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Is there a Variety (in the sense of Serre, that is given as a ringed space with affine covering) that is neither embeddable as (quasi) affine, nor (quasi) projective Variety? I believe for Schemes it's quite easy to give such an example, but I know none with Varieties.

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    $\begingroup$ There's an example due to Hironaka of a complete nonprojective 3-dimensional variety; you can read about it in Appendix B of Hartshorne. $\endgroup$
    – user64687
    Commented Jun 5, 2014 at 15:20
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    $\begingroup$ Every separated algebraic variety can be embedded in a proper (=complete) algebraic variety. Thus the question is whether a proper algebraic variety is always projective. The answer is non as in the comment of Asal Beag Dubh. $\endgroup$
    – Cantlog
    Commented Jun 5, 2014 at 17:01

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