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What are the best algebraic geometry textbooks for undergraduate students?

J W
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    Perhaps unnecessary to have this question here, since there's the MO question about the same thing. Unless it's the case that asking for undergraduate textbooks especially would result in different answers. – ShreevatsaR Aug 06 '10 at 19:50
  • @ShreevatsaR, I agree that there's overlap, although I think that specifying "undergraduate" does lead to different books being emphasized (e.g. Ideals, Varieties, and Algorithms). – J W Oct 17 '22 at 18:08
  • Not specified in the question - undergraduate in which country/countries. It can make a difference, as some textbooks would be final-year undergraduate in one country and first-year graduate, say, in another. – J W Oct 17 '22 at 18:14

11 Answers11

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(answer moved from closed duplicated question)

If you are interested in learning Algebraic Geometry I recommend the books of my Amazon lists. Most of them, at the beginning of the lists, are in increasingly difficulty and recommended order to learn from the beginning by yourself.

In particular, from those lists, a quick path to understand basic Algebraic Geometry would be to read Bertrametti et al. "Lectures on Curves, Surfaces and Projective Varieties", Shafarevich's "Basic Algebraic Geometry" vol. 1, 2 and Perrin's "Algebraic Geometry an Introduction" and the beautiful new Holme's "A Royal Road to Algebraic Geometry" . But then you are entering the world of abstract algebra. For that it is advisable to master the book of Miles Reid "Undergraduate Commutative Algebra" accompanied by the new Singh's "Basic Commutative Algebra". Both are very readable and thorough at their level, the former a geometry-oriented introduction and the latter a purely formal reference.

My personal opinion is to avoid as main text, books like Harris - "Algebraic Geometry: A First Course" or Cox et al., above all if you have limited time to learn. In my experience these kind of books will not get you very far within algebraic geometry by themselves, although they are good companions as sources for examples and computations. In particular Harris' is a very nice companion to the others as a more "literary" supplement.

Focusing only at the undergraduate level the best books in order of sophistication are: Smith et al. "An invitation to Algebraic Geometry", Reid "Undergraduate Algebraic Geometry", Hulek "Elementary Algebraic Geometry", Beltrametti et al. "Lectures on Curves, Surfaces and Projective Varieties".

There are several free online pdf courses. Concretely, the introductory notes by Dolgachev "Introduction to Algebraic Geometry" (along with his "Topics on Classical Algebraic Geometry" to be published soon) are very algebraic and may be read without problems after or along with a book like Hulek's "Elementary Algebraic Geometry". For introductions which cover very nicely just the basic up to even schemes and cohomology the very best are by far the notes by Gathmann's "Algebraic Geometry" and Holme's "A Royal Road to Algebraic Geometry". A nice introductory course focusing on algebraic curves is Fulton's "Algebraic Curves - An introduction to Algebraic Geometry".

When one dives into graduate level, it is very important to remember that Algebraic Geometry is a huge subject and there are different approaches to start. In my opinion, the word "geometry" is fundamental and foundational, so people should not forget that in the end there must be some geometric content or analogue. Therefore the bible book by Hartshorne "Algebraic Geometry" MUST be studied only after some mastering of the basics like Beltrametti et al., Shafarevich and Perrin and above all Mumford's "Algebraic Geometry: Complex Projective Varieties" which is a masterpiece (you can continue to schemes by Mumford with the unpublished notes here). In order to supplement Hartshorne's with another schematic point of view, the best books are Mumford's "The Red Book of Varieties and Schemes" and the three volumes by Ueno "Algebraic Geometry I. From Algebraic Varieties to Schemes", "Algebraic Geometry II. Sheaves and Cohomology", "Algebraic Geometry III. Further theory of Schemes". ONLY after all this materials one can understand the geometry behind the extremely algebraic but also good and interesting books like Liu Qing - "Algebraic Geometry and Arithmetic Curves" (another approaches into Arithmetic Geometry might be Lorenzini - "An Invitation to Arithmetic Geometry", and Hindry/Silverman "Diophantine Geometry" which starts with a review on algebraic geometry and proves Mordell and Faltings theorems among others without the use of schemes).

My personal learning path is this: Beltrametti et al.'s for classic and basic geometric foundations, Perrin's for a more algebraic introduction with a nice treatment of Riemann-Roch, Mumford's "projective varieties" for foundations along with a flavor on complex algebraic geometry, then study notes by Gathmann. After this, I would start approaching Mumford's red book and Ueno to supplement Hartshorne on schemes. With all this background mastering Hartshorne should not be a problem, BUT you must DO all the exercises you can since they are the most important stuff in Hartshorne's. Other sources on this regard is the new book by Görtz/Wedhorn "Algebraic Geometry I, Schemes wiith Examples and Exercises" and its future volume two. If one needs a different approach including some of the sheaf theory and homology needed along with Riemann surfaces, there is Harder's "Lectures on Algebraic Geometry vol. 1 & 2".

