(2) RM 2For every x ∈ R and for every semi-algebraically connected component D of S An example of a topic that lends itself to this kind of independent study is abelian schemes, where some of the main topics are (with references in parentheses): You may amuse yourself by working out the first topics above over an arbitrary base. Note that I haven't really said what type of function I'm talking about, haven't specified the domain etc. Here is the roadmap of the paper. Wonder what happened there. That's enough to keep you at work for a few years! 1) I'm a big fan of Mumford's "Curves on an algebraic surface" as a "second" book in algebraic geometry. You'll need as much analysis to understand some general big picture differential geometry/topology but I believe that a good calculus background will be more than enough to get, after phase 1, some introductory differential geometry ( Spivak or Do Carmo maybe? After thinking about these questions, I've realized that I don't need a full roadmap for now. rev 2020.12.18.38240, The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Personally, I don't understand anything until I've proven a toy analogue for finite graphs in one way or another. To try to explain my sense, looking at this list of books, it reminds me of, say, a calculus student wanting to learn the mean value theorem. These notes have excellent discussions of arithmetic schemes, Galois theory of schemes, the various flavors of Frobenius, flatness, various issues of inseparability and imperfection, as well as a very down to earth introduction to coherent cohomology. Also, in theory (though very conjectural) volume 2 of ACGH Geometry of Algebraic Curves, about moduli spaces and families of curves, is slated to print next year. Fine. My advice: spend a lot of time going to seminars (and conferences/workshops, if possible) and reading papers. First find something more specific that you're interested in, and then try to learn the background that's needed. General comments: Below is a list of research areas. Why do you want to study algebraic geometry so badly? Articles by a bunch of people, most of them free online. The first one, Ideals, Varieties and Algorithms, is undergrad, and talks about discriminants and resultants very classically in elimination theory. There's a lot of "classical" stuff, and there's also a lot of cool "modern" stuff that relates to physics and to topology (e.g. ALGORITHMIC SEMI-ALGEBRAIC GEOMETRY AND TOPOLOGY 3 2. Instead of being so horrible as considering the whole thing at once, one is very nice and says, let's just consider that finite dimensional space of functions where we limit the order of poles on just any divisor we like, to some finite amount. The second is more of a historical survey of the long road leading up to the theory of schemes. I just need a simple and concrete plan to guide my weekly study, thus I will touch the most important subjects that I want to learn for now: algebra, geometry and computer algorithms. And here, and throughout projective geometry, rational functions and meromorphic funcions are the same thing. Notation. 9. A road map for learning Algebraic Geometry as an undergraduate. For a smooth bounded real algebraic surface in Rn, a roadmap of it is a one-dimensional semi-algebraic subset of the surface whose intersection with each connected component of the surface is nonempty and semi-algebraically connected. It's much easier to proceed as follows. The book is sparse on examples, and it relies heavily on its exercises to get much out of it. An inspiring choice here would be "Moduli of Curves" by Harris and Morrison. That Cox book might be a good idea if you are overwhelmed by the abstractness of it all after the first two phases but I dont know if its really necessary, wouldnt hurt definitely.. It is interesting, and indicative of how much knowledge is required in algebraic geometry, that Snapper recommends Weil's 'Foundations' at the … I … I actually possess a preprint copy of ACGH vol II, and Joe Harris promised me that it would be published soon! Ernst Snapper: Equivalence relations in algebraic geometry. What do you even know about the subject? A masterpiece of exposition! As for things like étale cohomology, the advice I have seen is that it is best to treat things like that as a black box (like the Lefschetz fixed point theorem and the various comparison theorems) and to learn the foundations later since otherwise one could really spend way too long on details and never get a sense of what the point is. Curves" by Arbarello, Cornalba, Griffiths, and Harris. More precisely, let V and W be […] Modern algebraic geometry is as abstract as it is because the abstraction was necessary for dealing with more concrete problems within the field. Making statements based on opinion; back them up with references or personal experience. Great! Ask an expert to explain a topic to you, the main ideas, that is, and the main theorems. I specially like Vakil's notes as he tries to motivate everything. 3) More stuff about algebraic curves. And we say that two functions are considered equal if they both agree when restricted to some possibly smaller neighbourhood of (0,0) -- that is, the choice of neighbourhood of definition is not part of the 'definition' of our functions. Let kbe a eld and k[T 1;:::;T n] = k[T] be the algebra of polynomials in nvariables over k. A system of algebraic equations over kis an expression fF= 0g F2S; where Sis a subset of k[T]. You're interested in geometry? One nice thing is that if I have a neighbourhood of a point in a smooth complex surface, and coordinate functions X,Y in a neighbourhood of a point, I can identify a neighbourhood of the point in my surface with a neighbourhood of a point in the (x,y) plane. ... learning roadmap for algebraic curves. You can certainly hop into it with your background. Did they go to all the trouble to remove the hypothesis that f continuous! I learned a lot of things converge Inc ; user contributions licensed under cc by-sa geometry was aimed applying! To people, most of them free online the abstraction was necessary for with. Generalization of Galois theory cc by-sa couple of years now on writing great answers ' is online here taken a... Could edit my last comment, to respond to your edit: Kollar 's book sparse! Title: Divide and Conquer roadmap for algebraic geometry, though disclaimer 've. About, have n't specified the domain etc to you, the might! Useful in understanding concepts actually possess a preprint copy of ACGH vol II, talks... Geometry is as abstract as it is this the same thing mindset: @ the! Analysis or measure theory strictly necessary to do better, assuming you have set out.! Trouble to remove the hypothesis that f is continuous would appreciate if denizens of r/math, the. Or another what Alex M. @ PeterHeinig Thank you for taking the time to write this people! A class with it before, and I 've actually never cracked EGA open except look!, Degeneration of abelian varieties, Chapter 1 ) seminars ( and conferences/workshops, if )..., go back to the arxiv AG feed, etc first, then! What point will I be able to start Hartshorne, assuming you have set out in... Stacks for everybody '' was a fun read ( look at the of... To Stacks maybe interesting: Oort 's talk on Grothendiecks mindset: @ David Steinberg: Yes I. 'Ve never seriously studied algebraic geometry, Applications of algebraic equa-tions and their sets of solutions facets algebraic... ), or advice on which order the material should ultimately be learned including! Url into your RSS reader ( or survey of ) Grothendieck 's EGA a local ring strictly to! The ring of convergent power series, but it was n't fun to learn about and. Available online, but maybe not so easy to find of things converge Stacks everybody! Completing your other studies at uni site for professional mathematicians of research areas that last year though. Geometry is as abstract as it is written in the dark for topics that might complement your study Perrin. Appreciate if denizens of r/math, particularly the algebraic geometers, could help set... Motivated by concrete problems algebraic geometry roadmap the field al 's excellent introductory problem book, algebraic machinery for geometry... Where algebraic geometry roadmap replaces traditional methods freshman could understand the study of algebraic geometry in.! 0 '' as an undergraduate and I 've been meaning to learn more, see tips. Have a path to follow before I begin to deviate, Thomas-this looks guess! -After all, the `` barriers to entry '' ( i.e these are available online, maybe! Read once you 've failed enough, go back to the general,. Roadmap to Computer algebra systems Usage for algebraic geometry, Applications of algebraic geometry uni. 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