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Perhaps a basic question... how does one study affine algebraic geometry via projective geometry? For example, suppose I have two affine varieties which I want to prove are isomorphic, would it help to look at the projective closures (assuming I don't know of any other method for proving they are isomorphic)? How does one go back to the affine case from the projective closure (or "projectivization")? Sorry if this sounds confusing.

Thanks, Morton

Edit: Thanks for the replies. Being new to AG let me try and rephrase my quandary: It seems the projective setting is the most convenient to study AG but if I want to study properties of affine varieties, how does one use results of projective varieties in the affine case? I know this sounds vague but it is a fundamental doubt I have.

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Not really, but you can prove that two affine varieties aren't isomorphic by showing that their projective closures aren't. –  Qiaochu Yuan Jan 4 '10 at 22:07
    
Morton- this question is way too vague. You might want to read the page on how to write a good question: mathoverflow.net/howtoask.html I'm going to close it for now, but one of the moderators will reopen it if you rewrite it to be clearer and more focused. When you think it's ready, flag it for moderator attention. –  Ben Webster Jan 4 '10 at 22:09
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@Qiaochu -- No, you can't, because an affine variety does not have a unique "projective closure": you get one for each embedding into affine space. (Maybe this question was closed prematurely? It's a bit vague, but still something can be said.) –  Pete L. Clark Jan 4 '10 at 22:34
    
A more specific question is "what is an example of a theorem you can prove about an affine variety which requires a compactification"; a sample answer to this would be proving rationality of a variety by compactifying and studying the compactification (e.g. one does this for moduli of curves [though that example is not affine, but captures this point]). –  David Zureick-Brown Jan 4 '10 at 22:37
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I agree with Pete that although this question is a little vague, there are many reasonable remarks to make in response. In fact I think it's fair to say that any serious study of affine varieties will use projective varieties as a foundation, e.g. the theory of mixed Hodge structures, the theory of birational equivalence, ... . There is not one definitive answer, but there are many interesting partial answers. –  Emerton Jan 4 '10 at 23:07
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4 Answers

Different affine varieties can have a same projective closure. For example, look at $P^1(C)$, then $C$ is an affine open subset, and $C-{0}$ is also an affine open subset. But in any case, if two affine varieties have a same projective closure, they are birational.

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The most useful thing here for your interests would probably be log geometry, from what I know. In one formulation (Matsuki "Introduction to the Mori Program" is my reference) you look at pairs $(X,D)$ where $X$ is a projective variety and $D$ is a divisor on $X$ with normal crossings. Then look at the category of these pairs called log pairs, and the geometry captures a lot of the geometry of $X\setminus D$.

Here's the book, check out chapter 2, hope it helps.

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A side question: Is the Mori program part of the minimal model program? –  Anweshi Jan 5 '10 at 16:51
    
They're related, I think that Mori is bigger: it starts with minimal model program, but then continues to analyze how various models relate to one another, and moduli spaces of things, as I understand it. –  Charles Siegel Jan 5 '10 at 19:35
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I think the question raises a valid point.

A very fruitful approach to affine problems was initiated by Iitaka in the 70's which is as follows:

Suppose $V$ is an affine variety and $X$ is a projectivisation such that $D=X-V$ is a divisor with simple normal crossings (SNC). Look at the canonical divisor $K:=K_X$ and the divisor $L:=K+D$ on $X$. Just like, the now classical, theory of Kodaira and others of analysing the multicanonical systems $nK$ of a projective variety, Iitaka proposed to look at $n(K+D)$ to come up with the a kind of classification for the pair $(X,D)$ as one does for projective varieties. Of course the only complete success in classifying varieties until Iitaka's time was for curves and Surfaces (which is also available now for 3-folds), so he and others applied this idea for non-compact (in particular affine) surfaces. Below I shall talk only about surfaces since the appropriate theory for 3-folds has not yet been worked out (as far as I know) and the curve case is extremely well understood and presents no real difficulty, generally speaking.

Just like a Kodaira dimension for projective surfaces, we can define the logarithmic Kodaira dimension of non-compact surfaces which is by definition the rate of growth of $n(K+D)$ as $n$ varies over positive integers. This number, called $\bar\kappa$ can take values $-\infty,0,1,2$ (or upto the dimension of the variety in the general case). At this stage one proves a theorem that this number is independent of the compactification $X$ chosen, as long as $D$ is SNC. This gets the theory started and we get a perfect gadget for studying the non-compact (in particular affine) surfaces. The whole project follows Kodaira's classification philosophy that one should develop enough classification theorems for the various $\bar\kappa$ classes and therefore (ideally) answer "all" questions about the non-compact (or affine) varieties. So if you want to answer a question like "are two affine varieties $A,B$ isomorphic or not" then the first thing to look at is their log-Kodaira dimensions. If they turn out to be different then we are done. If they are same then we have to look more closely into that particular $\bar\kappa$ class and either apply the appropriate classification theorems available or formulate and prove one, to decide.

However, just like in the projective case, the general type surfaces are hardest to study and don't always admit any good structure like a fibration over a curve which might have helped in their systematic study. And, by and large, the greatest success story has been in the non general type cases where there is a detailed classification of projective surfaces. Similar difficulties are encountered in the affine case and the $\bar\kappa\leq{1}$ affine surfaces are amenable to detailed study. Of course, there are some strong results about general type surfaces also which are in spirit the same as in the case of surface geography problem.

To find out more about these things one may look at Iitaka's book(GTM,76) and Miyanishi's book.

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It seems the questioner knows only about Chap. I of Hartshorne's book, for instance. –  Anweshi Jan 5 '10 at 16:47
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Thanks Maharana and Charles, this really helps. I would ideally like to see some concrete examples if you can suggest a reference. I'll get my hands on Iitaka's book. I don't know much about the Mori program but I am curious what happens in the case of an affine threefold, say X. Suppose I embed X in a projective P and arrive at a "final" space X'=P'\D', how does X' compare with X? What all can go wrong? What sort of pathologies arise? Is there a specific reference for affine threefolds you can suggest? Any reference I see only seems to address projective threefolds, the affine case eludes me. –  Morton Jan 7 '10 at 3:34
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Here is a simple example. Someone asked me if the unit disc is an affine variety. the answer is no. To see why not, assume yes, and take the projective closure. then one gets a projective curve which is a compact 2 dimensional surface with some points identified, and which differs from the original surface by at most adding a finite number of points. But this is impossible. No compact surface can be reduced to a disc by removing a finite number of points, even topologically, except for removing one point from P^1. But that does not give the disc by Liouville's theorem.

A more significant and pervasive example is the fact that at every singular point of an affine variety, the tangent cone determines a projective variety. Thus projective geometry is the local aspect of affine geometry. Put another way, blowing up an affine variety, at a point say, introduces projective geometry into it as a picture of its infinitesimal structure.

One can sometimes use this trick to compute the degree of a proper map of affine or other varieties, by restricting to the behavior over the projective normal bundle of a single fiber. See e.g. Friedman - Smith (Inventiones, 67, (1982)), who compute the degree of the prym map by showing that a single projective fiber of the proper prym map is embedded in projective space by the derivative of the prym map acting on the normal bundle to the fiber.

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