Further references about algebraic curves, surfaces, higher-dimensional varieties and other subjects can be found in my Amazon lists.

For those interested in the Complex Geometric side (Kähler, Hodge...) I recommend Moroianu's "Lectures on Kähler Geometry", Ballmann's "Lectures on Kähler Manifolds" and Huybrechts' "Complex Geometry" above all. To connect this with Analysis of Several Complex Variables I recommend trying Fritzsche/Grauert "From Holomorphic Functions to Complex Manifolds" and also Wells' "Differential Analysis on Complex Manifolds". Or, to connect this with algebraic geometry, try, in this order, Miranda's "Algebraic Curves and Riemann Surfaces", or the new excellent introduction by Arapura - "Algebraic Geometry over the Complex Numbers", Voisin's "Hodge Theory and Complex Algebraic Geometry" vol. 1 and Griffiths/Harris "Principles of Algebraic Geometry".

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Javier Álvarez
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  • Hi, you seem to know a lot about this subject. I have a question. I have recentely returned to the university after being away for >10 years. And now im about to join a class in algebraic geometry. Being away for so long i have forgotten much of the basics and i wonder if you possibly can recommend some books to read before i dive into algebraic geometry. Books about subjects you need to know to understand algebraic geometry. I have forgotten things like rings, fields, most of complex analysis etc. An introductory book about algebraic structures maybe is a good start? Anything else? Thank you. – JKnecht Jan 14 '16 at 12:25
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    @JKnecht since you probably need to catch up fast, I would recommend a book on complex analysis, other on commutative algebra. Follow the first links in this other answer of mine http://math.stackexchange.com/a/257528/4058 For basic algebra review the excellent free books http://abstract.ups.edu/download.html and http://homepage.math.uiowa.edu/~goodman/algebrabook.dir/download.htm Pick any good book on complex analysis, my favourite is Freitag & Busam, the 2nd vol. even introduces Riemann surfaces and modular forms, then follow my 1st link for Gathmann and Kerr as brief but great intros to AG – Javier Álvarez Jan 14 '16 at 17:45
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    @JKnecht if you need to review basic topology a very short but comprehensive text is Runde's "A taste of Topology", for more rigour and algebraic topology having Bredon's "Geometry and Topology" as a reference is more than enough. Once "basic algebra" is mastered, any good comprehensive reference for adv. commutative algebra is enough, like Eisenbud or Sigh texts. – Javier Álvarez Jan 14 '16 at 17:50
  • Great, thanks ! – JKnecht Jan 14 '16 at 19:52
  • @JavierÁlvarez In your answer to the first comment, I see you recommended a book on Complex Analysis. Is much complex analysis needed for Algebraic Geometry? How big is complex analysis in AG? – user5826 Apr 18 '16 at 06:08
  • @AlJebr If you approach AG from a purely commutative algebra perspective (say via Hartshorne) then you just need a very basic conceptual understanding of complex analysis. However Complex AG, e.g. all the classical results about algebraic curves and surfaces, and the modern study of complex varieties with interconnections with the differential geometry approach, then complex analysis is a BIG important thing. Even in arithmetic geometry, complex analysis of modular forms related to e.g. elliptic curves is fundamental, so "some" complex analysis is a basic conceptual requirement for AG. – Javier Álvarez Apr 18 '16 at 21:38
  • @JavierÁlvarez I was looking at the book Lectures of Curves, Surfaces and Projective Varieties, and in the preface it lists projective geometry as a prerequisite. I have never taken a course in projective geometry. Do you have any books you can recommend? – user5826 Nov 22 '16 at 04:42
  • @JavierÁlvarez This is a great answer, but I'm very interested to know why you didn't suggest Kunz's book: "Introduction to plane algebraic curves" neither in this answer, nor in the other more general answer about books in algebraic geometry at every level. Thank you – Richard Mar 08 '17 at 21:33
  • @JavierÁlvarez Moreover, I do not get for example why it seems you prefer Fulton's book instead of Kunz. It seems to me that Kunz gives many details in every proof while many claims are left as exercises by Fulton. I know, in this way maybe you can appreciate some techniques, but the not proved claims maybe are too many for my tastes. Or are there any other reasons of your preference? I ask since I can't decide between these two books to start learning algebraic plane curves. – Richard Mar 08 '17 at 21:46
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    @JavierÁlvarez the amazon link at the beginning of your answer no longer works, could you please provide a new one? – QuasarChaser May 13 '19 at 11:33
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Since today (when I received it) I think that the new book Introduction to Algebraic Geometry is the best undergraduate textbook on the market.
The book has an extremely motivated presentation of concepts, many excellent examples and a lot of exercises, half of them solved at the back (a unique feature at this level).
This thick volume, more than 600 pages, starts from elementary algebraic subsets of $\mathbb A^n$ and goes on to schemes and projective varieties.
Another unique feature is that the book contains in its four appendices all the necessary prerequisites, including 149 pages of commutative algebra.
As an icing on the cake, the volume is dirt-cheap at $25 in the US (and 21.10 euros in France, where I bought it).
Moreover our friend @Justin Smith lets us download it for free from his site, even after the publication of his book (see link above): thank you for your generosity, Justin, and for having written such a great book .

