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I don't understand why it is so difficult to find a really beginner book in Algebraic Geometry. Let's take for example Fulton's algebraic curves: An introduction to Algebraic Geometry. I've already heard on MSE and some students of my university saying that it's very concise and leave the proofs as exercises to the students, which personally I agree with. The last thing I've heard about this book is from a professor of my university (he is a little bit old now, but he was a very prolific researcher in Algebraic Geometry), he said that Chapter 7 is very hard to understand and has a lot of hard calculations and he never fully understood this chapter.

There are other introduction books, but they are very few and miss some things that a good introduction book should have.

I think this doesn't happen in Analysis and Abstract Algebra, for example. There are a lot of good introduction books with very detailed proofs, solutions to exercises and so on. I can't imagine a very good researcher in Analysis saying "I've never understood this chapter about continuity of the introductory book of the author $X$".

Why does this happen with Algebraic Geometry? Is it maybe because this area is very recent? Or because there aren't many buyers to buy Algebraic Geometry books? Or because this area is indeed very hard to understand (and it will never have a good beginner book).

Thanks!

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    $\begingroup$ See math.stackexchange.com/questions/1748/…. $\endgroup$ – lhf May 25 '15 at 18:17
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    $\begingroup$ Also math.stackexchange.com/q/394896/264 $\endgroup$ – Zev Chonoles May 25 '15 at 18:25
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    $\begingroup$ @lhf as I said, they aren't so abundant as other areas and sometimes they miss something. Take for example Harris and Beltrametti. I don't think these books so easy to a beginner understand. $\endgroup$ – user42912 May 25 '15 at 18:26
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    $\begingroup$ Probably because algebraic geometry uses algebra and topology as pre-requisites. So they assume by the time you get there that you already have a certain amount of experience with proofs and such. $\endgroup$ – Gregory Grant May 25 '15 at 18:28
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    $\begingroup$ As Mumford says in one of his books, it's not a "primary" subject like analysis or algebra. I'm not saying this excuses the lack of a book that you like, just that the situation is different. Have you taken a look at Jim Milne's online book? I haven't read it cover to (virtual) cover and he rewrote a bunch of it recently, but he's usually pretty good about including background, proofs, solutions, etc. $\endgroup$ – Hoot May 25 '15 at 20:27
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1) "Maybe because this area is very recent?"
No, algebraic geometry goes back at least to Descartes and Fermat and is essentially as old as calculus.
2) "Because there aren't many buyers to buy algebraic geometry books?"
No, that's irrelevant.
Authors don't care, since they know very well that any book beyond the level of linear algebra or calculus won't bring in any money.
More surprisingly publishers don't seem to care either: I have at times been called upon to write a book but it was pretty clear that the representatives of publishers weren't too interested in the content but just wanted to add titles to their catalogue.

3) "...he said that the chapter 7 is very hard to understand and has a lot hard calculations and he never fully understood this chapter"
The candidness of this professor is much to be admired and I quite believe that, as you say, he did good research in algebraic geometry.
He pinpointed the heart of the problem: algebraic geometry is indeed extremely difficult to penetrate, precisely because it is such an old subject which people have attacked with such a variety of unrelated tools. For details, look at this answer .

4) A challenge.
There always exists a conic passing through five points in the projective plane.
Given six points in such a plane, what condition must they satisfy in order that a conic passes through all of them ?
I'd be interested in the thoughts of a non-professional algebraic geometer on this problem, which illustrates a point I'll make later.

Edit: Answer to the challenge
Consider the six points $p_1,\cdots,p_6$ and the hexagon they determine. The three pairs of opposing sides intersect in three points $$A_1=(p_1p_2)\cap (p_4p_5), A_2=(p_2p_3)\cap (p_5p_6),A_3=(p_3p_4)\cap (p_6p_1) $$ There exists a conic passing through these six points $p_i$ if and only if the points $A_1,A_2,A_3$ are collinear.
This extremely pretty result is Pascal's theorem (+its converse), which he found in 1640 at the ripe age of 16.
My point is that one couldn't come up with such a solution even after having read and understood the more than 10000 pages of EGA+SGA+Stacks Project.
That algebraic geometry is so vast, ranging from this theorem by Pascal to the moduli stack of curves, explains why writing a comprehensive introductory book on the subject is an essentially impossible mission.

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  • $\begingroup$ I'm really looking forward to someone answer your challenge. $\endgroup$ – user42912 May 26 '15 at 20:48
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    $\begingroup$ Dear user, I'll answer it soon if nobody does. Let's say within 36 hours. Remind me if I forget :-) $\endgroup$ – Georges Elencwajg May 26 '15 at 20:55
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    $\begingroup$ @GeorgesElencwajg: Hi Georges :) Given 5 points in general position in $P^2$ there is a unique conic $C$ that passes through them. So if we add one more point and ask that a conic passes through the 6 points, the only possibility is that this conic is $C$ and this can happen only if the sixth point is actually a point of $C$. More generally, a conic passes through 6 points if the rank of $V(x_1,\dots,x_6):= [v_2(x_1) \dots v_2(x_6)]$ is less than $6$; $v_2:P^2 \rightarrow P^5$ is the veronese map. So the answer is if and only if $Det\left[V(x_1,\dots,x_6)\right]=0$. Right? $\endgroup$ – Manos May 27 '15 at 0:55
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    $\begingroup$ Dear @Manos, this is absolutely correct and yields a perfect "analytic" solution to the problem, for which I have now provided a "synthetic" solution. The dichotomy analytic-synthetic was deemed quite important a century ago but is now completely obsolete. Anyway, +1 and bravo for bravely picking up the "challenge". $\endgroup$ – Georges Elencwajg May 28 '15 at 8:31
  • $\begingroup$ @GeorgesElencwajg: Ah yes, Pascal's theorem, i remember now reading it in Fulton :) $\endgroup$ – Manos May 28 '15 at 17:35
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For full disclosure, I am hardly an expert in this area (indeed, I'm learning from Fulton, just like you), but I thought I'd throw in my two cents.

First, a recommendation for a "really beginner book:" Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea. This is one of the friendliest math books I've ever read, and the emphasis on computation and algorithms ensures you see lots of concrete examples. The prerequisites are very minimal: they don't even assume that you know what an ideal is! Another useful resource, with many great visuals, is Donu Arapura's site.

Fulton is very terse, and sometimes he is a bit light on giving the reader intuition. But one has to appreciate this style--every sentence has a clear purpose, with no extraneous remarks. And many other books are so long as to be quite intimidating: just look at Ravi Vakil's nearly 800 pages of notes, or the countless, dense volumes of Grothendieck's EGA, SGA, and FGA.

But just because a chapter in Fulton is only a half-dozen pages does not mean that you shouldn't spend a week or more on it. I think your understanding will be much clearer if you spend the first day working with simple, easy examples to get a feel for the definitions and theorems. For me, it was often these examples that truly illustrated the meaning behind these definitions and results. For instance, in Cox, Little, and O'Shea, I felt like every chapter included a reference to the twisted cubic. Keeping this example in mind while reading the new material made things much more concrete and comprehensible.

From your question you sound a bit discouraged, so remember that a bit of motivation always helps, too!

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  • $\begingroup$ Thank you for your answer! I'm motivated, I'm only thinking if it's not better to choose something more direct without so many prerequisites as abstract algebra (group rings for example). $\endgroup$ – user42912 May 26 '15 at 20:53

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