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4 deleted 55 characters in body

I think the question raises a valid pointand is fine for MO in my opinion. Here is my take on it:

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.

3 added 249 characters in body

I think the question raises a valid point and is fine for MO in my opinion. Here is my take on it:

A very fruitful approach to this problem affine problems was initiated by Iitaka in the 70's which goes 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). Now look Look at the canonical divisor $K:=K_X$ of $X$ and the divisor $L:=K+D$ on $X$ again. X$. Just likethe, the now classical, theory of Kodaira and others of analysing the multicanonical system systems$nK$of a projective variety, Iitaka proposed to look at$n(K+D)$and to come up with the a kind of classification for the pair$(X,D)$as one does for projective varieties. But of Of course the only complete success in classifying varieties until Iitaka's time was for curves and Surfaces (and which is also available now for 3-folds). So , so he and others applied this idea for non-compact (in particular affine) surfaces. Below I will shall talk only about surfaces since the appropriate theory for 3-folds has not yet been worked out (as far as I know) and the curve case is extremely well understood and presents no real difficulty, generally speaking. So just Just like a Kodaira dimension for projective surfaces, we can define the logarithmic Kodaira dimension of non-compact surfaces which is by definition the rate of growth of$n(K+D)$as$n$varies over positive integers. This number, called$\bar\kappa$can take values$-\infty,0,1,2$(or upto the dimension of the variety in the general case). One At this stage one proves a theorem that this number will be is independent of the compactification$X$chosen, as long as$D$is SNC. This gets us the theory started and we get a perfect gadget for studying the non-compact (in particular affine) surfaces. The whole project follows Kodaira's classification philosophy that one should develop enough classification theorems for the various$\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$isomorphic or not" then the first thing is to look at is their log-Kodaira dimensiondimensions. If it turns they turn out to be different then we are done. If they are same then we have to look more closely into that particular$\bar\kappa$class and either apply the appropriate classification theorems available or formulate and prove one, to decide. However, just like in the projective case, the general type surfaces are hardest to study and don't always admit any good structure like a fibration over a curve which might help one to have helped in their systematic studythem systematically. And, by and large, the greatest success story has been in the non general type cases where there is a detailed classification of projective surfaces. Similar difficulties are encountered in the affine case and the$\bar\kappa\leq{1}$affine surfaces are amenable to detailed study. Of course, there are some strong results about general type surfaces also which are in spirit the same as in the case of surface geography problem. To find out more about these things one might may look at Iitaka's book(GTM,76) and Miyanishi's book. 2 added 101 characters in body A very fruitful approach to this problem was initiated by Iitaka in the 70's which goes as follows: Suppose$V$is an affine variety and$X$is a projectivisation such that$D=X-V$is a divisor with simple normal crossings (SNC). Now look at the canonical divisor$K:=K_X$of$X$and the divisor$L:=K+D$on$X$again. Just like the, now classical, theory of Kodaira and others of analysing the multicanonical system$nK$of a projective variety, Iitaka proposed to look at$n(K+D)$and to come up with the a kind of classification for the pair$(X,D)$as one does for projective varieties. But of course the only complete success in classifying varieties until Iitaka's time was for curves and Surfaces (and now for 3-folds). So he and others applied this idea for non-compact (in particular affine) surfaces. Below I will talk only about surfaces since the appropriate theory for 3-folds has not yet been worked out (as far as I know)know) and the curve case is extremely well understood and presents no real difficulty, generally speaking. So just like a Kodaira dimension for projective surfaces, we can define the logarithmic Kodaira dimension of non-compact surfaces which is by definition the rate of growth of$n(K+D)$as$n$varies over positive integers. This number, called$\bar\kappa$can take values$-\infty,0,1,2$(or upto the dimension of the variety in the general case). One proves a theorem that this number will be independent of the compactification$X$chosen as long as$D$is SNC. This gets us started and we get a perfect gadget for studying the non-compact (in particular affine) surfaces. The whole project follows Kodaira's classification philosophy that one should develop enough classification theorems for the various$\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$isomorphic or not" then the first thing is to look at their log-Kodaira dimension. If it turns out to be different then we are done. If they are same then we have to look more closely into that particular$\bar\kappa$class and the appropriate classification theorems to decide. However, just like in projective case, the general type surfaces are hardest to study and don't always admit any good structure like a fibration over a curve which might help one to study them systematically. And by and large the greatest success story has been in the non general type cases where there is a detailed classification of projective surfaces. Similar difficulties are encountered in the affine case and the$\bar\kappa\leq{1}\$ affine surfaces are amenable to detailed study. Of course there are some strong results about general type surfaces also.

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

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