Many students – myself included – have a lot of problems in learning scheme theory. I don't think that the obstacle is the extreme abstraction of the subject, on the contrary, this is really the strong point of modern algebraic geometry. I'm reading many books such those written by Hartshorne, Gortz & Wedhorn, Liu, Vakil (notes), Gathmann (notes), Shafarevich, Perrin and Milne (notes) and in my humble opinion the learning problems arise from the following considerations:
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It is enlightening to read about "the aim" of modern algebraic geometry, so I'm referring to: motivations behind schemes, the correpondence between algebraic and geometric entities (so the duality between the category of affine schemes and the category of rings), the importance of sheaves (so the concept of admissible functions) etc. But, despite this, when one goes into the actual construction of the new objects, all theorems, lemmas and propositions are missing details (that are left to the reader). For example the verification that certain presheaves are sheaves, functorial properties of assignments between categories and details about limit/colimit constructions are often missing. Even if the student has a solid background in algebra and geometry, generally they don't have the time or the capacity to complete all the statements. Basically taking a course in algebraic geometry implies that one must take many statements as acts of faith. I realize that authors and professors may have the same difficulties (especially lack of time) in writing down all the boring details, and moreover that a book with all proofs may include thousands of pages, but in this way students are encouraged (read discouraged) to simply memorize the most important results without really understanding the constructions. Finally, a book or a course characterized by explanations and by motivating as complete proofs is much more instructive than a book or a course which covers many advanced arguments IMHO.
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In mathematics when two object are isomorphic, it is a common practise to "identify" them. Basically if $A\cong B$ but $A$ has a simple description we write $A$ instead of $B$, but formally we are thinking of $B$. This procedure is used very often in algebraic geometry, but in some cases without explaining the isomorphisms and in other cases the two objects in question are considered "really the same" even if this can provoke formal problems (look for example here). This "abuse of identifications" often makes one lose sight of the essence of what one is studying and once again the "stupid student", exhausted, tends to simply memorize things. I point again that the problem is not the abstraction, but the fact that the excessive tendency to simplify notation, often leads to inconsistencies.
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Enough importance is not given to the following: the process of successive generalizations, put in place by the great mathematicians across history, which marked the birth of modern algebraic geometry. This process is fundamental in learning because it probably represents the most natural way whereby the human mind can deal with the subject.
In summary, because of the above issues (principally the first two), rigorous mathematical statements, incredibly seem to be informal dissertations at the eyes of the student that is eager for formalism.
In your opinion, what are the most common difficulties that a student encounters during their learning process of algebraic geometry? If my problems do arise precisely from the above considerations, can you give me some advice to solve them?
Best Answer
1) Algebraic geometry is indeed vast and difficult.
But don't be discouraged: professors and experts only know parts of it and you would be surprised to discover how little they know outside of their narrow domain of expertise.
This can be a strength: Grothendieck only knew Serre's article FAC and the content of a few Cartan seminars when he began to transform algebraic geometry by the introduction of scheme theory, in accordance with his awesome prophetic vision.
His correspondence with Serre has been published by Leila Schneps and is one of the most exciting documents in the history of mathematics.
His ignorance and his genius are displayed there, to our greatest delight.
2) Yet you should aim at knowing all of it.
There are many approaches to algebraic geometry:
-Classical in the style of the books by Fulton, Harris, Hodge-Pedoe, Kendig, Reid, Seidenberg, Walker, ...
-Complex analytic like in Grauert-Fritzsche, Griffiths-Harris, Huybrechts, Taylor, ...
-Scheme-theoretic like Bosch, Hartshorne, Görtz-Wedhorn, ... -Especially praiseworthy are books mixing several points of views, the best by far being Shafarevich, but there are others: Danilov-Shokurov, Perrin,...
Ideally you should learn all points of view.
As I wrote this is the aim: there are many hours in a life and knowing that it is impossible to reach this impossible goal should not prevent you from trying.
Willem van Oranje Nassau said it very well:
Point n'est besoin d'espérer pour entreprendre, ni de réussir pour perséverer.
[One need not have hope to begin an undertaking, nor a guarantee of success to persevere]
3) Solve little problems on a napkin while sipping coffee with a friend.
But actually the books you read are not so important.
The most important advice I can give is to solve little concrete problems, which you can find in books, invent yourself or read on this site.
It is no use spending much time on some equivalence of categories involving affine schemes while being incapable of exhibiting a birational isomorphism between a smooth quadric in projective space and a projective plane.
And for explaining why the two-codimensional union of two transverse planes in $\mathbb A^4$ cannot be defined by less than four equations, the equivalence of said category with that of commutative rings will not lead you very far ...
4) Also, draw doodles on that napkin.
Another important aid to understanding scheme theory is to invent conventions that will enable you to draw schemes so as to follow or invent proofs by visualization.
The best way is to start from Mumford's wonderful sketches in his Red Book: the way he draws spaghetti-like generic points (for example) is priceless!
Vakil's wonderful notes are even more graphic : for example, he explains again and again how the "fuzz" in his numerous drawings is the visual translation of algebraic notions like nilpotents, primary decomposition,...
Geometry has been for more than two thousand years the art of reasoning correctly on incorrect figures.
There is no reason why this should stop now.
5) And finally: you can do it! Good luck!