Reference for Algebraic Geometry

Question

I tried to learn Algbraic Geometry through some texts, but by Commutative Algebra, I left the subject; many books give definitions and theorems in Commutative algebra, but do not explain why it is needed.

Can one suggest a good reference to learn this subject geometrically, which would also give ways to translate geometric ideas in algebraic manner, possibly through examples?

Particularly, I am interested in differentials on algebraic curves, Riemann-Roch theorem, various definitions of genus and their equivalences, and mainly groups related to complex algebraic curves such as group of automorphisms, monodromy group etc.

Answer 1

I think the best books for what you are looking for are the following:

• Kirwan - "Complex Algebraic Curves".

This is an introduction to algebraic geometry through Riemann surfaces/complex algebraic curves which does the kind of thing you want: it proves the equivalence of the different notions of genus, proves Riemann-Roch and explains differentials.

• Beltrametti et al. - "Lectures on Curves, Surfaces and Projective Varieties"

This is a new book translated and expanded from the Italian original which is highly overlooked. It develops CLASSICAL algebraic geometry using a minimal amount of commutative algebra (you barely need some notions of abstract algebra). It is developed in the most similar way to how the subject was done before the Grothendieck revolution which seems to have permeated every modern title. There you can find old definitions of genus as an "excess", a nonstandard (noncohomological) treatment of Riemann-Roch for curves, classical important examples and constructions for curves, surfaces and morphisms, and even the development of multiplicities and Bézout's theorem using old good elimination theory and proving its projective invariance arriving at the formula most modern books use to define multiplicity (and thus forgetting all classical motivation whatsoever, the dimension of a particular vector space).

• Perrin - "Algebraic Geometry"

Along with the previous two books, this wonderful title may help you do the transition to modern abstract algebraic geometry like no other. I think it is a great companion and complement to Beltrametti's for learning the basics of the modern style along the way. It has good examples and lots of (doable) exercises (like some very interesting and easy motivations for cohomology to solve conceptually geometric problems). This is a more algebraic treatment but which develops almost all needed concepts along the way and it is one of the best introductions to the language of sheaf theory in algebraic geometry without the need of schemes (but even defines finite schemes to define multiplicities, letting you understand the relationship of how Beltrametti does things and why jumping into schemes is a good idea in the end). So eventually, you end up understanding degrees, genus, Bézout and Riemann-Roch among other things through exact sequences.

• Harris - "Algebraic Geometry, A First Course"

Actually I think one cannot learn much from this book alone if it is not supplemented by standard theoretical titles like the ones above. Nevertheless it is a great source for classical examples and results which does a good job at complementing the other books as a companion. However I do not find it more useful than that as many other people do.

• Hartshorne - "Algebraic Geometry"

In my opinion one should try to learn Hartshorne's book (not to mention Liu's which is heavily less geometrically motivated) ONLY AFTER one has understood and worked through the kind of classical algebraic geometry done in Beltrametti's book and related that geometry with an algebraic foundation like in Perrin's. Hartshorne's is the best book around IF one works through almost all the exercises, but that is no easy task. He does not motivate from a classical perspective most of the constructions, definitions and results so one must be able to do so by knowing beforehand a small deal of classical geometry. Most modern standard books work through schemes and there you are leaving the world of classical geometry and entering commutative algebra with a geometric interpretation, which is what today is called "algebraic geometry". It is nevertheless a fascinating subject.

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Along the way you will need to learn enough commutative algebra to understand modern algebraic geometry. For this I would recommend the pre-published version of the recent book by Kemper - "A Course in Commutative Algebra" since its originally freely downloadable draft version included lots of exercises and problems with all their solutions (the published version does not). A great book indeed for self-study if you can get it. For a much more complete reference dealing with all the material needed for Hartshorne, Eisenbud's book is the great, but lengthy and wordy, option. A nice new succinct textbook is Singh - "Basic Commutative Algebra", to use as a purely formal reference (but with doable exercises). A good geometry-motivated introduction for those titles is Reid - "Undergraduate Commutative Algebra".

It seems to me most people have forgotten what "classical" algebraic geometry really is for which you really do not need much commutative algebra. There are two kinds of people when approaching algebraic geometry: those fascinated by the algebraic/categorical modern approach, the algebrists, and those fascinated by the original very geometrical concepts and problems, the goemeters. A typical easy question for the old geometer school is "how many surfaces of degree $d$ contain a given spatial projective curve"; the algebrists do not bother about the visual geometric interpretation and just wonder "what is the dimension of the space of global sections of the degree $d$ ideal sheaf of an irreducible algebraic set of dimension 1" (which they solve trivially through an exact sequence of sheaves). The problem to me is that many modern books (and students) are completely blinded by the algebraic language/setting, and forget what geometry is about. The perfect student of algebraic geometry should care as much for, say, derived functors as for the old geometric school that started it all (above all the geometric problems that they formulated for curves, surfaces and varieties, many of which are still unsolved whereas too many students get lost in the world of schemes for too long without knowing any geometry at all).

Answer 2

Fulton's Algebraic Curves: An Introduction to Algebraic Geometry which is freely available seems to fit your description. I haven't started reading it yet, but I'm planning to do so shortly: I'm in a situation similar to yours.

