When will the volume of "Algebraic Geometry III: Schemes" be written?

[fzyzcjy]

Hi thanks for the book, I wonder when will "Algebraic Geometry III: Schemes" be written - it looks quite interesting!

[vEnhance]

Could be a while LOL. Turns out that I had way more free time at age 20 than at age 27. Pull requests are certainly welcome of course ;)

[fzyzcjy]

I see. Take your time and looking forward to it!

I am happy to make a PR, but as you know, I am not a mathematician so definitely not have the ability :/

[Incompleteusern]

Do probability chapters first >:(.

[fzyzcjy]

+1 for probability - that one also looks quite interesting, looking forward to it as well!

[user202729]

The book recommends a few resources to learn algebraic geometry.

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Some comments unrelated to the Napkin.

Personally, I think Vakil is written in a relatively "friendly" language, although of course it's difficult.

Take some examples:

• Explanation of what fiber product should mean.

  

  Another (more rough but mostly equivalent) way that I like to think of it is "for each point z ∈ Z, compute the fiber f⁻¹(z) and g⁻¹(z), multiply them together, and glue them all together to get ⋃_z f⁻¹(z) × g⁻¹(z)" -- which explains the term "fiber product".

  That also give an intuitive explanation on why the fiber product of two open set is their intersection -- the number of "points" in each fiber is either 0 or 1, and the product of two numbers either 0 or 1 is 1 if and only if both are 1.

• Valuative criterion for separatedness:

  

  I think this is very clear. (although personally speaking, even though both this and the "diagonal morphism are closed" properties can be topologically motivated, I find this one more intuitive to think of, so you might as well think of this as the definition of separatedness with no harm)

• Definition of properness.

  although for this one I'd prefer the author to just explain the answer to me, because I find the remark unfortunately quite opaque (as it turns out, trying to reinterpret the exercise 10.3.A right below in the context of topological space is quite enlightening)

Nevertheless, I find a few points not that clear, to me at least. Perhaps the author find it too obvious...?

• what it means for f: X → Y to be separated, if Y is not "a single point" (read: Spec ℂ).

  Personally I interpret this as that: if you go backwards from Y to X by inverse image, it does not create any more inseparatedness -- for example, a map from the line with doubled origin to a line is not separated, but a map from a plane with a line doubled to a line with the origin doubled is separated.

  It's in fact true that if f is separated then every fiber of f is separated (but the converse does not hold).

• what the correspondence of "proper" in topological context is. (as I figured out later, it corresponds to compact Hausdorff space, which is why proper morphism are required to be separated, instead of just spaces where every open cover have finite subcover)

• how exactly quasicoherent sheaves of schemes correspond to vector bundles. While the book does explain for a bit, it takes until much later to explain that a locally free sheaf and a vector bundle is not the same thing -- you apply Spec Sym^● (–^∨) on a locally free sheaf to get a vector bundle.

  

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Overall, I think Vakil did a really nice job of teaching the content (and also explain the intuition, most of the time) that it would be rather difficult to improve upon it.

[user202729]

Speaking of which, Evan does have a lecture note on algebraic geometry https://web.evanchen.cc/notes/Harvard-137.pdf .
I haven't read it fully, but -- as you may expect it's of napkin-style in terms of formatting and wording of stuff. (unfortunately, through the whole thing, there's exactly one and only one green \begin{moral}...\end{moral} box.)

There's some sections with more intuition (outside green box) though.

and some sections with... anti-intuition.

Anyway, this is actually the first time I heard of the hyperplane bundle as the dual of the tautological bundle. Let's see if I can finally get what sheaf of module really is (I got quite a bit of trouble with this all the time)