Showing papers on "Castelnuovo–Mumford regularity published in 1999"

Asymptotic Behaviour of the Castelnuovo-Mumford Regularity

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.

Asymptotic behaviour of Castelnuovo-Mumford regularity

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

Irrational Asymptotic Behaviour of Castelnuovo-Mumford Regularity

Abstract: We give examples of nonsingular curves in projective 3 space such that the regularity of powers of their ideal sheaves are highly nonlinear. This is in constrast to the case of an ideal I in a polynomial ring, where the regularity of I^n is a linear polynomial in n for large n.

Castelnuovo-Mumford regularity bound for smooth threefolds in P-5 and extremal examples

Abstract: Abstract Let X be a nondegenerate integral subscheme of dimension n and degree d in ℙN defined over the complex number field ℂ. X is said to be k-regular if Hi(ℙN, ℐX (k – i)) = 0 for all i ≧ 1, where ℐX is the sheaf of ideals of ℐℙN and Castelnuovo-Mumford regularity reg(X) of X is defined as the least such k. There is a well-known conjecture concerning k-regularity: reg(X) ≦ deg(X) – codim(X) + 1. This regularity conjecture including the classification of borderline examples was verified for integral curves (Castelnuovo, Gruson, Lazarsfeld and Peskine), and an optimal bound was also obtained for smooth surfaces (Pinkham, Lazarsfeld). It will be shown here that reg(X) ≦ deg(X) – 1 for smooth threefolds X in ℙ5 and that the only extremal cases are the rational cubic scroll and the complete intersection of two quadrics. Furthermore, every smooth threefold X in ℙ5 is k-normal for all k ≧ deg(X) – 4, which is the optimal bound as the Palatini 3-fold of degree 7 shows. The same bound also holds for smooth regular surfaces in ℙ4 other than for the Veronese surface.

Non-commutative Castelnuovo–Mumford regularity

Abstract: We define Castelnuovo–Mumford regularity for graded modules over non-commutative graded algebras. Two fundamental commutative results are generalized to the non-commmutative case: a vanishing-theorem by Mumford, and a theorem on linear resolutions and syzygies by Eisenbud and Goto. The generalizations deal with sufficiently well-behaved algebras (i.e. so-called quantum polynomial algebras).We go on to define Castelnuovo–Mumford regularity for sheaves on a non-commutative projective scheme, as defined by Artin. Again, a version of Mumford's vanishing-theorem is proved, and we use it to generalize a result by Martin, Migliore and Nollet, on degrees of generators of graded saturated ideals in polynomial algebras, to quantum polynomial algebras.Finally, we generalize a practical result of Schenzel which determines the regularity of a module in terms of certain Tor-modules.

Sharp bounds on Castelnuovo-Mumford regularity

Abstract: The Castelnuovo-Mumford regularity is one of the most important invariants in studying the minimal free resolution of the defining ideals of the projective varieties. There are some bounds on the Castelnuovo-Mumford regularity of the projective variety in terms of the other basic measures such as dimension, codimension and degree. In this paper we consider an upper bound on the regularity reg(X) of a nondegenerate projective variety X, reg(X) ≤ d(deg(X)− 1)/ codim(X)e+ k · dim(X), provided X is k-Buchsbaum for k ≥ 1, and investigate the projective variety with its Castelnuovo-Mumford regularity having such an upper bound.

On Castelnuovo-Mumford regularity of projective curves

Abstract: We give an effective method to compute the regularity of a satu- rated ideal I defining a projective curve that also determines in which step of a minimal graded free resolution of I the regularity is attained.