Let $f:X \rightarrow Y$ be a morphism between two smooth projective varieties $X,Y$ which are defined over an algebraically closed field $k$. I am looking for some criteria which guaranties the projectivity of $f$.

For instance if $f$ is finite it is projective. Here we don't need the projectivity of the varieties $X,Y$.

Is the morphism $f$ projective if

Question1: The fibers of $f$ are finite?

Question 2: $f$ is one-to-one?

Question 3: $f$ is onto?

Does the assumption $k=\mathbb{C}$ makes the questions easier?

Let $f:X \rightarrow Y$ be a morphism between two smooth projective varieties $X,Y$ which are defined over an algebraically closed field $k$. I am looking for some criteria which guaranties the projectivity of $f$.

For instance if $f$ is finite it is projective. Here we don't need the projectivity of the varieties $X,Y$.

Is the morphism $f$ projective if

Question1: The fibers of $f$ are finite?

Question 2: $f$ is one-to-one?

Question 3: $f$ is onto?

Does the assumption $k=\mathbb{C}$ make the questions easier?