road map to learn algebraic geometry

I was fascinated by hearing about the clan of algebraic geometers in India,

Sridharan $\rightarrow$ parimala $\rightarrow$ {Sujatha, Suresh},

Seshadri $\rightarrow$ {C.S.Musili, Vikraman Balaji,},

M.S.Narasimhan $\rightarrow$ {M.S.Raghunathan,V.K.Patodi, T.Ramadas(ICTP), Nitin Nitsure(TIFR)}

and many more when i was in my 3rd year Integrated M.Sc mathematics course.I have completed My M.Sc course in July2013.

I was not having enough Background (i was not sure of what background i should have to learn algebraic geometry) and my teachers suggested me to wait for an year or two and learn good amount of groups, rings commutative algebra and then try for Algebraic geometry.

I waited till i learn some ring theory and tried looking at Algebraic geometry. For obvious reasons, I could not understand a bit.after a year, I had courses in Commutative algebra and algebraic geometry (very Basic) courses simultaneously. I was comfortable with commutative algebra but i was not very comfortable with algebraic geometry. I was not very comfortable with geometric notions. I wanted through get out of this trouble but could not make it due to spending time in other courses.

Now I want to learn "Geometric" view of algebraic geometry. I have fair amount of idea in commutative algebra.

I did not had any course in Geometry after my schooling. I know only high school geometry and I strongly feel I should first learn Geometry(projective, affine) and then look for algebraic geometry.

I would be thankful if some one can give an idea for me to the question "Where to start" ...

I am planning to learn simultaneously some affine varieties and solve problems in Daniel Bump's Algebraic Geometry Book.

every time when i ask this question, It will not reach to other person properly and it end up with some thing else. I hope that would not happen now :)

P.S : I strongly feel somehow that Algebraic Geometry is Not Just "Advanced/ well structured Commutative Algebra". That is the reason why i am trying to see the geometric point (if any, I strongly feel there is such) of Algebraic geometry.

• – Martin Brandenburg Aug 20 '13 at 10:40
• @MartinBrandenburg: I have seen those sites, but my question is geometric approach and not about "Commutative Algebra Towards Algebraci Geometry" :(.. I am comfortable with Commutative algebra approach.. algebraic sets, Zariski topology, affinie varieties and some more.. But, I am looking for geometric viewpoint.. – user87543 Aug 20 '13 at 10:48
• I'm far from an expert in this field but I might look here amazon.com/Basic-Algebraic-Geometry-Varieties-Projective/dp/… – Wintermute Aug 20 '13 at 11:41
• Yes, Shafarevich's book has to be my favorite place to start studying algebraic geometry. It has a very geometric flavor. – rfauffar Aug 20 '13 at 12:44
• @Praphulla: Yes, but in the linked discussions you will find enough books which fit your needs. The answers here will just repeat them again ... the most geometric introduction is probably Geometry of schemes by Eisenbud-Harris. – Martin Brandenburg Aug 20 '13 at 13:53

First of all, for a discussion of the fact that algebraic geometry is not just "Advanced/well structured Commutative Algebra", you might want to read this.

As to how to learn some algebraic geometry, you might start with the book Undergraduate Algebraic Geometry by Miles Reid. After that, there are several good books to look at next, such as: Basic Algebraic Geometry by Shafarevic, Algebraic Geometry I: Complex Projective Varieties by Mumford, the first chapter of Hartshorne's Algebraic Geometry (which is something like a summary of Shafarevic's book), The Red book of varieties and schemes by Mumford (it is hard to use this as a stand-alone text, but it can be inspiring and provide motivation), and Algebraic Geoemtry: A First Course by Joe Harris.

To learn to think geometrically takes time and practice, especially if your training has been to think in a more algebraic/algorithmic way. All the book mentioned above should help you with this goal.

Also, it doesn't make too much sense to dwell on affine varieties before you move to projective varieties: with affine varieties there is always the temptation to convert everything back to commutative algebra, whereas with projective varieties (which are in the some sense the more fundamentally geometric objects) you are forced from the beginning to be a bit more geometric. Finally, although of course both affine and projective varieties can be greatly generalized to schemes and so on, there is no need to worry about this when you are first learning the subject (unless you have specific research goals which require you to move at a rapid pace). Just focusing on varieties makes perfect sense.

• I don't completely agree with the last paragraph. There are many classical nontrivial theorems about varieties which become almost trivial within the language of schemes (there was a discussion on MO about this, cannot find it right now). – Martin Brandenburg Aug 20 '13 at 14:00
• @Matt E: I strongly feel that, "with affine varieties there is always the temptation to convert everything back to commutative algebra".. I have experienced this every time :D... you said, "unless you have specific research goals which require you to move at a rapid pace".. I would like to work on problems related to algebraic geometry.. SO, I should move at rapid pace as you said... what should be the next plan.. – user87543 Aug 20 '13 at 15:48
• @PraphullaKoushik: Dear Praphulla, If you want to do research in algebraic geometry, you should look at the books in my answer, but then at the same time, or soon after, start looking at Vakil's notes and at Hartshorne (staring with Chapter I, and focusing on the exercises). If you find the scheme theory and cohomology in Hartshorne too daunting or off-putting to read through without movitation, skip from Ch. I to Chs. IV and V (on curves and surfaces --- both quite geometric, with great exercises), and then move back into Ch. II and Ch. III as you need to to fill in details. Regards, – Matt E Aug 26 '13 at 2:51
• @MartinBrandenburg: Dear Martin, I'm not sure what you have in mind, but I'm not sure I really agree with your statement, although presumably it all hinges on the use of the word "non-trivial", and perhaps we'll end up simply disagreeing about that. Regards, – Matt E Aug 26 '13 at 2:53
• @MattE : Thanks for your suggestion. I would prefer to follow the way you suggested.. – user87543 Aug 28 '13 at 15:10

I think that Miranda's Algebraic Curves and Riemann Surfaces might be a very good start at a more geometric flavored algebraic geometry. I recommend looking through the reviews at amazon to get a feeling of what the book is like. Basically this book will give you many basic tools in algebraic geometry, while keeping a very geometric viewpoint.