  We derive new bounds for the Castelnuovo-Mumford regularity of the ideal
sheaf of a complex projective manifold of any dimension. They depend linearly
on the coefficients of the Hilbert polynomial, and are optimal for rational
scrolls, but most likely not for other varieties. Our proof is based on an
observation of Lazarsfeld in his approach for surfaces and does not require the
(full) projection step. We obtain a bound for each partial linear projection of
the given variety, as long as a certain vanishing condition on the fibers of a
general projection holds.
