Regularity bounds by minimal generators and Hilbert function

Let $\rho_C$ be the regularity of the Hilbert function of a projective curve $C$ in $\mbox {P}^n_K$ over an algebraically closed field $K$ and
$\beta_1, \ldots, \beta_{n-1}$ be degrees for which there exists a complete intersection of type ($\beta_1, \ldots, \beta_{n-1}$) containing properly $C$. Then the Castelnuovo-Mumford regularity of $C$ is bounded above by max $\{\rho_C +1, \beta 1 + \ldots + \beta_n-1 -(n-1)\}$ .We investigate the sharpness of the above bound, which is achieved by curves algebraically linked to ones having degenerate general hyperplane section.