0

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.

Junaid Aftab
  • 1,582
  • 2
    (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... – DonAntonio Mar 30 '17 at 15:30
  • @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. – Junaid Aftab Mar 30 '17 at 15:35
  • 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) – Sebastian Schulz Mar 30 '17 at 15:40
  • @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. – Junaid Aftab Mar 30 '17 at 15:43
  • Yes, it seems like an undergraduate text book would be the best start for you then. I've heard that Allenby's book is good but I haven't personally looked at it, so I'll take no responsibility – Sebastian Schulz Mar 30 '17 at 15:46
  • On a side note, maybe your primary focus for now should really not be on topics such as Galois theory, but really on rings, modules, polynomial rings and eventually some category theory – Sebastian Schulz Mar 30 '17 at 15:48
  • @SebastianSchulz All right. But I'm still unclear of the road map that one could/needs to follow to dive into AG? Also, any particular texts (even undergraduate texts) that'll do me more good on this front, based on some topics that they emphasize. – Junaid Aftab Mar 30 '17 at 15:48
  • Well, this is getting lengthy so here are my 3 final remarks: 1) I can't recommend any literature of the kind you're looking for as I've never looked at any undergraduate algebra books myself. 2) If you're interested in AG, you'll find that you'll always need more algebra, so it can't do harm to do more in the first place. 3) The first thing you'll encounter in AG is a sort of duality between geometric spaces and the algebra of functions on them, hence your first and foremost friend is a solid background on rings, modules and algebras. – Sebastian Schulz Mar 30 '17 at 16:03
  • Javier Álvarez (user here) has compiled a couple of big lists on learning AG; this one in particular seems helpful. But check out his other answered tagged with [tag:algebraic-geometry] too. – pjs36 Mar 30 '17 at 16:12
  • 2
    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. – Viktor Vaughn Mar 30 '17 at 18:01
  • @Quasicoherent and others: would you recommend keeping Knapp's Basic Algebra as the first textbook I should go through? It contains loads of linear algebra, group, ring, field and galois theory. It'll get me up to date with the basics; I can then pick up an advanced book for my purposes. – Junaid Aftab Mar 31 '17 at 03:17
  • I'm not familiar with Knapp's book. Again, if you just want the bare bones algebra necessary to do basic algebraic geometry, you mostly need results on polynomial rings, which are covered in the first two chapters of Cox, Little, and O'Shea. If you want to undertake a broader study of algebra, there are many, many books, of which my favorite is Dummit and Foote. They also develop classical algebraic geometry and the beginnings of scheme theory in Ch. 15. – Viktor Vaughn Mar 31 '17 at 03:54
  • @JunaidAftab : I did read the paper you did put online. This looks pretty hard with no background in algebraic geometry ! You probably need to read a commutative algebra book (Atiyah and McDonald say), basic notes in algebraic geometry (the book of Cox, Little and O'Shea is probably too slow). The best notes you can read is probably Reid's one about algebraic surfaces, it's on arxiv. Along the way you'll probably need to learn about representation theory, singularity, and algebraic topology, scheme, sheaf theory ... –  Mar 31 '17 at 07:18
  • @N.H. Oh well. As of now, I know the material for linear algebra, at the level of a second course, and some basic group theory. What line of course would you recommend? I'm thinking of going through an algebra book (Knapp's textbook for example); I should then follow up on Commutative Algebra (Atiyah and McDonald, say) and start with algebraic geometry. :/ – Junaid Aftab Mar 31 '17 at 07:24
  • Ok, so for sure first you need to have a solid background in commutative algebra. Atiyah and McDonald is nice, says read a lot the first 40 pages and come back if necessary. For more details you can look Dummit and Foote. For algebraic geometry you probably need to know surfaces at least at the level of this : https://arxiv.org/abs/alg-geom/9602006 (especially part with singularities). But I think this will probably be pretty hard so you should talk with your teacher for know what does he expect from you. –  Mar 31 '17 at 07:58
  • @N.H. Perfect. One last thing: have you seen Knapp's Basic Algebra? I think it seems like a good book I can use a precursor to a book on commutative algebra? I'll get to review a lot of the material; the first few chapters are exclusively on linear algebra and group theory. The text then discusses advanced topics in group theory before moving on to rings, modules and commutative rings in detail. Here's the amazon link: https://www.amazon.com/Basic-Algebra-Cornerstones-Anthony-Knapp/dp/0817632484. What do you think? – Junaid Aftab Mar 31 '17 at 08:02
  • This looks pretty ok but algebraic geometry is mainly about ring, so probably reading chapter VIII is as good as read Atiyah and McDonald. Anyway I think I'm gonna study these kind of topic in a few month, so if you have question I can try to answer (I don't claim I will be able to answer but I'll certainly be interested to think about it). –  Mar 31 '17 at 08:20

0 Answers0