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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?

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  • $\begingroup$ 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.) $\endgroup$ – user64687 Oct 17 '13 at 12:40
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No: there is the famous Hironaka example of a non-projective complete variety in dimension 3. This is explained in Hartshorne: Appendix B, Example 3.4.1.

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