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1. Algebraic Geometry (Graduate Texts
2. Elementary Algebraic Geometry
3. Basic Algebraic Geometry 1: Varieties
4. Lie Algebras and Algebraic Groups
5. Introduction to Algebraic Geometry
6. Using Algebraic Geometry (Graduate
7. Algebraic Geometry and Arithmetic
8. Fourier-Mukai Transforms in Algebraic
9. Computational Methods in Commutative
10. Hodge Theory and Complex Algebraic
11. An Invitation to Algebraic Geometry
12. Real Algebraic Geometry (Ergebnisse
13. Positivity in Algebraic Geometry
14. Complex Analysis and Algebraic
15. Rudiments of algebraic geometry
16. Algebraic and Analytic Geometry
17. Principles of Algebraic Geometry
18. The Geometry of Syzygies: A Second
19. Algebraic Geometry: A First Course
20. Basic Algebraic Geometry 2: Schemes

1. Algebraic Geometry (Graduate Texts in Mathematics)
by Robin Hartshorne
Paperback: 516 Pages (2010-11-02)
list price: US$74.95 -- used & new: US$60.14
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Asin: 1441928073
Average Customer Review: 4.0 out of 5 stars
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An introduction to abstract algebraic geometry, with the only prerequisites being results from commutative algebra, which are stated as needed, and some elementary topology. More than 400 exercises distributed throughout the book offer specific examples as well as more specialised topics not treated in the main text, while three appendices present brief accounts of some areas of current research. This book can thus be used as textbook for an introductory course in algebraic geometry following a basic graduate course in algebra.
Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. He is the author of "Residues and Duality", "Foundations of Projective Geometry", "Ample Subvarieties of Algebraic Varieties", and numerous research titles. ... Read more

Customer Reviews (8)

5-0 out of 5 stars this is a wonderful book by a master
Robin Hartshorne is a master of Grothendieck's general machinery for generalizing the tools of classical algebraic geometry to apply to families of varieties, and more broadly to number theory.A fundamental difficulty is to grapple with algebro geometric objects such as doubled lines, or surfaces with embedded curves and points in them,that arise as "limits" of simpler varieties.Here the algebra is essential as the naive set of points does not reveal the antecedents of the limiting object.Even more in number theory, when the rings of coefficients used may not admit solutions, the structure of the rings themselves is all you have to go on.For the most basic invariants, when we leave the complex numbers and Riemann's topological and integration techniques are not available, sheaf cohomology is the abstract substitute.

These esoteric developments did not arise spontaneously, but out of classical problems that should be approached first in order to motivate and appreciate the power of the tools in chapters 2,3 of this book.Professor Hartshorne says himself that he taught the chapters out of order when he first was writing the book.The average reader should probably read the chapters in the order he taught them in, not the order they appear in this book.Thus first read chapters 4 and 5 on curves and surfaces, or possibly read 1,4,5, to get first a general introduction, then study curves and surfaces.Only then delve into chapters 2 and 3 for the sophisticated stuff.

If you really want to start with the classical roots, begin instead with Rick Miranda's book on Algebraic curves and Riemann surfaces.Of course there are hardy souls who can wade right through Hartshorne's book in order, but for many that is a prescription for losing heart and losing interest in the subject.When all is said and done, there are very valuable ideas and tools in this book that are not available as easily anywhere else.You just have to learn how to get at them.You might want to read in whatever order appeals to you.But do not feel obligated to just plow from page 1 on.Or try the first volume of Shafarevich and then this, or bounce back and forth as the spirit moves you.Kempf also has a book on Algebraic varieties with sheaf cohomology but not schemes, which may ease the abstraction level, and there is also Serre's original paper FAC in that vein.

4-0 out of 5 stars A Necessary and Useful Pain
Algebraic Geometry is the first textbook on scheme-theoretic algebraic geometry. Scheme theory was created in the 1960's by Alexander Grothendieck. Grothendieck also co-authored an extremely well-written, 1800-page reference manuscript on scheme theory called "Éléments de Géométrie Algébrique" (EGA). However, EGA is unsuitable as a textbook because it had no examples or motivation and proved every theorem in great detail and maximal generality.

Algebraic Geometry has 5 chapters. The first chapter summarizes algebraic geometry before schemes. The next two chapters compress EGA to 230 pages(!). The last two chapters show how well scheme theory can solve classical problems from algebraic geometry.

That should be a hint that Algebraic Geometry is one of the most dense and difficult math textbooks ever written.

To achieve that kind of compression, Hartshorne's writing is extremely terse. He assumes a solid understanding of commutative algebra and point-set topology. He often gives one or two-sentence proofs and explanations that, when fleshed out and made complete, would need both many pages and new techniques that are never mentioned in the text. He also gives almost no motivation throughout Chapters II and III, because Chapters IV and V fill this role. When he does give motivation, it is usually relegated to the exercises, many of which, again, require techniques that are never mentioned in the text. Finally, he assigns the proofs of many essential and extremely difficult theorems as exercises.

