Bounding Castelnuovo-Mumford regularity for varieties with good general projections

Let X subset of P-C' be a smooth variety of dimension n and degree d. There is a well-known conjecture concerning the k-regularity, saying that X is k -regular if k greater than or equal to d - r + n + 1. We prove that X is k- regular if k greater than or equal to d - r + n + 1 + (n - 2)(n - 1)/2 when n less than or equal to 14 (or, more generally, when X admits a general pr ojection in Pn+1 which is "good"), recovering the known results for curves, surfaces, threefolds (when r > 5), and improving the known results for fou rfolds and higher-dimensional varieties of codimension > 2. (C) 2000 Elsevi er Science B.V. All rights reserved.