# Is every variety (defined as separated prevariety) a locally closed subset of some projective space?

In Hartshorne Ch1, variety is defined to be a affine, quasi-affine, projective or quasi-projective variety.

In Mumford's Red book, it was defined to be separated prevariety(gluing of a finite number of irreducible varieties).

Is every separated prevariety isomorphic to some variety defined as affine, quasi-affine, projective or quasi-projective variety?

• Regarding your first sentence: note that Hartshorne changes his tune in Chapter II --- Remark 4.10.1 says "From now on we will use the word "variety" to mean "abstract variety" in the sense just defined." (Which is to say, the same as Mumford's definition.) – user64687 Oct 17 '13 at 12:40