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In his textbook Algebraic Geometry, he wrote in p. 58:

Now that we have seen a litle bit of what algebraic geometry is about, we should discuss the degree of generality in which to develop the foundations of the subject. In this chapter we have worked over an an algebraically closed field, because that is the simlest case. But there are good reasons for allowing fields which are not algebraically closed.

However, in Chapter 4 and 5 treating curves and surfaces, which are the "meat" of algebraic geometry, he worked only over an algebraically closed field. Hence my title question.

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3 Answers 3

Chapters 2 and 3 comprise the majority of the five chapters in Harshorne's text -- something like $233$ out of $423$ pages -- and they treat schemes in much more generality than just varieties over an algebraically closed field. So taking out the schemes would remove more than half of the text!

[As an aside: the first algebraic geometry class I took -- in my first semester of grad school -- used Hartshorne's text but did not discuss scheme theory. Trying to read Hartshorne and restricting to varieties instead of schemes was very confusing: imagine trying to read about "separated and proper morphisms" when you only know about varieties!]

If you mean to ask why does he assume the ground field is algebraically closed in Chapters 4 and 5 when much of what he does is true without that hypothesis: well, that's a choice that he made, and no one but he could definitively answer it. As an arithmetic geometer I certainly wish that he had treated more general ground fields. To a large degree Qing Liu's recent text Algebraic geometry and arithmetic curves exists because of this restriction in Hartshorne's text. (More precisely it exists because Hartshorne did not treat arithmetic surfaces.) Still Hartshorne does more with cohomology of schemes than Liu does, so a student of arithmetic geometry should spend time reading both texts.

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The arguments in chapters 4 and 5 use the scheme-theoretic machinery of chapters 2 and 3.

See e.g. the proof of Castelnuovo's criterion, which among other things uses the theorem on formal functions.

Basically, even when the ground field is algebraically closed, it is easy to encounter non-trivial schemes (e.g. as thickened neighborhoods of closed sub varieties), and these come up in the arguments and discussion of chapters 4 and 5.

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As someone who hasn't worked through Hartshorne yet: do the scheme-theoretic arguments significantly simplify the exposition in these latter chapters? (Could roughly the same theorems be proved without much extra difficulty in a course that didn't use the scheme-theoretic or cohomological machinery Hartshorne develops?) –  Mike Miller Apr 23 at 6:49
Dear Mike, It's not just a question of simplification. The proof of the more substantial theorems rely on both the cohomological and scheme-theoretic tools. My understanding is that the book began life as Chapters 4 and 5, based on courses that Hartshorne taught, and that the material of Chapters 2 and 3 were added as essential background material. I'm not sure how Castelnuovo's criterion (to take one key example) was proved in earlier days, but the proof Hartshorne gives uses the full panoply of tools. Regards, –  Matt E Apr 23 at 11:47
Thanks, Professor Emerton! That was very informative. –  Mike Miller Apr 23 at 14:35

Schemes allow one to do things that you can't do with varieties even over algebraically closed fields. For instance, the tangent space is very interesting scheme-theoretically at cusps but not as a variety. It is generally operations like taking the tangent space that lead one to scheme theory, and not neccessarily weird fields or other rings.

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But this doesn't answer the question. Hartshorne doesn't really discuss tangent spaces, right? –  Martin Brandenburg Dec 27 '13 at 10:40

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