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I have just finished my undergraduate studies and i want to study some algebraic geometry. i have taken 1 year analysis, topology, algebra courses and finished 1 semester complex analysis, homological algebra, fourier analysis, differential geometry.

Hi there. Just finished up my first semester as a graduate student focusing heavily on geometry and topology at Colorado State University. Didn't know any algebraic geometry until this past spring and have...grown since then haha.

Given your background, you would probably appreciate Rick Miranda's Algebraic Curves and Riemann Surfaces. It is almost entirely done in the context of complex analysis, but all the things you want to be true in the algebraic geometry context are true if you just replace "holomorphic" and "meromorphic" with "regular" and "rational." (In fact, Dr. Miranda notes this in his "Algebraic Sheaves" section toward the end.) You get a great introduction to curves, divisors, sheaves, line bundles, etc., all with examples using the Riemann Sphere (Complex projective line), tori, and a general complex projective (or affine) curve. You also might be able to find this book as a PDF if you know how to Google well. And you don't need all the deep commutative algebra to understand this book, because it leaves everything mostly complex analytic (in a fun way) and really just relies on a little differential topology in some spots that you might not have. If you do choose that book, feel free to get ahold of me if you want some discussion on the concepts or a bit of help with some of the exercises. I'm just finishing up my first readthrough of that one over the winter break and have been doing the problem in that book for a couple years now. (I think my contact info should be on my Google Drive. If not, you can ask me for it on here or probably find it under the CSU graduate students list.)

After a final review of the rest of my recommendations, I really would put Dr. Miranda's book at the top for you. That book makes you feel like you're doing complex analysis with some differential topology but sneaks you into algebraic geometry until you realize that's what you were doing the whole time. And, in fact, there is a paper called Algebraic Geometry and Analytic Geometry--GAGA because French--which states that basically these two are one in the same; complex analytic geometry and complex algebraic geometry really aren't different. (Cf. Chow's Theorem and Dr. Miranda mentions this paper as well.) I think this book would be ideal for your background.

If, however, you are wanting to do some algebra before anything else, check out Dummit and Foote's Algebraic Geometry part/chapter/section (whatever they call it) which goes through the Hilbert Basis Theorem and other very basic results which we use(d) in a classical algebraic geometry intro. class. You also learn enough about localization (especially the correspondence between prime ideals in a localization and prime ideals in your original ring).

Other popular algebraic geometry books include Milne's Notes (just google the words "Milne Algebraic Geometry"--freely available online by the author), obviously Hartshorne (though I get that that's a bit tough to read), and maybe some Harris and/or Eisenbud stuff (e.g. Geometry of Schemes which you might be able to find online). You might be able to find Shafarevich's Basic Algebraic Geometry online, too, and that one has plenty of examples. You really want a lot of examples when studying algebraic geometry.

A book with some heavy category theory and basically all the algebraic geometry in the world, you could look at Vakil's The Rising Sea. He publishes these (I hesitate to call them) notes online every year or so after edits, and this book is just a massive outpouring of alg. geo. with a 100-page grounding in category theory (and some commutative algebra in category theoretic terms) as the foundation for sheaves and schemes. Along the way, Ravi Vakil gives some great pictures of what we are trying to describe with Spec(-) under the Zariski topology and the affine scheme. I honestly have only gotten through maybe 150 pages of that book after about 4 or 5 months of time with it, because of its density and thoroughness. There are examples throughout, and you should really do all the examples you can. The only way I've found to get through alg. geo. is by examples--otherwise you're lost in a sea of abstract meaninglessness.

One note that cannot be stressed enough: DO PROBLEMSDo problems while you're awake. Do problems while you're half-awake. Do problems on the train. Do problems on the bus. Do problems in your sleep. Do problems at the gym. Do problems literally all of the time every time all day every day you possibly can. And if you're having trouble with some problems, ask the community! They've been great while I learn algebraic geometry, and I'll be around to post when I can as well! :)

One of the main research programs in Algebraic Geometry is the classification of varieties. Towards this goal two methodologies arose, the first is classifying varieties up to isomorphism which leads to the study of moduli spaces and the second is classifying varieties up to birational equivalences which leads to the study of birational geometry. Part of the engine of the birational classification is the Minimal Model Program which, given a variety, seeks to find "nice" birational models, which we call minimal models. Towards this direction much progress has been made but there is also much to be done. One aspect of interests is the role of algebraic fiber spaces as the end results of the Minimal Model Program are categorized into Mori fiber spaces, Iitaka fibrations over canonical models and varieties of general type. A natural problem to consider is, starting with an algebraic fiber space, how might it behave with respect to the Minimal Model Program. For case of elliptic threefolds, it was shown by Grassi, that minimal models of elliptic threefolds relate to log minimal models of the base surface. This shows that minimal models, in a sense, have to respect the fiber structure for elliptic threefolds. In this dissertation, I will provide a framework towards a generalization for higher dimensional elliptic fibration and along the way recover the results of Grassi for elliptic fourfolds with section.