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>>10019947
Basic Algebraic Geometry 1: Varieties in Projective Space by Shafarevich
Basic Algebraic Geometry 2: Schemes and Complex Manifolds by Shafarevich
An Invitation to Algebraic Geometry by Karen Smith, Lauri Kahanpää, Pekka Kekäläinen, and William Traves

>>10020528
is there any better book than shafarevich? It reads terribly with the shitty translation

>>10020694
Fulton algebraic curves

Autism.

>>10021129
fuck you

>>10019947
Algebraic Geometry is literally Rêddit:the math subfield.

>>10021146
I'll be here enjoying my apple and my grapes fagget.

>>10020704
is there an easier one

>>10019947

>>10019947
the only maths you will USE heavily are commutative algebra and cohomology. you can pick up the second as you go, but it would be helpful to know cohomology somewhere you might have more intuition, even simplicial cohomology (maybe take a break at some point to learn it when you start needing it).

From CA: Before you start scheme theory, you should basically be absolutely sure you know what things like local, normal, integrally closed, and valuation rings are. Be very comfortable with localizations of rings. If you want to see scheme theory in its full power, do a bit of algebraic and arithmetic geometry, which just requires basic number theory knowledge (being familiar with things like the Dedekind class group will be very helpful because they are generalized by the Picard group).

Know the basics of classical algebraic geometry (over C^n) and classical projective algebraic geometry (over Pn(C)) cold.

otherwise, it's good to have a strong intuition of what a sheaf is and what categorical operations on them look like (esp. pullback). While def not a prereq, I've found learning bits of differential geometry helpful for analogical purposes. For example, there are some analogies to be made between (line) bundles in AG and DiffGeo.

>>10021908
also obviously a bit of topology, but like just the very basics. Might be helpful to know some of the elementary separation conditions and finiteness conditions like "compactness." You will rarely use the topology for much other than record-keeping.

>>10021574
HOWLING

>>10021910
I've always wondered why the Zariski topology is mentioned in like every text but then it is never used? Or have i just not advanced far enough?

>>10024166
It's because it is a relatively natural (in hindsight), widely used example of non-hausdorff topology.
Now, since topological spaces are too large a class for nontrivial results, you have to restrict to interesting subclasses for interesting results.
And since topology is usually first introduced for the needs of analysis, the spaces that you will meet the most will be spaces of interest to analysts (so usually, metric spaces or function spaces).
Still, the Zariski topology is definitely used, but since it is only of use to algebraic geometers, it is generally discussed in introductory AG texts.

>>10024166
bc the topology is necessary to get off the ground, but isn't that topologically interesting in the usual senses. The open sets are very "coarse".

For the simplest HS example, rational functions over C are typically undefined at some finite set of points (the poles/zeroes of the denominator). So, these rational functions aren't functions on C, but on some subset of C minus some finite set of points. So you need the Zariski topology to have a useful sheaf of rational functions. But the topology itself only comes up in more subtle ways, and things like separation axioms are better behaved when defined scheme-theoretically. The underlying space comes up more in things like density of maps, dimension, and finiteness conditions like compactness. Which is why i said you'd only really need basic topology.

Real niggas do additive combinatorics.

>>10019947
>abs algebra

>>10025200
go to bed Terry

>>10021565
There is no royal road to algebraic geometry...except the royal road to algebraic by Holme. You could also try Beltrametti's lectures on curves and surfaces for classical algebraic geometry, or the last chapters of D&F. Also these notes are pretty great
http://math.stanford.edu/~vakil/216blog/index.html