Prerequisites for Algebraic Geometry

Solution 1:

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 :) ).

Solution 2:

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.)