This is a good book on important ideas. But it competes with Hartshorne ALGEBRAIC GEOMETRY and that is a tough challenge. It has roughly the same prerequisites as Hartshorne and covers much the same ideas. The three volumes together are actually a bit longer than Hartshorne. I had hoped this would be a lighter, more easily surveyable book than Hartshorne's. The subject involves a huge amount of material, an overall survey showing how the parts fit together could be very helpful, and the IWANAMI SERIES has some terrific, brief, easy to read, overviews of such subjects--which give proof techniques but refer elsewhere for the details of some longer proofs.

But it turns out that Ueno differs from Hartshorne in the other direction: He gives more explicit nuts and bolts of the basic constructions. Overall it is easier to get an overview from Hartshorne. Ueno does also give a lot of "insider information" on how to look at things. It is a good book. The annotated bibliography is very interesting. But I have to say Hartshorne is better.

If you get stuck on an exercise in Hartshorne this book might help. If you are working through Hartshorne on your own, you will find this alternative exposition useful as a companion. You might like the more extensive elementary treatment of representable functors, or sheaves, or Abelian categories--but you could get those from references in Hartshorne as well.

Someday some textbook will supercede Hartshorne. Even Rome fell after enough centuries. But here is my prediction, for what it is worth: That successor textbook will not be more elementary than Hartshorne. It will take advantage of progress since Hartshorne wrote (almost 30 years ago now) to make the same material quicker and simpler. It will include number theory examples and will treat coherent cohomology as a special case of etale cohomology---as Hartshorne himself does briefly in his appendices. It will be written by someone who has mastered every aspect of the mathematics and exposition of Hartshorne's book and of Milne's ETALE COHOMOLOGY, and like both of those books it will draw heavily on Grothendieck's brilliant, original, but thorny Elements de Geometrie Algebrique. Of course some people have that level of mastery, notably Deligne, Hartshorne, and Milne who have all written great exposition. But they can't do everything and no one has yet boiled this down to a textbook successor to Hartshorne. If you write this successor *please* let me know as I am dying to read it.

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