Answer 3

It was mentioned by Mariano in a comment, but it deserves to be in its own answer:

Miles Reid's book Undergraduate algebraic geometry is a really good introduction to the subject. (Don't get put off by the title: I routinely recommend this book to beginning graduate students.)

At a more advanced level, a text which goes quite some way in algebraic geometry, but uses no commutative algebra to speak of, is Griffiths and Harris. (The prerequisites are familiarity with algebraic and differential topology at the beginning graduate level.)

Answer 4

Let me reproduce here a response of mine to a very similar question asked on mathoverflow:

There is an excellent book on algebraic geometry entitled Algebraic Geometry: A First Course by Joe Harris. This book, however, emphasizes the classical roots of the subject but if you have not yet seen too much of algebraic geometry, it is worthwhile getting this book and reading a few lectures. (The book is split into "lectures" rather than "chapters".) There are many beautiful constructions in classical algebraic geometry that can be understood without too much background (and which lay the foundations for some aspects of modern algebraic geometry) and this can perhaps give you a rough indication of the geometric intuitions in algebraic geometry. And in my opinion, the book does an excellent job of conveying the beauty and elegance of algebraic geometry.

The prerequisites for reading this book (according to Harris) are: linear algebra, multilinear algebra and modern algebra. However, since this is a "Graduate Texts in Mathematics" book, there are some places where it is very helpful (but not essential to the point that you cannot read the book otherwise) to have a basic knowledge of commutative algebra, complex analysis and point-set topology. (E.g., basic facts about topological spaces, local rings, basic constructions in commutative algebra, holomorphic functions etc.) Atiyah and Macdonald's An Introduction to Commutative Algebra should furnish more than enough preparation. (You can also concurrently read commutative algebra if that is your preference.)

Since you are a beginning student, you should not worry too much about learning "background material" just yet before at least seeing what classical algebraic geometry is about. If at some point you decide to specialize in the subject, you will need to learn the "modern tools" such as, for example, schemes, sheaves and sheaf cohomology. The "classic book" for this is Robin Hartshorne's Algebraic Geometry but since that does require a solid background in commutative algebra (or at least the mathematical maturity to accept facts without proofs), you might want to try other books. (But this is, I hasten to add, an excellent book if you do have the background to understand it.)

As Bcnrd (on MathOverflow) recommended to me, Qing Liu's Algebraic Geometry and Arithmetic Curves seems to be an excellent book on the subject. Most of the background material in commutative algebra is developed from scratch, and the first six chapters furnish a good introduction to the "modern tools". The last three chapters focus more on the arithmetic side of algebraic geometry, but you can always omit that if you so desire. (But if you are interested in number theory, definitely take a look at that!)

Succinctly, I recommend: Take a look at Atiyah and Macdonald and at least read the first few chapters. (The book is roughly 120 pages so covering the first few chapters is not too hard. Though be warned: Some people say that Atiyah and Macdonald is "dense", but I personally found it a very readable book and I think the majority find that so as well.) Then you should have the right background to read Harris and I hope that that will show you how fascinating the subject of algebraic geometry is. Good luck!

Answer 5

Let me also recommend Daniel Bump's Algebraic Geometry. This is an excellent textbook on the subject and it seems to fit some of your requirements. For example:

(1) The textbook discuss many notions in commutative algebra with reference to geometry and culminates in a discussion of the theory of curves in the final six chapters.

(2) The Riemann-Roch theorem and differentials on algebraic curves are discussed.

(3) The prerequisites for the textbook are fairly minimal; I believe these include "a level of algebra that one expects from a first or second year graduate student" (in the words of the author). In particular, results from commutative algebra such as Noether normalization, Hilbert's nullstellensatz, valuation rings etc. are proven in the text. However, Galois theory is assumed.

I think that this is exactly the sort of book that might be good for you based on what you have written. Although it does discuss commutative algebra, there is a flavor of geometry pervasive throughout the entire text.

Answer 6

I recommend reading Liu's book and Hartshorne's book at the same time. This got me started. There are many other books which you'll be able to digest much easier after reading these books. Of course, you shouldn't read everything in these books at first. You'll have to figure out what parts are important to what you're trying to learn.

If the exercises in Hartshorne bother you, you could look up solutions available on the web. [Edit: See the below discussion for why this is not such good advice.]

Edixhoven once told me that doing algebraic geometry should consist of a healthy combination of category theory, commutative algebra and geometry. Following his philosophy, you should try to develop a bit of feeling for these subjects.

Concerning the things you mention in your question, they are explained very well in Liu's book. I believe Chapter 7 is what you need.

If you find these books too difficult at first, try reading the syllabus of the following course by B. Edixhoven and L. Taelman available here

http://www.math.leidenuniv.nl/~edix/teaching/2010-2011/AG-mastermath/index.html

The course aims at proving the Riemann hypothesis for curves over finite fields. The proof uses Riemann-Roch, Serre duality and intersection theory on surfaces. The first six lectures are meant as motivation and lectures 7-14 prepare and give the proof of Weil's theorem.