Regularity bounds by minimal generators and Hilbert function

Let p_C be the regularity of the Hilbert function of a projective curve C in the projective space Pn of dimension n over an algebraically closed field K and let a_1, . . . , a_(n−1) be degrees for which there exists a complete intersection of type (a_1, . . . ,a_( n−1)) containing properly C. Then the CastelnuovoMumford regularity of C is bounded above by max {p_C +1, a_1 +. . .+a_(n−1) −(n−1)} .We investigate the sharpness of the above bound, which is achieved by curves algebraically linked to ones having degenerate general hyperplane section.