Readers of MO are probably aware of the pedagogic need that would provoke such a query. It's around 60 years since Serre's FAC, and I imagine some people would say "you still have to read the original papers". There are well-known issues with the EGA/SGA material, and they can be met only by extensive rewriting. While there are some highly competent (IMO) authors dealing with aspects, an integrated treatment seems a long way off.

So, who has an overview? And who might be able to pull together a treatment of the whole area? We've come a long way since Zariski's Algebraic Surfaces, in particular, which de facto drew a line under the Italian school.


closed as not a real question by J.C. Ottem, Dan Petersen, Felipe Voloch, Allen Knutson, Angelo Mar 28 '12 at 14:46

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question.

locked by François G. Dorais Dec 9 '12 at 19:05

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    $\begingroup$ @Martin. Yes, you don't understand the question, because you don't understand why EGA/SGA, a project started more than 50 years ago, should be rewritten for pedagogic reasons. That is because it has always been very difficult to learn from, for most mathematicians. $\endgroup$ – Charles Matthews Mar 28 '12 at 12:47
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    $\begingroup$ Dear Charles, I'm not sure that it's realistic to expect a unified treatment of the type your question is yearning for. In addition to the texts already mentioned in comments, there are the recent books by Lazarsfeld, on Positivity, and by Kollar, on Resolution of Singularities. I think that good books like these on slightly more specialized topics are more realistic to hope for, and may also be of real pedagogical benefit: by presenting a less monolithic front, they may make the subject accessible to a wider range of students. Best wishes, Matthew $\endgroup$ – Emerton Mar 28 '12 at 13:09
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    $\begingroup$ I voted to close this as subjective and argumentative; I really don't like the tone of the question, nor of the subsequent comments by the OP. As to the substance, there are introductory texts at several levels; among the new ones, Liu's book comes to mind as a particularly good example. After that, one can very well read EGA and SGA, and a host of other books. $\endgroup$ – Angelo Mar 28 '12 at 14:53
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    $\begingroup$ I'm not a geometer, but I think Angelo's comment and the closing of the question are both completely reasonable. $\endgroup$ – Peter Samuelson Dec 9 '12 at 7:18
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    $\begingroup$ To Shanmukha: I am sorry you felt that my comment was offensive, but I stand by it, and feel that there is not need to reopen this thread. And I do take offense at your implication that I get pleasure out of closing down questions. $\endgroup$ – Angelo Dec 9 '12 at 8:51