A Remark on Lazarsfeld's Approach to Castelnuovo-Mumford Regularity

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.