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I'm looking to work with a professor who has a research problem for students in Algebraic Geometry. Unfortunately, given my unconventional background, I haven't had a course in Algebra as of yet. I was wondering if anyone could tell me the textbooks I should read in the next few months to get myself apace to the point where I can start learning Algebraic Geometry. I'm looking for recommendations that'll not only help me understand the basics but also aid in my goal of starting with Algebraic Geometry soon.

Edit:

Here's the paper a former student of the professor sent to me: https://arxiv.org/pdf/math/0410281.pdf. He said the problem was loosely based around this paper, though I should get in touch with the most recent student of the professor to get more details on the problem.

It'd be great if someone could recommend a path through algebra to algebraic geometry by taking the content of this paper into consideration.

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    $\begingroup$ (1) The prof. has a problem with Alg. Geometry? Perhaps you don't want to study that course with her/him...(2) Haven't you studied any algebra at all? Because you're going to need a hefty background in group theory, ring theory, algebras, polynomial rings and etc. How many months do you have to do this? (3) To begin with, try Atiyah-MacDonald's "Introduction to Commutative Algebra", or the book with the same name by Zariski-Samuel, or Reid's, or Eisenbud's... $\endgroup$ – DonAntonio Mar 30 '17 at 15:30
  • $\begingroup$ @DonAntonio (1) I have edited the post. I have some background in algebra. I have seen most of the material in group theory, but I haven't studied algebra from inside out so I need to start covering the material in a sequel from the textbook. Shouldn't I be going through some abstract algebra book before going through books on commutative algebra? Also, from that I gather from various posts over here, it seems as if students cover commutative algebra after covering algebraic geometry. $\endgroup$ – Junaid Aftab Mar 30 '17 at 15:35
  • $\begingroup$ If you've seen group theory before, you should probably get your hands on some ring theory next. There's plenty of resources for that, I remember Kaplansky; Dummit and Foote; Hungerford; Goldhaber. It's really a matter of taste as well and how well you're prepared to pick up a graduate level text book (which you will have to before thinking about diving into AG) $\endgroup$ – Sebastian Schulz Mar 30 '17 at 15:40
  • $\begingroup$ @SebastianSchulz Would it be a bad idea to pick up an undergraduate text first and cover group, ring and field theory first? The upshots are that I have seen (but not necessarily mastered) most of the material in group theory, and I have done a second course in Linear Algebra up till the Jordon Form. So I can most probably go through these units rather quickly, I suppose, even though the presentation and the proofs may be new. $\endgroup$ – Junaid Aftab Mar 30 '17 at 15:43
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    $\begingroup$ I think your best bet is Cox, Little, and O'Shea's Ideals, Varieties, and Algorithms. It's an undergraduate textbook, and they assume very little algebra and teach you most of it along the way. Really though, you should just ask the professor him- or herself. Only they know the required background for the research they have in mind, so they would be best suited to recommend a book. $\endgroup$ – André 3000 Mar 30 '17 at 18:01

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