Mathematics Subject Classification (1991)." 13D45, 14B15, 13H10 I IntroductionLet X c_ PT~ be a subscheme over an infinite field K. We denote by Jx the ideal sheaf of X in P~. It is classical that Hi(p~x, Jx(r -i)) = 0 for i > 1 and r~>O. The Castelnuovo-Mumford regularity reg(X) of X is the smallest such r. Bounds on the regularity of X are important in connection with algorithms for computing syzygies. There are some good bounds in some particular cases ifX is assumed to be smooth, see [1] and the references there.Our interest is to consider the case X being locally Cohen-Macaulay, i.e., the homogeneous coordinate ring R of X is a generalized Cohen-Macaulay (abbr. C-M) ring. Under this assumption there is a nonnegative integer k such that (to ..... t,) ~ | = 0 for 1 < i < dimX =: d-1 (see Definition 2.2). In this case X is called a k-Buchsbanm scheme. A quasi-Buchsbaum scheme is just the 1-Buchsbaum scheme. For a subvariety X, it was shown in [17] and [8] that reg(X) is bounded above by invariants which depend exponentially on d. In [9] a bound depending on d 2 is given. Note that for arithmetically Buchsbaum subvarieties X there are some sharp bounds on reg(X) given in [16] and [2].The purpose of this paper is to give bounds on reg(X) which depend linearly on d. We also consider the case tnk being an R-standard ideal for some positive integer k, where 111 is the maximal homogeneous ideal of R. The main method in [2,8,9,16,17,13] is using hyperplane (or hypersurface) sections. Our method here is quite different. Using the structure theory of generalized C-M rings we can give in Sect. 2 a bound on reg(X) in terms of the socalled index of speciality of X. This investigation gives a new bound even for * Supported by a grant from the

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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) ≤ (deg(X) − 1)/ codim(X) + 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.

This paper investigates the cohomological property of vector bundles on biprojective space. We will give a criterion for a vector bundle to be isomorphic to the tensor product of pullbacks of exterior products of differential sheaves. Mathematics Subject Classification. 14J60.

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