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This isn't a fun answer, but you basically can't without substantial background in undergrad mathematics. The subject has some of the highest prerequisites of a
by throwawaymath 8y ago
This isn't a fun answer, but you basically can't without substantial background in undergrad mathematics. The subject has some of the highest prerequisites of any upper undergrad/lower grad level math course, as it touches on (and uses) just about everything you'd learn through an undergrad math degree.
I don't want to discourage you, I'm just being realistic. If you want to work towards algebraic geometry, you can certainly do that. You'll need to first master linear algebra and abstract algebra. You should have a strong understanding of fields, groups, rings, vector spaces and modules. Someone else mentioned commutative algebra - that is more of a circular dependency with algebraic geometry than a hard one. It's good to have walking in, but realistically you can't master the subject without knowing algebraic geometry.
You'll also need analysis, in particular complex analysis for curves. Real analysis and topology should also be covered but I suppose with tenacity you could get by without them.
To translate these into concrete suggestions, in your position I'd try to work through the following, in order:
1. Linear Algebra Done Right (Axler)
2. Abstract Algebra (Dummit & Foot)
3. Complex Analysis (Ahlfors)
4. Algebraic Curves (Fulton)
The last one is a standard upper undergraduate introduction to the subject.
If possible you should organize a study group or take a class though, because trying to learn math on your own from a textbook is rough.
- NiftyWalrus 8y agoCould you describe what Grothendieck was doing in algerbiac geometry? Would studying the above get you up to the point of his work? If not, what are concrete suggestions that would get you there?
- obastani 8y agoThe "Background and history" section of the following article gives a very high-level idea of Grothendieck's contribution: https://en.wikipedia.org/wiki/Weil_conjectures https://en.wikipedia.org/wiki/Weil_conjectures
- NiftyWalrus 8y agoWhat is the modern path to study this area now? I’m sure it’s better understood now and one wouldn’t have to follow the historical approach to study the same concepts.
- obastani 8y agoI don't know myself, but here are some possible answers: https://mathoverflow.net/questions/114034/learning-path-for-the-proof-of-the-weil-conjectures https://mathoverflow.net/questions/114034/learning-path-for-... Edit: Here is another overview that seems good: https://www.math.ucdavis.edu/~osserman/math/pcm.pdf https://www.math.ucdavis.edu/~osserman/math/pcm.pdf
- MaysonL 8y agoTake a look at the stacks project: https://stacks.math.columbia.edu https://stacks.math.columbia.edu A cross between a wiki and a collaborative textbook.
- sdenton4 8y agoYeah, it's still incredibly difficult terrain, despite all the time that's passed. Wrestling with Hartsthorne is still a rite of passage for students in this area. I am (very) intrigued to know what would happen if someone made a serious effort to make the ideas more accessible. It remains an area where the depth of knowledge required is legitimately deep: would be cool to figure out which ideas are actually independent of other parts of the stack.
- Grustaf 8y agoEveryone should also read Modern Geometry book by Dubrovin, Novikov and Fomenko, it starts pretty basic and covers a lot of ground, and is extremely well written.
- johncarlosbaez 8y agoThat book is great, but not mainly about algebraic geometry: it's more about differential geometry.
- Grustaf 8y agoI know, but itäs an excellent introduction to geometry in general, like some of the other books listed. It’s probably not even possible to get from 0 to algebraic geometry in a single book...