Projective curves with next to sharp bounds on Castelnuovo-Mumford regularity (Q2456185)

Let \(C \subset \mathbb {P}^n\) be an integral non-degenerate degree \(d\) curve. Assume that \(C\) is not ACM and let \(k(C)\) be the minimal integer \(t>0\) such that any degree \(t\) homogeneous form kills the Hartshorne-Rao module of \(C\). If the index of regularity reg\((C)\) of \(C\) is maximal, then it is known that \(C\) lies on a rational normal surface scroll. Here (in characteristic zero) the author proves the next step, i.e. he proves that if \(d \geq n^2+2n+1\), \(d \geq 13\) and \(\text{reg}(C) = \lceil (d-1)/(n-1)\rceil + k(C)-1\), then \(C\) lies either on a rational normal surface scroll or on a del Pezzo surface.