scispace - formally typeset
Search or ask a question
Topic

Castelnuovo–Mumford regularity

About: Castelnuovo–Mumford regularity is a research topic. Over the lifetime, 327 publications have been published within this topic receiving 5557 citations.


Papers
More filters
Journal ArticleDOI
TL;DR: In this article, the question of whether a polynomial ring over a field can be generated by elements of degree Qp + j for all j = 0, l,..., n is investigated.

561 citations

Journal ArticleDOI
TL;DR: In this paper, the asymptotic behavior of the Castelnuovo norm and the Mumford norm of the integral closure of the powers of a homogeneous ideal I is studied.
Abstract: In this paper the asymptotic behavior of the Castelnuovo$ndash;Mumford regularity of powers of a homogeneous ideal I is studied. It is shown that there is a linear bound for the regularity of the powers I whose slope is the maximum degree of a homogeneous generator of I, and that the regularity of I is a linear function for large n. Similar results hold for the integral closures of the powers of I. On the other hand we give examples of ideal for which the regularity of the saturated powers is asymptotically not a linear function, not even a linear function with periodic coefficients.

282 citations

Journal ArticleDOI
06 Jul 1999
TL;DR: In this article, it was shown that the Castelnuovo-Mumford regularity of a polynomial ring over a field is bounded by a linear function with leading coefficient at most P.
Abstract: Let S be a polynomial ring over a field For a graded S-module generated in degree at most P, the Castelnuovo-Mumford regularity of each of (i) its nth symmetric power, (ii) its nth torsion-free symmetric power and (iii) the integral closure of its nth torsion-free symmetric power is bounded above by a linear function in n with leading coefficient at most P For a graded ideal I of S, the regularity of I' is given by a linear function of n for all sufficiently large n The leading coefficient of this function is identified Let S = k[xl, , Xd] be a polynomial ring over a field k with its usual grading, ie, each xi has degree 1, and let m denote the maximal graded ideal of S Let N be a finitely generated non-zero graded S-module The Castelnuovo-Mumford regularity of N, denoted reg(N), is defined to be the least integer m so that, for every j, the Jth syzygy of N is generated in degrees F -> o N --> 0 where Fi = 1 S(-aij) for some integers ai -which we will refer to as the twists of Fi Then, reg(N) < maxij {aij i} with equality holding if the resolution is minimal For other equivalent definitions and properties of this invariant, see [Snb] For a graded ideal I in S, the behaviour of the regularity of I' as a function of n has been of some interest If I defines a smooth complex projective variety, it is shown in [BrtEinLzr, Proposition 1] using the Kawamata-Viehweg vanishing theorem that reg(In) < Pn + Q where P is the maximal degree of a minimal generator of I and Q is a constant expressed in terms of the degrees of generators of I In [GrmGmgPtt, Theorem 11] and in [Chn, Theorem 1] it is shown that if dim(R/I) < 1, then reg(In) < n reg(I) for all n E N In [Chn, Conjecture 1], this is conjectured to be true for an arbitrary graded ideal Supporting this conjecture is the result of [Swn, Theorem 36] that reg(In) < Pn for some constant P and for all n E N The method of proof makes it difficult to explicitly identify such a constant For monomial ideals, such a P is explicitly calculated in [SmtSwn, Theorem 31] and improved upon in [HoaTrn, Corollary 32] We show that with S and N as above, the regularity of Symn(N) and related modules is bounded above by a linear function of n with leading coefficient at Received by the editors October 28, 1997 and, in revised form, April 15, 1998 1991 Mathematics Subject Classification Primary 13D02; Secondary 13D40 (?1999 American Mathematical Society

272 citations

Journal Article
TL;DR: In this article, it was shown that the Castelnuovo-Mumford regularity of a module can be doubly exponential in the degrees of the minimal generators and in the number of variables.
Abstract: The Castelnuovo-Mumford regularity reg$(M)$ is one of the most important invariants of a finitely generated graded module $M$ over a polynomial ring $R$. For instance, it measures the amount of computational resources that working with $M$ requires. In general one knows that the regularity of a module can be doubly exponential in the degrees of the minimal generators and in the number of the variables. On the other hand, in many situations one has or one conjectures a much better behavior. One may ask, for instance, whether the Castelnuovo-Mumford regularity reg($IM$) of the product of an ideal $I$ with a module $M$ is bounded by the sum reg($I$) + reg($M$). In general this is not the case. But we show that it is indeed the case if either dim $R/I\leq 1$ or $I$ is generic (in a very precise sense). Further we show that products of ideals of linear forms have always a linear resolution and that the same is true for products of determinantal ideals of a generic Hankel matrix.

168 citations

Network Information
Related Topics (5)
Algebraic geometry
8.7K papers, 205K citations
87% related
Cohomology
21.5K papers, 389.8K citations
84% related
Simple Lie group
8.3K papers, 204.2K citations
83% related
Ring (mathematics)
19.9K papers, 233.8K citations
82% related
Representation theory
8.6K papers, 266.4K citations
82% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20221
202118
202022
201919
201812
201720