I was reviewing the relation of a base point free linear system $|D|$ on a curve $X$ with the associated morphism $\phi: X \to P^{n}$. I know that this $\phi$ may not be an embedding in general, but when it is? If $D$ is very ample, is this $\phi$ an embedding?
In particular, if $\phi: X \to \mathbb{P^{1}}$, this has to be a dominant morphism, is it always the case that $\phi$ is actually an isomorphism? Or do we require any other other condition on $|D|$ to obtain that isomorphism (like degee of $\phi=1$)?
Any idea or answer will be helpful!