vEnhance
When will the volume of "Algebraic Geometry III: Schemes" be written?
Hi thanks for the book, I wonder when will "Algebraic Geometry III: Schemes" be written - it looks quite interesting!
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 ;)
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 :/
+1 for probability - that one also looks quite interesting, looking forward to it as well!
The book recommends a few resources to learn algebraic geometry.
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:
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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.
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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)
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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...?
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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).
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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)
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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.
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.
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)