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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.

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?

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?

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This depends on what you mean by trivial. For example, $\mathbb{A}^n$ is certainly quasi-affine; but it's also quasi-projective, as it is an open subset of $\mathbb{P}^n$ (consisting of coordinates $[1, a_1, ..., a_n]$ for example). – Daniel Litt Jul 7 '10 at 16:16
Thanks, to me this is a non-trivial example. But I would still appreciate an answer to my main question (even though it's quite vage). – Rasmus Bentmann Jul 7 '10 at 16:30
There have been a couple of questions about teaching oneself mathematics, one asking what precisely is better about learning from a human than a book, say. This is an excellent example of how a book can contain a statement that is logically correct but which can cause a student much wasted time that could be quickly resolved by the presence of a human teacher. – Dan Piponi Jul 7 '10 at 18:13
Hartshorne's Algebraic Geometry, in my opinion, is an excellent book, but not for chapter I. I recommend skipping entirely the first chapter and beginning with chapter II, Schemes, directly. Varieties are better learned either before, from an other book, or why not, after, as a special case of scheme. – Joël Jan 30 '14 at 11:46
up vote 17 down vote accepted

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.

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Ah, so the definition is Hartshorne is phrased that way for didactical reasons? – Rasmus Bentmann Jul 7 '10 at 17:04
Not all quasi-projective varieties are quasi-affine, so sometimes it makes sense to consider quasi-affine varieties (not that they are paricularly special as far as I know). But when it comes to define a category in wich to work, quasiprojective varieties is the smallest that contains all basic examples. But don'r worry: if you are studying Hartshorne, you will change category very soon. – Andrea Ferretti Jul 7 '10 at 17:48

I would just comment, but I'm a new user so I can't.

I believe you are referring to Hartshorne's definition on p. 15:

"A variety over $k$ is any affine, quasi-affine, projective, or quasi-projective variety as defined above."

The reason he defines things this way is that in section I.1 he defined affine and quasi-affine varieties; in section I.2 he defined projective and quasi-projective varieties. You are right that he hasn't made very clear the relation between the two. Exercise I.2.9 on projective closures hints at the relation, but it hasn't been made completely precise (he just uses the terminology "identify"). To make this precise, we need to say what the morphisms between any two different varieties are.

Suffice it to say, the identification of Exercise I.2.9 given by projective closure is in fact an isomorphism in the sense of section I.3, so all varieties are isomorphic to quasi-projective varieties.

If this is your first encounter with algebraic geometry, however, I'm not sure I would recommend chapter I of Hartshorne very highly. I made the mistake of thoroughly studying chapter I myself, and I think there are far better ways to spend your time.

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"I think there are far better ways to spend your time..." for example...? – eventually Jan 30 '14 at 1:37
@pmath: "for example...?" Answer: Chapters II and III. – Jason Starr Feb 26 '14 at 13:52

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