Is the graph of morphism of projective varieties $X \rightarrow Y$ closed in $X \times Y$?

Solution 1:

The graph of a morphism $f: X \to Y$ is the pull-back under the product map $f\times 1: X \times Y \to Y \times Y$ of the diagonal $\Delta(Y) \subset Y \times Y.$ Thus for the graph to be closed, what you need is the diagonal $\Delta(Y)$ to be closed in $Y \times Y$. This is true for all quasi-projective varieties, and so in particular for projective varieties (as well as affine varieties, as you noted in the question).

In general, a variety (or more generally, a scheme) is called separated if the diagonal is closed. Although there are non-separated objects, in practice it is hard to find them if you don't deliberately go looking for them.