Hartshorne's "Algebraic geometry" begins with the definition of (quasi-)affine and (quasi-)projective varieties over some fixed algebraically closed field. At a first glance, these seem to be quite different, so that I would have expected that one would pose questions *either* on quasi-affine *or* on quasi-projective varieties.


However, Hartshorne then defines a *variety* to be either a quasi-affine or a quasi-projective variety. These varieties (together with certain continuous and in some sense regular maps) then form the category of varieties.



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> Here is my question: Is the above definition natural in the sense that we really want to compare quasi-affine and quasi-projective varieties or at least study them both at the same time?
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For instance, is there a (non-trivial) example of a quasi-affine variety which is isomorphic in the above category to a quasi-projective variety? If not, isn't this "unifying" definition a bit artificial?



Is Hartshorne's definition of the category of varieties natural?
A quasiaffine variety IS quasi projective. Indeed it is an open set in an affine variety, which in turn is open in its projective closure. So one only considers quasiprojective varieties.


