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

Abstract

This paper is devoted to the study of the next extremal case for a Castelnuovo-type bound reg C ⩽ ⌈ ( deg C − 1 ) / codim C ⌉ + 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.