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The graph of a morphism $X \rightarrow Y$ is closed in $X \times Y$ if $X$ and $Y$ are affine varieties. What if $X$ and $Y$ are projective varieties?

I am still not quite familiar with projective varieties. So I need some help. Thanks very much.

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    @Zhen: Thank you very much for editting and adding the tags. I can put only one tag to my question, and I will take care of it from now on. Many thanks.2011-07-09

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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.

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    @MatttE Why is $f\times 1$ a morphism of quasi projective varieties?2017-05-01