Hartshorne or Vakil's notes

Question

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!

Answer 1

In my humble opinion, the Vakil notes (also known as FOAG) are very complete with regards to scheme theory; they include all prerequisites (category theory, commutative algebra, topology, etcetera omissis [e.o.]) to scheme theory, an extensive bibliography, and also information about the "art status" of algebraic geometry.

But this completeness is an overload of information, so I use FOAG only for when I want a detailed study of some argument. On the other hand, the Harthshorne book (I write about his "Algebraic Geometry", with emphasis on Chapters II and III) is an underload of information; because it recaps Éléments de géométrie algébrique by Grothendieck and Dieudonné (which is exactly 1800 pages of scheme theory, not a page more, not a page less), it is not very easy to read. In the opinion of someone who has studied 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 recommendation is that you continue your 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;

You can use FOAG for some more detailed study as these books complement it well. For example, IMHO the Bosch book is poor on cohomology theory, so I studied cohomology from FOAG; Görtz and Wedhorn's book is poor on commutative algebra; Eisenbud and Harris's book is rich with examples, but less so compared to FOAG; Mumford's book does not contain exercises; e.o.

Answer 2

It's always good to be acquainted with several texts. And although the page count may be large, it's not as though you need to memorise every page or every argument.

What is important is to see how the whole hangs together.

I was impressed by Hartshorne because when I was looking for a description of algebraic bundles - in both bundle and sheaf termonology - I only discovered Hartshorne gave the full duality between the two pictures.