Georges Elencwajg
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See the MathOverflow question about exactly this: Best Algebraic Geometry text book?

Personally, from what little I've seen, Ideals, Varieties and Algorithms by Cox, Litle and O'Shea is a very nice book for undergraduates. It requires little background, and has a computational perspective that's especially good for those with concrete or computational interests. I don't know how good it is as a preparation for further "serious" study in algebraic geometry, but it's certainly engaging.

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ShreevatsaR
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Cox, Little, and O'Shea is an excellent book. I'd also like to recommend Miles Reid's Undergraduate Algebraic Geometry, which unlike Hartshorne or many other standard references was written specifically with undergraduates in mind. You should also check out his book on commutative algebra; I found both very enlightening.

Qiaochu Yuan
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Everyone is linking to that MO question, so allow me to link to another MO question which is more relevant to this specific request.

Also, I should mention that it partially depends on what you are looking for. Should this be for self study with little help from professors? Then your best bet would be to follow Ravi Vakil's course next semester or use his old notes here.

If you intend on having a professor over see the self study then I think that Ueno's books: Alg Geom 1 and 2, which I think are great for self study.(I should mention that his lower level book is not nearly as good and I would NOT recommend it.) Another strong candidate for led self study would be Harris' nice intro.

If you are going to offer a course or take a course then I think everybody's buddy would be nice augmented by this masterpiece would be a sweet course, but likely too challenging for undergrad.

I think that covers the basics. :D

EDIT: Two more comments, this book is frustrating and I do not think it is appropriate to look at for undergrads trying to learn algebraic geometry, especially if they are moving to modern scheme theory. And another book with a very different feeling, but a really fun way to start thinking about algebraic geometry is Katz' Enumerative Geometry book. It was written especially for undergrads, and covers some really neat material that can give some good motivation to think about algebraic geometry and topology.

