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I'm looking into studying Algebraic Geometry alongside some Algebraic Topology and was wondering what book would be useful. I was looking along the lines from Igor's Basic Algebraic Geometry, Cox, Little and O'Shea's Ideals, Varieties, and Algorithms, and Justin Smith's Introduction to Algebraic Geometry. I've heard that Igor's book Basic Algebraic Geometry is fantastic but the problems are known to be impossible.

What is the best text to use for study on algebraic geometry? I have an understanding in Real Analysis (only single variable), Topology, Abstract Algebra (up to Ring theory).

J W
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Alexander King
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    I think the book Algebraic Curves by Fulton would be a good start. You likely don't know enough commutative algebra yet, and Fulton introduces the necessary commutative algebra as he goes along – Alex Mathers Apr 23 '17 at 01:43
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    Fulton himself has a free, modified version of the text on his website: http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf – Ben West Apr 23 '17 at 02:01
  • Is Igor's Basic Algebraic Geometry any good for a first run through of algebraic geometry? – Alexander King Apr 23 '17 at 02:02
  • I was wondering if an introduction to complex geometry may be more useful: e.g. Griffiths-Harris, Voisin or the easier Miranda. This will give you the right ideas and intuitions; in the meanwhile one can look at abstract commutative algebra and after that any introduction to algebraic geometry will do (e.g. Vakil, Reid, Hartshorne). This is at least my experience.. – User3773 Apr 23 '17 at 14:06
  • @AlexanderKing You're forgetting the author's last name! You're probably referring to Igor Shafarevich. – user49640 Apr 26 '17 at 03:16
  • Here is an excerpt from the Mathscinet review by Werner Kleinert of the second edition of Shafarevich's book: "The first edition of this book appeared (in Russian) in 1972. At that time, this textbook was the first and the only one which built bridges between the geometric notions, the classical origins and achievements, the modern concepts and methods, and the complex-analytic aspects in algebraic geometry. The English translation of this outstanding textbook was published in 1977 under the title Basic algebraic geometry. In the meantime, it has become one of the most valuable, – user49640 Apr 26 '17 at 03:24
  • "recommended and used textbooks on algebraic geometry worldwide, together with the other standard textbooks of R. Hartshorne, D. Mumford, and P. A. Griffiths and J. Harris. The special feature of the author’s book, in comparison to the other textbooks, is provided by the fact that it really conveys the many different aspects of contemporary algebraic geometry, without focussing on any particular approach and without requiring any advanced prerequisites. In this sense, it – user49640 Apr 26 '17 at 03:25
  • "has proved an extremely useful addition to the other (here and there) more thorough-going textbooks, a recommendable introduction to them and to current research, and in any case, an excellent invitation to algebraic geometry." – user49640 Apr 26 '17 at 03:25
  • Here is what Shafarevich says about prerequisites in his preface: "The nature of the book requires the algebraic apparatus to be kept to a minimum. In addition to an undergraduate algebra course, we assume known basic material from field theory: finite and transcendental extensions (but not Galois theory), and from ring theory: ideals and quotient rings. In a number of isolated instances we refer to the literature on algebra; these references are chosen so that the reader can understand the relevant point, independently of the preceding parts of the book being referred to. Somewhat more – user49640 Apr 26 '17 at 03:30
  • "specialised algebraic questions are collected together in the Algebraic Appendix at the end of Book 1." – user49640 Apr 26 '17 at 03:30
  • Thanks. For some reason I just wasn't entirely sure if it was a better intro than Cox's book. My bad for forgetting to mention the last name. – Alexander King Apr 26 '17 at 15:29
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    Here are a bunch of past threads on this topic, all with many answers: 1, 2, 3, 4, 5, 6, 7 – Viktor Vaughn Nov 15 '18 at 03:20

2 Answers2

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Thomas Garrity's Algebraic Geometry: A Problem Solving Approach.

The book starts with establishing the equivalence of conics in the complex projective plane and then moves on smoothly to discussing tangents and singularities, elliptic curves, Bezout's theorem, Riemann-Roch, affine and projective varieties, and -- finally -- a brief intro to sheaves and cohomology.

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You don't have enough background in commutative algebra. Try to finish Atiyah & McDonald or Eisenbud first. Then uyou can start Liu Qing or Hartshorne.

user439714
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    I disagree. This depends what you want to study : I did read the book of Fulton with very few prerequisites in commutative algebra. And for example, the book of Miranda as already mentioned in the comment required nothing except complex analysis and basic algebra. –  Apr 23 '17 at 17:39