  Castelnuovo-Mumford regularity is an important invariant of projective
algebraic varieties. A well known conjecture due to Eisenbud and Goto gives a
bound for regularity in terms of the codimension and degree,i.e.,
Castelnuovo-Mumford regularity of a given variety X is less than or equal to
deg(X)-codimension(X)+1. Generic projection methods proved to be effective for
the study of regularity of smooth projevtive varieties of dimension at most
four(cf.[BM},[K2],[L],[Pi] and [R1]) because there are nice vanishing theorems
for cohomology of vector bundles (e.g. the Kodaira-Kawamata-Viehweg vanishing
theorem) and detailed information about the fibers ofgeneric projections from X
to a hypersurface of the same dimension. Here we show by using methods similar
to those used in [K2] that $\reg{X}\le(deg(X)-codimension(X)+1)+10$ for any
smooth fivefold and $\reg{X}\le(deg(X)-codimension(X)+1)+20$ for any smooth
sixfold.
  Furthermore, using similar methods we give a bound for the regularity of an
arbitrary (not necessarily locally Cohen-Macaulay) projective surface X in P^N.
To wit, we show that $\reg{X}\le(d-e+1)d-(2e+1)$, where d=deg(X) and
e=codimension(X). This is the first bound for surfaces which does not depend on
smoothness.
