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, i.e., 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 1.1] 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 3.6] 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 3.1] and improved upon in [HoaTrn, Corollary 3.2]. 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