I believe Hartshorne and Vakil's notes are two most popular text currently, so my question is about how to choose the text.

I have worked through the first 4 chapters of Vakil's notes and now I am thinking whether should I continue or try to study Hartshorne.

Vakil's notes are very well-organized. Especially, the exercises appear just in the right time, and there are more explanation of the exercises, so that I know what I am doing. But the problem is most arguments are given in the form of exercises, which means I am always stuck. The typical situation is after 2 hours work, maybe I am still in the same page. But the book has almost 800 pages! Hartshorne has some proof, the exercises also have some explanation. So maybe I should try to work Hartshorne?

Another question is about exercises. How long should I spend for an exercise that stick me. Should I look up a solution after maybe struggling for half hour? There are solutions for Hartshorne, so maybe study Hartshorne is more convenient since it is easier to look up solution?

Also, what is the right pace to learn the stuff? I mean should I worry if every day I spend 3 hours to learn the stuff but I only finish 1 page?( I know maybe I should spend more time, but unfortunately I am teaching myself algebraic geometry and I have other classes currently)

I appreciate any advice, thanks!

  • $\begingroup$ Are you a beginner in algebraic geometry and\or scheme theory? $\endgroup$ – Armando j18eos Mar 1 '16 at 22:36
  • $\begingroup$ @Armandoj18eos I am a beginner of scheme theory, but know some basic algebraic geometry. $\endgroup$ – user198206 Mar 2 '16 at 2:45

In my humble opinion (IMHO), the Vakil's notes (also known as [aka] FOAG) are a very complete text of scheme theory; they include all prerequisite (category theory, commutative algebra, topology, etcetera omissis [e.o.]) to the scheme theory, an ample bibliography, and more informations about the "art status" in algebraic geometry;

but this completeness is an over loading of informations, therefore I use the FOAG only for a detailed study of some argument (nextly, I shall expose my experience).

On the other hand, the Harthshorne's book (I write about "Algebraic Geometry" of Hartshorne) is an under loading of informations; because this book is a recap of the Éléments de géométrie algébrique of Grothendieck and Dieudonné (exactly 1800 pages of scheme theory, not a page in plus, not a page in minus; aka EGA) it is not very easy; rather, in opinion of who had study it: the essence of Hartshorne's book is in the exercises, and the exposition of the theory is not very clear (for obvious reasons).

After all this, my council is: you continue the study of algebraic geometry from another textbook; I suggest:

  1. Bosch - Algebraic Geometry and Commutative Algebra,
  2. Eisenbud and Harris - The Geometry of Schemes,
  3. Gathmann's lecture notes (classical Algebraic Geometry and Scheme Theory),
  4. Görtz and Wedhorn - Algebraic Geometry I,
  5. Mumford - The Red Book of Varieties and Schemes;

and you can use FOAG for some detailed study.

For example, IMHO: the Bosch's book is poor of cohomology theory, so I was study the cohomology from FOAG; Görtz and Wedhorn's book is poor of commutative algebra; Eisenbud and Harris's book is rich of examples, less with respect to FOAG; Mumford's book does not contain exercises; e.o.


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