Consider $\mathbb CP^3$ and smooth algebraic curve $C$. What is the minimal degree of such a curve which is not on a cubic surface?

Consider a fixed smooth algebraic curve $C$ over $\mathbb C$. It is well-known that $\mathbb CP^3$ contains curves that are abstractly isomorphic to $C$. What is the minimal degree of a curve in $\mathbb CP^3$, abstractly isomorphic to $C$, which is not on any cubic surface?

Consider $\mathbb CP^3$ and smooth algebraic curves. What is the minimal degree of such a curves which is not on a cubic surface?

Consider $\mathbb CP^3$ and smooth algebraic curve $C$. What is the minimal degree of such a curve which is not on a cubic surface?