Let $X$ be a smooth projective variety and $\phi: X \to \mathbb P^n$ be a map. If $\phi$ is an embedding then $E=\phi^*(O(1))$ is very ample. But can one say something if $\phi$ is birational (but not isomorphism) to its image? Is it ample?
Suppose $\phi$ is a morphism (i.e., defined everywhere) which is birational, but not an embedding. Then there are two cases: