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Is it possible to get into algebraic geometry by just knowing calculus and linear algebra or is this too far of a stretch?

If not could anyone give me a list of book/lecture notes recommendations in chronological order to dive into algebraic geometry with my state of knowledge?

Thanks

mannequin
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    No way, you definitely need a solid course in Abstract Algebra. – Gregory Grant Aug 03 '16 at 20:25
  • I suggest reading Hungerford's Algebra, and also pay particular attention to "Commutative Algebra". If you can get through Hungerford then you can get through Algebraic Geometry. – Gregory Grant Aug 03 '16 at 20:26
  • In truth, I don't know much about Algebraic Geometry, aside from the fact that all my friends are always talking about it. From what I've gathered, its one of the hardest (if not THE hardest) topics in modern mathematics. Don't rush into it: take your time and learn some basics first. – Alekos Robotis Aug 03 '16 at 20:46
  • I'd have to agree that Algebraic Geometry is relatively difficult compared to most branches of math. But by far the hardest has to be Number Theory because to do Number Theory you need to know Algebraic Geometry pretty well, as well as a bunch of other really difficult things like Modular Forms and Elliptic Curves. "Don't rush into it" is excellent advice. – Gregory Grant Aug 03 '16 at 20:55
  • As Gregory Grant says, you need to know the basic ideas of abstract algebra (rings, modules, fields), and it would be beneficial to know a little commutative algebra.

    If you're really keen, though, once you know what a polynomial ring over a field is, and what an ideal is, I think you could profitably start reading Volume 1 of Basic Algebraic Geometry by Shafarevich. Of course you'll have to pick up lots more algebraic and categorical ideas on the way, but that's ok...

     – Nefertiti Aug 03 '16 at 21:17
  • At the very least, you'll want to have taken a course in group theory and ring theory; in particular, commutative ring theory as @GregoryGrant mentioned. Of course remember your calculus and linear algebra, as those are also used.

    Fulton's Algebraic Curves assumes a course in group theory and ring theory, and builds up the necessary commutative algebra as the book goes on (free here: http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf ); however the theory is devoted solely to curves.

     – cat Aug 03 '16 at 21:21
  • More specifically I have to disagree with the suggestion to "get through" Hungerford. As I recall Hungerford has a wide (and somewhat idiosyncratic) selection of topics, some of them only tangentially related to introductory algebraic geometry (e.g. algebras over division rings). – Nefertiti Aug 03 '16 at 21:23
  • Fair enough, there's more to Hungerford than you need, but you do need a surprising amount of it for algebraic geometry, you really need a high level of comfort with algebra for that application. It's not enough to read one introductory book, you should probably give yourself a full year of algebra before tacking Fulton. – Gregory Grant Aug 03 '16 at 22:48
  • @cat Thanks for that link, I learned from Fulton as well and I really liked it but the version I read was typeset on a regular typewriter which made it painful to read, so cool to have an updated version like that! – Gregory Grant Aug 03 '16 at 22:49
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    Ok I see thanks for all the comments and sorry for the noob question :p I guess I will start with A Book of Abstract Algebra by Pinter since I heard it is a good book on Abstract Algebra. – mannequin Aug 04 '16 at 08:00
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    Try Brendan Hassett's Introduction to Algebraic Geometry. It's very down to earth, and covers important computational ideas that tend to be glossed over or left to the reader. – Tabes Bridges Aug 04 '16 at 20:46

2 Answers2

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Googling will lead you to various roadmaps for learning alg. geom., both on this site and on MO, for grad students but also for undergrads.

One place to start, if you are an undergrad, is Miles Reid's book Undergraduate Algebraic Geometry. Not everyone likes it, but I do, and routinely recommend it to both undergrads and beginning grad students.

(By the way, I work in algebraic geometry, arithmetic geometry, modular forms, elliptic curves, and related topics mentioned in the comments above. I think that viewing things as difficult, or the most difficult, etc., area of math is not very helpful. If you are interested in something, and motivated to learn it, try learning it! Just keep your common sense about you, make sure you do well in your regular classes too, and ideally find a nearby faculty member, grad student, post-doc, or even just more experienced undergrad to act as mentor. Also, although algebraic geometry, once it gets going, relies on other areas of math for background, including various areas of algebra, topology, and geometry, you can try getting into it directly, and then use it as motivation to learn something about those other areas.)

tracing
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I guess it is technically possible, if you have a strong background in calculus and linear algebra, if you are comfortable with doing mathematical proofs (try going through the proofs of some of the theorems you used in your previous courses, and getting the hang of the way you reason in such proofs), and if you can google / ask about unknown prerequisite material (like fields, what $k[x, y]$ stands for, what a monomial is, et cetera) efficiently...

...but you will be limited to pretty basic reasoning, computations and picture-related intuition (abstract algebra really is necessary for anything higher-level than simple calculations in algebraic geometry).

Nevertheless, you can have a look at the following two books:

  • Ideals, Varieties and Algorithms by Cox, Little and O'Shea. This book actually assumes only linear algebra and some experience with doing proofs, and I think it goes through things in a very easy-to read fashion, with many pictures and motivations of what is actually going on. However, expect mostly computation-related stuff in here (but I think that is good as well :) ) https://www.amazon.com/Ideals-Varieties-Algorithms-Computational-Undergraduate/dp/0387356509
  • Algebraic Geometry, A Problem Solving Approach, by...a lot of people, link here: https://web.archive.org/web/20151022025819/http://math.lssu.edu/bsnyder/PCMI/MathFest/compilemeforamsbook.pdf This is a rather unique book, because it begins with very basic intuition behind algebraic geometry, and successively moves deeper into the heavier stuff. The whole book is just one big list of problems, and each problem takes you one step closer to understanding algebraic geometry. I think you should already be able to at least do a lot of the problems in the beginning chapter(s).

Both of these books are designed to be easy on the reader when it comes to prerequisites, unlike most other books who are written for "pros", a.k.a. "people with a lot of background in Abstract Algebra". I think / hope that your knowledge in Calc. + Linear Algebra is enough for this to get you going (but be warned, it might be pretty hard to understand all the new concepts in one go, so take it easy :) ).

Cheong Sik Feng
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MonadBoy
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