TITLE: Finite map from quasi-projective variety
QUESTION [7 upvotes]: Suppose I have a quasiprojective variety $X$ and a finite surjective map
$$f: X \rightarrow Y$$
to a scheme $Y$.  Is it true that $Y$ is quasiprojective as well?  It seems like the answer could be no, but I don't know enough examples of non-projective schemes.

REPLY [11 votes]: It is true if $Y$ is normal. Let me explain why.
I will say that a variety has the Chevalley-Kleiman (CK) property if every finite subset is contained in an affine open. By Corollary 48 of Kollar's "Quotients by finite equivalence relations" arXiv:0812.3608, if $f:X\to Y$ is finite and surjective, then if $X$ has the CK property, $Y$ has it too.
Now, it is clear that a quasi-projective variety has the CK property, and a normal variety with the CK property is quasi-projective by Corollary 2 of "Quasi-projectivity of normal varieties" arXiv:1112.0975.
Putting this together, we see that if $f:X\to Y$ is a finite surjective map of varieties with $X$ quasi-projective and $Y$ normal, then $Y$ is quasi-projective.