Projective curves with next to sharp bounds on Castelnuovo–Mumford regularity

This paper is devoted to the study of the next extremal case for a Castelnuovo-type bound regC (degC−1)/codimC +max{k(C),1} for the Castelnuovo–Mumford regularity for a nondegenerate projective curve C, where k(C) is an invariant which measures the deficiency of the Hartshorne–Rao module of C. We show that a projective curve with next to the maximal regularity lies on either a Hirzebruch surface or a normal del Pezzo surface.