Bounding Castelnuovo–Mumford regularity for varieties with good general projections

Let X PCr 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≥d−r+n+1. We prove that X is k-regular if k≥d−r+n+1+(n−2)(n−1)/2 when n≤14 (or, more generally, when X admits a general projection in Pn+1 which is “good”), recovering the known results for curves, surfaces, threefolds (when r>5), and improving the known results for fourfolds and higher-dimensional varieties of codimension >2.