Let $X$ be a projective variety. Why is any surjective morphism from $X$ to itself finite?



Surjective morphism from $X$ to itself is finite
Let $f$ be your surjective endomorphism; it is generically finite, say of degree $d$. Then $f\_\*:H^\*(X,\mathbb{Q})\rightarrow H^\*(X,\mathbb{Q})$ is surjective, because $\ f\_\*f^\*=d.\mathrm{Id}\ $ (use $\mathbb{Q}\_\ell$ instead of $\mathbb{Q}$ in characteristic $p$). Therefore $f\_\*$ is bijective. But then $f$ cannot contract any positive dimensional subvariety $Z$, because we would have $f\_\*[Z]=0$. 


