modules with respect to different graded irrelevant ideals. 1.
Multigraded regularity: coarsenings and resolutions
"... Abstract. Let S = k[x1,..., xn] be a Z rgraded ring with deg(xi) = ai ∈ Z r for each i and suppose that M is a finitely generated Z rgraded Smodule. In this paper we describe how to find finite subsets of Z r containing the multidegrees of the minimal multigraded syzygies of M. To find such a se ..."
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Abstract. Let S = k[x1,..., xn] be a Z rgraded ring with deg(xi) = ai ∈ Z r for each i and suppose that M is a finitely generated Z rgraded Smodule. In this paper we describe how to find finite subsets of Z r containing the multidegrees of the minimal multigraded syzygies of M. To find such a set, we first coarsen the grading of M so that we can view M as a Zgraded Smodule. We use a generalized notion of CastelnuovoMumford regularity, which was introduced by D. Maclagan and G. Smith, to associate to M a number which we call the regularity number of M. The minimal degrees of the multigraded minimal syzygies are bounded in terms of this invariant. 1.
CURVILINEAR BASE POINTS, LOCAL COMPLETE INTERSECTION AND KOSZUL SYZYGIES IN BIPROJECTIVE SPACES
, 2006
"... Let I = 〈f1,f2,f3 〉 be a bigraded ideal in the bigraded polynomial ring k[s, u; t, v]. Assume that I has codimension 2. Then Z = V(I) ⊂ P 1 × P 1 is a finite set of points. We prove that if Z is a local complete intersection, then any syzygy of the fi vanishing at Z, and in a certain degree range, ..."
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Let I = 〈f1,f2,f3 〉 be a bigraded ideal in the bigraded polynomial ring k[s, u; t, v]. Assume that I has codimension 2. Then Z = V(I) ⊂ P 1 × P 1 is a finite set of points. We prove that if Z is a local complete intersection, then any syzygy of the fi vanishing at Z, and in a certain degree range, is in the module of Koszul syzygies. This is an analog of a recent result of Cox and Schenck (2003).
A case study in bigraded commutative algebra
 In: Peeva, I. (Ed.), Syzygies and Hilbert functions. Lecture Notes in Pure and Applied Mathematics
"... Abstract. We study the commutative algebra of three bihomogeneous polynomials p0, p1, p2 of degree (2, 1) in variables x, y; z, w, assuming that they never vanish simultaneously on P 1 ×P 1. Unlike the situation for P 2, the Koszul complex of the pi is never exact. The purpose of the article is to i ..."
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Abstract. We study the commutative algebra of three bihomogeneous polynomials p0, p1, p2 of degree (2, 1) in variables x, y; z, w, assuming that they never vanish simultaneously on P 1 ×P 1. Unlike the situation for P 2, the Koszul complex of the pi is never exact. The purpose of the article is to illustrate how bigraded commutative algebra differs from the classical graded case and to indicate some of the theoretical tools needed to understand the free resolution of the ideal generated by p0, p1, p2.