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BBischof
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    I absolutely do not agree that Griffiths and Harris is frustrating. In my opinion it is actually one of the best algebraic geometry texts around, and by far one of the most geometric. I also think that having a firm grasp of the classical setting before moving to scheme theory will help a lot. – Andrea Ferretti Aug 06 '10 at 21:10
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    You should title your links because I can't tell what they are just by hovering over them. – Qiaochu Yuan Aug 06 '10 at 21:22
  • @Andrea: I tried Griffiths and Harris, but I couldn't understand it unless I had a stronger (as in, nontrivial) background in several complex variables and complex manifolds (though that may be the case soon). By contrast, Hartshorne and EGA only presuppose commutative algebra. (Hartshorne has a chapter with the classical setting in an algebraic fashion.) – Akhil Mathew Aug 07 '10 at 00:56
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    It depends on your background. In Italy it is much more common to have a basic understanding of differential geometry than commutative algebra. It seems to me that the prerequisites of GH are just some differential geometry, complex analysis in one variable and some topologu. The rest is developed in Chapter 0, including more complex variables. – Andrea Ferretti Aug 07 '10 at 01:43
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    By the way "presupposes only some commutative algebra" seems no little requirement to me. David Eisenbud has written a textbook explicitly designed to give the background to study Hartshorne, and the result is that Eisenbud's book is thicker than Hartshorne's. You will not get through Hartshorne with Atiyah-MacDonald. – Andrea Ferretti Aug 07 '10 at 01:45
  • Yeah, Eisenbud is great. But I made the mistake of taking too literally the phrase "self-contained" in Griffiths and Harris. The material is contained in Chapter 0, but it's very sketchy (the same way Hartshorne does Kahler differentials, I suppose). – Akhil Mathew Aug 07 '10 at 02:48
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    I would like to remark that studying Hartshorne won't really help one understand Siu's proof of the birational invariance of plurigenera (a major fairly recent result in AG), whereas studying Griffiths and Harris will. This is not intended as a comment on the relative merits of the two books (which have very different viewpoints), but just to point out that one shouldn't study scheme theory simply because it is "more modern"; which book to choose is very much a matter of taste and background. – Matt E Aug 07 '10 at 05:51
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    Incidentally, would it be silly to recommend EGA? I know a lot of people recommended against EGA on MO. But for self-study, Hartshorne is really dense (at least for me), while EGA is much more detailed complete, and probably has fewer prerequisites (except for some familiarity with sheaves, I suppose). Also, I find EGA mesmerizing (to use Ravi Vakil's word), and for an undergraduate, there's really no hurry to get research done (or so I would hope for my own sake). – Akhil Mathew Aug 07 '10 at 13:03
  • @Akhil have you looked at Eisenbud and Harris? I started it recently, and I was wondering if you got more out of ega that it. I have never looked at Ega because it has been a bit intimidating. – BBischof Aug 07 '10 at 13:41
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    @BBischof: Not yet, probably will though. But I don't think that EGA should be any more intimidating than Hartshorne, which leaves many basic things as an exercise (or doesn't mention them). For instance, I'm not really sure why Hartshorne uses the valuative criteria to prove facts about separatedness and properness when they could be deduced directly and naturally (and then without noetherian hypotheses) as in EGA. At least, I thought these more general proofs would be worth an exercise. – Akhil Mathew Aug 07 '10 at 14:05
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    @Akhil Mathew: Dear Akhil, The basic problem with EGA is that it does not illustrate what the subject is about; everything happens very slowly, at a high technical level, and in a vacuum. One should note that even Grothendieck had Serre to keep him in communication with geometric reality. (See the Grothendieck--Serre correspondence.) If an undergrad is in the position of having enough technical facility to read EGA, then I would certainly not discourage them from doing so, but would very strongly encourage them to also read other, more basic and more geometrical treatments ... – Matt E Oct 04 '10 at 04:34
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    ... of the subject, such as Miles Reid's undergraduate book, Serre's FAC paper, Zariski's report on FAC in the Bulletin, other papers of Zariski (he invented many of the modern foundational concepts, such as regular local rings corresponding to smooth points, and wrote short nice papers about them which can be found in his collected works), Mumford's Red Book, Mumford's Complex Projective Varieties book, Mumford--Oda (available on Chai's web-page at UPenn). One could also look at Weil'f foundations, or Lang's precis thereof in his algebraic geometry text, which give some insight of a ... – Matt E Oct 04 '10 at 04:37
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    ... different nature. In short, when you have a lot of time to spend, as an undergraduate does, you have a chance to really gain some experience and insight into the subject, and a good way to do this is to read many different treatments, to derive the maximum possible insight into the subject. – Matt E Oct 04 '10 at 04:39
  • Dear @Matt E, Thank you for the detailed comments and references! I will bear them in mind when I look further at the subject. (Incidentally, if you posted as an answer, I would certainly upvote it.) – Akhil Mathew Oct 05 '10 at 01:13
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karen smith wrote a little book an invitation to algebraic geometry that has lots of pictures and is readable by an undergraduate.

yoyo
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Hulek's «Elementary algebraic geometry» is a really nice text for undergraduates. (It's part of the AMS's student mathematical library.)

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Bix - Conics and Cubics is probably the most accessible. I think a high school student who did the BC calculus syllabus could read this and get a lot out of it. But I'd agree with the others about Cox, Little, O'Shea - Ideals Varieties and Algorithms being the 'best' undergraduate text.

WWright
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My favorite book at undegraduate level is Harris' Algebraic Geometry.

For something more advanced, with a strongly geometric point of view, I advise Griffiths and Harris Principles of algebraic geometry. It is far more demanding, though.

Andrea Ferretti
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  • Whether Harris' Algebraic Geometry: A First Course is undergraduate depends very much on the country (and student). I've seen some recommend it for a second course, despite the title. – J W Oct 17 '22 at 19:52
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There also is a quite readble old introductory book by Jenner Rudiments of algebraic geometry.

physmath
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If you are also interested in the computational aspects, the book

"Rational Algebraic Curves, A Computer Algebra Approach" by Sendra, J. Rafael, Winkler, Franz, Pérez-Diaz, Sonia

and the new book "Computational Commutative Algebra and Algebraic Geometry Course and exercises with solutions" by I. Yengui

might interest you.

Adam Car
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