Regularity bounds for curves by minimal generators and Hilbert function

Let ${\rho }_C$ be the regularity of the Hilbert function of a projective curve $C$ in ${\mathbb{P}}_K^n$ over an algebraically closed field $K$ and ${\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_{n-1}$ be minimal degrees for which there exists a complete intersection of type $({\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_{n-1})$ containing the curve $C$. Then the Castelnuovo-Mumford regularity of $C$ is upper bounded by $\max \{{\rho }_C+1,{\alpha }_1+\mathrm{.}\mathrm{.}\mathrm{.}+{\alpha }_{n-1}-(n-2)\}$. We study and, for space curves, refine the above bound providing several examples.