Is the image of a projective variety projective?

Question

Suppose $X$ is a projective variety and $f:X\to Y$ a morphism, is the image $f(X)$ projective?(the schematic image is well-defined in this case, or the induced reduced structure on the closed subset $f(X)$) Hartshorne says this property holds for properness. Usually how can we claim a variety(scheme) is projective?

Answer 1

No, the image of a projective variety is not always projective.

Indeed Chow's lemma states that given any complete irreducible variety $X$, there exists a projective variety $X'$ and a birational surjective morphism $f:X'\to X$.
Choosing for $X$ a complete irreducible variety that is not projective thus gives an example of a projective variety $X'$ whose image $X$ is not projective.

To be complete (!), let me mention that an example due to Hironaka of a complete but not projective smooth $3$-dimensional variety is given in Shafarevich's Basic Algebraic Geometry, volume 2, page 75.