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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, Ideal's Varieties and algorithms by Cox and Shea and Justin smiths 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).

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    $\begingroup$ 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 $\endgroup$ – Alex Mathers Apr 23 '17 at 1:43
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    $\begingroup$ Fulton himself has a free, modified version of the text on his website: math.lsa.umich.edu/~wfulton/CurveBook.pdf $\endgroup$ – Ben West Apr 23 '17 at 2:01
  • $\begingroup$ Is Igor's Basic Algebraic Geometry any good for a first run through of algebraic geometry? $\endgroup$ – Alexander King Apr 23 '17 at 2:02
  • $\begingroup$ 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.. $\endgroup$ – User3773 Apr 23 '17 at 14:06
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    $\begingroup$ Here are a bunch of past threads on this topic, all with many answers: 1, 2, 3, 4, 5, 6, 7 $\endgroup$ – André 3000 Nov 15 '18 at 3:20
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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.

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    $\begingroup$ 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. $\endgroup$ – user171326 Apr 23 '17 at 17:39

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