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.
Everyone 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.
I 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...
Could 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?
What 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.
Yeah, 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.
The difficulty with learning 'modern' algebraic geometry is not only is it very dense and general, but that means the original motivation can become lost.
So I think understanding Weil conjectures are key for modern algebraic geometry. And it's always easier to understand algebraic curves (algebraic geometry with dimension 1) and their connection to Riemann surfaces (algebraic curves over the complex numbers with analytic rather then algebraic structure), as they provide motivation for many of the results and constructions.
A good introduction to Algebraic Curves and the Weil conjectures I've found is following
My favorite intros to algebraic geometry are these:
Igor R. Shafarevich, Basic Algebraic Geometry, two volumes, third edition, Springer, 2013.
Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry, 1994. (Especially nice if you like complex analysis, differential geometry and de Rham theory.)
I highly recommend the book "An Invitation to Algebraic Geometry" by Karen E. Smith et. al., as an introduction. It'll give you a taste of the field, without delving too deeply into technicalities.
You'll need a good background in commutative algebra to learn algebraic geometry, if you have that and want to see the modern approach, schemes and everything else that Grothendieck did, I suggest Vakil's notes, they're freely available from his homepage. (Disclaimer: I only studied the first 15 chapters and the one on Kähler differentials which should be the 21st, but I suppose the second half is as good as the first, I'll find out for sure next term)
Vakil's notes have an interesting epigram from Grothendieck:
I can illustrate the … approach with the … image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!…
A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration … the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it … yet finally it surrounds the resistant substance.
If you do have some math background, "An Invitation to Algebraic Geometry" by Karen Smith et al is a lovely, slim volume that gets to important ideas right away.
"Ideals, Varieties, and Algorithms" is an "invitation to computational geometry" by Cox, Little, and O'Shea. I think parts of it might really appeal to HN readers and it's supposed to be for an undergrad math major audience.
Atiyah-Macdonald is a small but dense book with most of the commutative algebra you'll need to start learning algebraic geometry. You'll need to know some abstract algebra as a prerequisite (groups, rings, fields; covered in undergraduate algebra courses).
A less dense book for commutative algebra is Miles Reid's Undergraduate Commutative Algebra, but it doesn't cover tensor product of modules which will definitely be needed for algebraic geometry. Also for everyone interested in Atiyah-Macdonald you MUST do the exercises, half of the book is in the exercises!
