I do not know much on abelian varities but I am working on Mumford's Abelian Varities and in the second chapter the arose some questions to me. Actually, it seems to me as I am lacking of a lot of background here, so please apologize if me question is just foolish.
We are in the following situation: $X$ is a complete variety and we consider a map $\psi \colon X\times X \to X\times X$ such that $\dim(Image(\psi))=\dim(X\times X)$. Mumford now claims that this implies, as $X\times X$ is complete, that $\psi$ is surjective.
In the context of vector spaces, this is a well-known result for me, but I have no real clue how to show this in this setting.
Thanks in advance