  The Castelnuovo-Mumford regularity reg(X) of a projective scheme X was
introduced by Mumford by generalizing ideas of Castelnuovo. The interest in
this concept stems partly from the fact that X is m-regular if and only if for
every p \geq 0 the minimal generators of the p-th syzygy module of the defining
ideal I of X occur in degree \leq m + p. There are some bounds in the case that
X is a locally Cohen-Macaulay scheme. The aim of this paper is to extend and
improve these results for so-called (k,r)-Buchsbaum schemes. In order to prove
our theorems, we need to apply a spectral sequence. We conclude by describing
two sharp examples and open problems.
