Showing papers on "Castelnuovo–Mumford regularity published in 2009"
TL;DR: In this paper, it was shown that A is a Koszul AS-regular algebra if and only if the Castelnuovo-Mumford regularity and the Ext-regularity coincide for all finitely generated A -modules.
11 citations
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TL;DR: It is proposed that the Castelnuovo–Mumford regularity of the sheaf of differential p-forms on X is bounded by p(em+1)D, where e, m, and D are the maximal codimension, dimension, and degree of all irreducible components of X.
Abstract: We describe a parallel polynomial time algorithm for computing the topological Betti numbers of a smooth complex projective variety $X$. It is the first single exponential time algorithm for computing the Betti numbers of a significant class of complex varieties of arbitrary dimension. Our main theoretical result is that the Castelnuovo-Mumford regularity of the sheaf of differential $p$-forms on $X$ is bounded by $p(em+1)D$, where $e$, $m$, and $D$ are the maximal codimension, dimension, and degree, respectively, of all irreducible components of $X$. It follows that, for a union $V$ of generic hyperplane sections in $X$, the algebraic de Rham cohomology of $X\setminus V$ is described by differential forms with poles along $V$ of single exponential order. This yields a similar description of the de Rham cohomology of $X$, which allows its efficient computation. Furthermore, we give a parallel polynomial time algorithm for testing whether a projective variety is smooth.
7 citations
TL;DR: In this article, the authors extended the bounds for the Castelnuovo-Mumford regularity of a graded ideal module over a homogeneous Cohen-Macaulay ring with a local base ring R 0.
Abstract: We extend the “linearly exponential” bound
for the Castelnuovo-Mumford regularity of a graded ideal
in a polynomial ring K[x1, . . . , xr] over a field (established by Galligo and Giusti in characteristic 0 and recently, by Caviglia-Sbarra for abitrary K) to graded submodules of a graded module over a homogeneous Cohen-Macaulay ring R = ⊕n≥0Rn with artinian local base ring R0. As an application we get a “linearly exponential” bound for the Castelnuovo-Mumford regularity of a graded R-module M in terms of the degrees which occur in a minimal free presentation of M.
6 citations
01 Jan 2009
TL;DR: In this paper, a notion of Castelnuovo-Mumford regularity was introduced to prove two splitting criteria for vector bundles with arbitrary rank in the space P n 1!···! P n s!Q m1!·· ·! Qmq.
Abstract: Here we consider the space P n 1 !··· ! P n s !Q m1 !··· ! Qmq. We introduce a notion of Castelnuovo-Mumford regularity in or- der to prove two splitting criteria for vector bundles with arbitrary rank.
4 citations
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TL;DR: In this paper, a bound for the Castelnuovo-Mumford regularity of a homogeneous ideal in terms of the degrees of its generators is given. But this bound assumes that the ideal defines a local complete intersection with log canonical singularities.
Abstract: In this note, we give a bound for the Castelnuovo-Mumford regularity of a homogeneous ideal $I$ in terms of the degrees of its generators. We assume that $I$ defines a local complete intersection with log canonical singularities.
3 citations
TL;DR: In this article, it was shown that reg ( K i (M ) ) is bounded in terms of gendeg (M) and the diagonal values d M j ( − j ) with j = 0, …, d − 1.
3 citations
TL;DR: In this article, it was shown that for r ⊆ 4 and d ⌉ r + 4, a rational rational curve X of degree d in P r has a unique (d − r + 1 ) -secant line if and only if X fails to be ( d − r ) -regular.
2 citations
TL;DR: In this paper, the concept of L-regularity for coherent sheaves on the Grassmannian G(1,4) was introduced, which is a generalization of Castelnuovo-Mumford regularity on Pn.
Abstract: Here we define the concept of L-regularity for coherent sheaves on the Grassmannian G(1,4) as a generalization of Castelnuovo–Mumford regularity on Pn. In this setting we prove analogs of some classical properties. We use our notion of L-regularity in order to prove a splitting criterion for rank 2 vector bundles with only a finite number of vanishing conditions. In the second part, we give the classification of rank 2 and rank 3 vector bundles without "inner" cohomology (i.e. for any i = 2,3,4) on G(1,4) by studying the associated monads.
2 citations
TL;DR: In this article, the maximum degree of a minimal Grobner basis of simplicial toric ideals with respect to the reverse lexicographic order is given, which is close to the bound stated in Eisenbud-Goto's Conjecture on the Castelnuovo-Mumford regularity.
Abstract: Bounds for the maximum degree of a minimal Grobner basis of simplicial toric ideals with respect to the reverse lexicographic order are given. These bounds are close to the bound stated in Eisenbud-Goto’s Conjecture on the Castelnuovo-Mumford regularity.
2 citations
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TL;DR: In this article, the authors studied the structure of residual intersections via constructing a finite complex which is acyclic under some sliding depth conditions on the cycles of the Koszul complex.
Abstract: In this article we study the structure of residual intersections via constructing a finite complex which is acyclic under some sliding depth conditions on the cycles of the Koszul complex. This complex provides information on an ideal which coincides with the residual intersection in the case of geometric residual intersection; and is closely related to it in general. A new success obtained through studying such a complex is to prove the Cohen-Macaulayness of residual intersections of a wide class of ideals. For example we show that, in a Cohen-Macaulay local ring, any geometric residual intersection of an ideal that satisfies the sliding depth condition is Cohen-Macaulay; this is an affirmative answer to one of the main open questions in the theory of residual intersection. The complex we construct also provides a bound for the Castelnuovo-Mumford regularity of a residual intersection in term of the degrees of the minimal generators.
2 citations
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TL;DR: In this paper, the Castelnuovo-Mumford regularity of Ext modules, over a polynomial ring over a field, is given in terms of the initial degrees and the number of generators involved.
Abstract: Bounds for the Castelnuovo-Mumford regularity of Ext modules, over a polynomial ring over a field, are given in terms of the initial degrees, Castelnuovo-Mumford regularities and number of generators of the two graded modules involved. These general bounds are refined in the case the second module is the ring. Other estimates, for instance on the size of graded pieces of these modules, are given. We also derive a bound on the homological degree in terms of the Castelnuovo-Mumford regularity. This answers positively a question raised by Vasconcelos.
01 Jan 2009
TL;DR: In this article, the authors generalize the classical Castelnuovo-Mumford regularity of a finitely generated graded module and use the generalized regularity to revisit piecewise-Koszul objects.
Abstract: We generalize the classical Castelnuovo-Mumford regularity (CM-regularity for short) of a finitely generated graded module and use the ‘generalized regularity’ to revisit piecewise-Koszul objects. Moreover, for a special commutative graded piecewise-Koszul algebra A and ( ). A M N ∈ We prove that, there exists an integer r, such that [] r M r − ≥ is a piecewise-Koszul module, which is related to a conjecture of G. Kempf ’s.