There are other, much more user-friendly introductions to scheme theory than Algebraic Geometry---For example, The Red Book of Varieties and Schemes, The Geometry of Schemes, and Algebraic Geometry and Arithmetic Curves. These books, along with EGA, can also serve as complements to Algebraic Geometry when Hartshorne's writing becomes too dense to learn from.

However, Algebraic Geometry is unique in that no other textbook on scheme theory covers nearly as much material as it does. Also, for all of its density, Algebraic Geometry is very well-written and an excellent reference, especially considering how much it covers and the length and complexity of its source material. Because of this, I cannot foresee any significantly better replacement for it being written in the near future. Algebraic Geometry will probably continue to be a necessary and useful pain to learners of scheme theory, just as it has been for the past 30 years.

2-0 out of 5 stars Unfortunately a better book on the subject doesn't exist.
The motivation is nonexistent, and the examples are trivial. If you want to learn anything you have to trudge through exercises which require techniques that are not addressed in the text. I don't mind working to learn a subject, but spending two hours trying to understand what a question is asking is a bit much.

The best, and most concise review I have ever heard was, "Hartshorne is the worst book on Algebraic Geometry, except for all the others".

5-0 out of 5 stars Nice selection of exercises
Here's my impression after doing the first 30 pages: What makes this a really good book is the exercises. Not too hard, always interesting. If you are new to the subject you need to look up results from commutative algebra somewhere else. It can be a little strange getting used to working with the Zariski topology. All open sets are dense, so you don't have the notion of a small neighborhood of a point. For instance any bijection between two curves is a homeomorphism.

5-0 out of 5 stars THE book for the Grothendieck approach
This is THE book to use if you're interested in learning algebraic geometry via the language of schemes.Certainly, this is a difficult book; even more so because many important results are left as exercises.But reading through this book and completing all the exercises will give you most of the background you need to get into the cutting edge of AG.This is exactly how my advisor prepares his students, and how his advisor prepared him, and it seems to work.

Some helpful suggestions from my experience with this book:
1) if you want more concrete examples of schemes, take a look at Eisenbud and Harris, The Geometry of Schemes;
2) if you prefer a more analytic approach (via Riemann surfaces), Griffiths and Harris is worth checking out, though it lacks exercises. ... Read more

2. Elementary Algebraic Geometry (Student Mathematical Library, Vol. 20) (Student Mathematical Library, V. 20)
by Klaus Hulek
Paperback: 213 Pages (2003-02-13)
list price: US$37.00 -- used & new: US$33.00
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Asin: 0821829521
Average Customer Review: 3.0 out of 5 stars
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This is a genuine introduction to algebraic geometry. The author makes no assumption that readers know more than can be expected of a good undergraduate. He introduces fundamental concepts in a way that enables students to move on to a more advanced book or course that relies more heavily on commutative algebra.

The language is purposefully kept on an elementary level, avoiding sheaf theory and cohomology theory. The introduction of new algebraic concepts is always motivated by a discussion of the corresponding geometric ideas. The main point of the book is to illustrate the interplay between abstract theory and specific examples. The book contains numerous problems that illustrate the general theory.

The text is suitable for advanced undergraduates and beginning graduate students. It contains sufficient material for a one-semester course. The reader should be familiar with the basic concepts of modern algebra. A course in one complex variable would be helpful, but is not necessary. It is also an excellent text for those working in neighboring fields (algebraic topology, algebra, Lie groups, etc.) who need to know the basics of algebraic geometry. ... Read more

Customer Reviews (1)

3-0 out of 5 stars Trees, not forest
First, a calibration: I am a total neophyte to algebraic geometry, and haven't taken a university algebra course since a few decades ago when I was a physics major. This book is one of several on the subject (along with some books on commutative algebra) that I'm using to get an amateur's orientation.

As so often happens, this book looked great in the bookstore. It is thin, reasonably well-illustrated compared to other books in the field, and even helps you gets your toes wet in sheaves, category theory and some other neat topics.

That said, I believe the prerequisites in the preface (university algebra, with a complex variables course optional) are understated; e.g. it helps to know something about fibres, lifts and other topics from geometry. It might be relevant that these notes were prepared at a German university; you should consider that "undergraduates" there are heading toward the equivalent of a US M.S. degree, not B.S./B.A.

More detrimental is that the presentation slogs from one proof to another and too rarely pauses for breath to consider the "big picture" significance of what you're proving. Notwithstanding that Joe Harris's "Algebraic Geometry: A First Course" is even less of a piece of cake for me than it might be for you, his style is a breath of fresh air when it comes to enlightening you as to some geometric context and payoff for all this effort. Other supplements I found helpful include Reid and Schenck.

