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(undergraduate) Algebraic Geometry Textbook Recomendations

I am planning to self-study one of these two subjects: Algebraic geometry , Algebraic topology. I can borrow books from library, but I don't know which books to borrow. Please suggest books for me.

Thank you

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Amr
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  • Related http://math.stackexchange.com/questions/1748/undergraduate-algebraic-geometry-textbook-recomendations/24443#24443

    http://math.stackexchange.com/questions/998/best-algebraic-geometry-text-book-other-than-hartshorne/24447#24447

     – Mohan Jan 31 '13 at 12:26
  • And another: http://math.stackexchange.com/questions/89843/learning-roadmap-for-algebraic-topology – Seirios Jan 31 '13 at 12:29
  • @Seirios I never studied homological algebra. Do I need to study homological algebra before studying algebraic topology – Amr Jan 31 '13 at 12:33
  • You can study many subjets in algebraic topology without homological algebra. For example, Massey's book doesn't mention it. – Seirios Jan 31 '13 at 12:38
  • @Seirios ,@Mohan Thanks everyone. Lets close the question! – Amr Jan 31 '13 at 12:40
  • You don't really need to study homological algebra before doing either. The deepest things you'll most likely use in AT are 5-lemma or snake lemma, but only for abelian groups. Can you clarify what level of AG you want to learn? – John Stalfos Jan 31 '13 at 12:40
  • @John Stalfos I never studied AG before, thus I guess I am looking for an undergraduate book – Amr Jan 31 '13 at 12:43
  • @JohnStalfos However if there are good graduate AG books for beginners, then please mention them – Amr Jan 31 '13 at 12:43
  • Based on my experience I'd suggest Reid's Undergraduate Algebraic Geometry. You can probably do every problem in the book within a few months, and best of all it has some actual geometry in it. From there I don't really know. I took a first semester graduate class out of Shafarevich, but I found the book quite frustrating, as did most of my classmates. Hartshorne(what we moved too) is also frustrating at times, but for different reasons. – John Stalfos Jan 31 '13 at 12:47

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A very beautiful introduction to algebraic geometry is the book "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra" in the Springer undergraduate texts in mathematics series. (It only requires basic abstract algebra as a prerequisite).

The second "half" of Munkres' 2000 Topology textbook is an introduction to algebraic topology.

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Algebraic Geometry

For algebraic geometry there are a number of excellent books. Hartshorne's Algebraic Geometry is widely lauded as the best book from which to learn the modern Grothendeick reformulation of Algebraic Geometry, based on his Éléments de géométrie algébrique. This is also, however, considered one of the most challenging textbooks ever written on any mathematical subject ever. This makes it a poor first choice. He leaves critical and very much non trivial results as exercises and has very few motivating examples outside of the exercises. But it is well worth the effort.

Others are The Red Book of Varieties and Schemes by Momford which covers the topic from a more classical viewpoint. This is an excellent book. Also from the classical and historical view is Basic Algebaric Geometry I by Shafarevich which is well written and also an excellent book.

I would strongly reccoment reading one of the last two to get a grounding in the historical motivation and classical theory, then read Hartshorne. Be warned though, reading Hartshorne may, and should, take years.

Algebraic topology

There is an excellent book by Allen Hatcher called Algebraic Topology that is available for free on his website, and also as a hard copy on Amazon. This is an excellent geometrically oriented book on the subject that contains much of what you would learn in a graduate course on the subject plus a large number of additional topics. This is where I reccommend you first go.

A second excellent text on the subject is Topology and Geometry by Bredon. This book isn't titled as if it an Angebraic topological work but it is, and it is particularly well written, although requires a bit of maturity and knowledge of category theory.

Finally we have J.P. May's A Concise Cource in Algebraic Topology. This is probably the best text for a second exposure to the subject, or to be used along with one of the two I have just described. It doesn't give a lot of exaples and proceeds very quickly through the subject (hence concise) but it places the field in context and provides motivation for many of the objects of study. This one comes highly recommended as well.

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    I wonder why Hartshorne is still recommended. A better choice would be Liu, Görtz-Wedhorn or Bosch. – Martin Brandenburg Jan 31 '13 at 14:18
  • @DoctorBatmanGod Is Munkres topology good to study algebraic topology ? – Amr Feb 03 '13 at 16:09
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    @Amr Absolutely. The first half covers general topology so well that it effectively has no competition as a book on general topology, and the second half is an excellent introduction to algebraic topology. Munkres is an excellent writer and teacher. – guest196883 Feb 03 '13 at 17:59