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It seems that Robin Hartshorne's Algebraic Geometry is the place where a whole generation of fresh minds have successfully learned about modern algebraic geometry. But is it possible for someone who is out of academia and has not much background, except typical undergraduate algebra and some analysis, to just go through the book, page by page? If not, what is the proper route for entering a serious algebraic geometry book, like Hartshorne's?

J W
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Hooman
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    With just a typical undergrad algebra course as background, I think Hartshorne would be out of reach. David Eisenbud's "Commutative Algebra: with a View Toward Algebraic Geometry" might make a better starting point (this text was written sort of as background for Hartshorne -- notice the pun in the title). – Bill Cook Sep 26 '12 at 18:05
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    @Bill: where is the pun? – mlbaker Jun 07 '13 at 09:37
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    Hartshorne's book is entitled "Algebraic Geometry". Eisenbud says in his introduction that he started writing Commutative Algebra to fill in background for Hartshorne's book, and so he considers the name "Commutative Algebra: with a View Toward Algebraic Geometry" a kind of pun. – Bill Cook Jun 07 '13 at 13:51

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Hartshorne's book is an edulcorated version of Grothendieck and Dieudonné's EGA, which changed algebraic geometry forever.
EGA was so notoriously difficult that essentially nobody outside of Grothendieck's first circle (roughly those who attended his seminars) could (or wanted to) understand it, not even luminaries like Weil or Néron .
Things began to change with the appearance of Mumford's mimeographed notes in the 1960's, the celebrated Red Book, which allowed the man in the street (well, at least the streets near Harvard ) to be introduced to scheme theory.
Then, in 1977, Hartshorne's revolutionary textbook was published.
With it one could really study scheme theory systematically, in a splendid textbook, chock-full of pictures, motivation, exercises and technical tools like sheaves and their cohomology.
However the book remains quite difficult and is not suitable for a first contact with algebraic geometry: its Chapter I is a sort of reminder of the classical vision but you should first acquaint yourself with that material in another book.

There are many such books nowadays but my favourite is probably Basic Algebraic Geometry, volume 1 by Shafarevich, a great Russian geometer.
Another suggestion is Milne's excellent lecture notes, which you can legally and freely download from the Internet.
The most elementary introduction to algebraic geometry is Miles Reid's aptly named Undergraduate Algebraic Geometry, of which you can read the first chapter here .
Miles Reid ends his book with a most interesting and opinionated postface on the recent history and sociology of algebraic geometry: it is extremely profound and funny at the same time, in the best tradition of English humour.

Holdsworth88
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Georges Elencwajg
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    In what sense is Hartshorne's book an "edulcorated" version of EGA? I might have written reduced (maybe you will say "caramelized"?), or abridged, or condensed instead. Also, and I am sure you know this, the thing that makes EGA difficult to read is not that it is dense, but rather that it is gigantic. Line by line it is very easy to read---easier than Hartshorne! – Stephen Sep 26 '12 at 21:25
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  • Edulcorated comes from the Latin dulcis meaning pleasant, sweet, dear...Hartshorne and his pictures and exercises is certainly sweeter than EGA which hasn't a single one of either. 2) I am not so lucky as you to find EGA very easy to read line by line.I can only say I deeply admire anyone who finds, say the seven-page proof of $EGA IV_4, 19.7$ Critère de platitude normale de Hironaka, easy to follow.
    • – Georges Elencwajg Sep 26 '12 at 21:47
  • Plus one. Do you have any opinion on Griffiths and Harris? Unfortunately, it comes without exercises. – Rudy the Reindeer Sep 26 '12 at 21:47
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    Dear Matt, I have browsed that book a lot. It is very rich and contains a lot of geometry. On the other hand I find its organization less than ideal: forcing the reader to ingurgitate Sobolev spaces and hard results in partial differential equations (like regularity of the Laplacian) before defining the Plücker embedding is a bit discouraging... – Georges Elencwajg Sep 26 '12 at 22:03
  • I have to disagree with you regarding EGA. For me it's like a solutions manual to Hartshorne, which is almost unreadable in comparison. –  Sep 27 '12 at 06:07
  • The remark about (IV, 19.7) seems unfair since Hartshorne as far as I know doesn't cover this at all. –  Sep 27 '12 at 06:17
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    Dear @Adeel: there is nothing unfair in my remark since Steve wrote "the thing that makes EGA difficult to read is not that it is dense, but rather that it is gigantic". All the professional mathematicians (I included) and advanced students around me find EGA difficult, and it shows when we hear someone give a talk on that work. Anyway, I don't find it very constructive to explain how easy someone finds EGA. People who think so are very welcome instead to make use of their expertise in giving good answers to questions here or on MathOverflow. – Georges Elencwajg Sep 27 '12 at 07:14
  • Dear @GeorgesElencwajg Thank you! I've been thinking about reading the book cover to cover (like I did with Atiyah-MacDonald) but on second thought this doesn't seem to be a good idea. I got it from the library and it's massive, too. – Rudy the Reindeer Sep 27 '12 at 09:12
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    I am not saying that EGA is easy either. The point as Steve wrote is that the difficulty is in the length, not the density; and your example about a long proof falls exactly in line with this. This also echoes some discussions on MathOverflow like http://mathoverflow.net/questions/3041/the-importance-of-ega-and-sga-for-students-of-today where Ravi Vakil writes: "I had assumed that it would be like Hartshorne, only more so, with huge heavy machinery constantly being dropped on my head. Instead, each statement was small and trivial, yet they inexorably added up to something incredibly powerful." –  Sep 27 '12 at 11:59
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    Dear Georges, So "edulcorated" comes from the Latin "dulcis"? That changes everything! I apologize for my ignorance. – Stephen Sep 28 '12 at 01:01
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    Dear @Steve: no problem. Actually the adjective is dulcius: I apologize for my ignorance to Monsieur Deliège, my fondly remembered Latin teacher of so long ago... – Georges Elencwajg Sep 28 '12 at 06:42
  • I've just noticed a few days ago a new undergraduate "book" by Ravi Vakil with his students, that seem to be promising. I wonder what you would say about it. Here is the page of the course: http://virtualmath1.stanford.edu/~vakil/17-145/ and here are the lecture notes https://www.overleaf.com/read/jhbrjrsxfdcv – agleaner Feb 11 '20 at 12:05
  • @aglearner "I wonder what you say about it". I say that Vakil has written a wonderful document and that it will be very profitable to study it! – Georges Elencwajg Feb 11 '20 at 12:07