TITLE: Morphism between projective varieties
QUESTION [8 upvotes]: 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?

REPLY [11 votes]: A morphism $X \to Y$ factorizes as an embedding $X \to X\times Y$ (the graph of $f$) followed by the projection $X\times Y \to Y$. The first is a closed embedding (and hence is projective) if $Y$ is separable and the second is projective if $X$ is projective.