# Non (quasi) affine or projective Variety

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.

• There's an example due to Hironaka of a complete nonprojective 3-dimensional variety; you can read about it in Appendix B of Hartshorne. – user64687 Jun 5 '14 at 15:20
• 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. – Cantlog Jun 5 '14 at 17:01