PS in 2008: I very belatedly found the terrific "An Invitation to Algebraic Geometry," by Karen E. Smith &al. (Springer 2000, corrected printing 2004). This is the hands-down best introduction to the subject, IMHO. ... Read more

3. Basic Algebraic Geometry 1: Varieties in Projective Space
by Igor R. Shafarevich
Paperback: 304 Pages (1994-08-08)
list price: US$84.95 -- used & new: US$64.79
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Asin: 3540548122
Average Customer Review: 5.0 out of 5 stars
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This book is a revised and expanded new edition of the first four chapters of Shafarevich’s well-known introductory book on algebraic geometry. Besides correcting misprints and inaccuracies, the author has added plenty of new material, mostly concrete geometrical material such as Grassmannian varieties, plane cubic curves, the cubic surface, degenerations of quadrics and elliptic curves, the Bertini theorems, and normal surface singularities.

... Read more

Customer Reviews (3)

5-0 out of 5 stars the most geometric, user friendly book on algebraic geometry
In the 1950's algebraic geometry was tedious and hard to grasp because it was mostly commutative algebra, developed by Zariski and Weil andtheir schools to fill logical gaps in the Italian arguments of the previous half century.The rich geometric texture of the italian school was lost.In the 1960's Serre and Grothendieck introduced homological algebra to the subject and greatly expanded and enhanced it to embrace also arithmetic, but the abstraction level went WAY up, so again it was hard to grasp and relate to geometry.Hartshorne is a member of both Zariski and Grothendieck's schools and appreciates down to earth objects like space curves, but his book has a long beginning section on schemes and cohomology that can definitely throw a beginner off the horse.

Pardon the delay in getting here, but the point is that Shafarevich's book has none of the tediousness of the previous generation, yet benefits from the rigorous foundations via commutative algebra of Zariski's works.I would say Shafarevich's book, is a geometrically oriented explanation of the material that can be explained using Zariski's methods.I.e., it has a rich geometric feel, is very well explained, includes many easy examples, and is rigorous in its use of commutative algebra.This book allowed many of us who were stymied by the huge amount of algebra needed for 1960's Grothendieck style AG, to finally gain admission to the subject.I love this book.

There are three themes one can mention in algebraic geometry, 1) projective varieties, 2) schemes 3) cohomology.The first topic concerns the objects most geometers are interested in.The second one is of more interest to number theorists, but also has value for geometers in understanding limits of varieties.The third is a powerful and abstract technical tool which is valuable to both schools.

So here is the difference between Shafarevich and Hartshorne:Shafarevich focuses mainly on varieties, as the title reveals, has nothing at all in volume I on schemes, and neither volume treats cohomology.Hartshorne quickly reviews varieties then goes straight to schemes and cohomology, presenting schemes before cohomology.Thus Shafarevich is much more elementary than Hartshorne, and is a good introduction to that book.For a book treating cohomology before schemes, try George Kempf's algebraic varieties, but hartshorne will eventually be essential.

A few negative aspects to Shafarevich:it has a large number of typos and outright errors, in spite of numerous editions and reprints.One of the worst mathematical mistakes is the false assertion in the section I.6.3 of the first edition, that for any regular map f:X-->Y the set of points of Y over which the fibers of f have dimension at least r, is closed in Y. One needsthe map f to be closed, e.g. proper, for this to hold, and unfortunately this false result is used in the treatment of the tangent bundle of a variety.The proof of thenormalization for curves seems also flawed to me.All these mistakes can be fixed, and even in the present flawed form, this book still has no peer in readability and geometric richness with the same scope (although the undergraduate book by Miles Reid, Shafarevich's translator, is excellent for what it covers).I recommend it highly.

I would give it 4 stars for the errors, to leave room for a higher score for a book not only well conceived and well written but also well edited, but there is no other book that approaches this one in its strengths so how can one give it only 4 stars?But how can this book go through so many printings and not repair the most egregious mistakes?(There are other famous books out there with more and worse errors, but we who do not write books should not be too harsh in this regard.What if no one had the courage to attempt to explain this difficult stuff to others?)

5-0 out of 5 stars Someone hasn't read the first page of the index!!!!!!
I have been a student of AG for the past six years and I have come to the conclusion that Shafarevich is a great place to start.Having said this, one must have the necessary background in algebra and topology.I disagree with the other reviewer about doing this after Hartshorne--start here then do Hartshorne!!!Oh ya, the index refers to both volumes 1 and 2; read the first page of the index!!!

5-0 out of 5 stars After Hartshorne!!!
This book is very good for the secondary course after learning with Harshorne's Algebraic geometry. ... Read more

4. Lie Algebras and Algebraic Groups (Springer Monographs in Mathematics)
by Patrice Tauvel, Rupert W. T. Yu
Paperback: 653 Pages (2010-11-30)
list price: US$115.00 -- used & new: US$115.00
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Asin: 3642063330
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Devoted to the theory of Lie algebras and algebraic groups, this book includes a large amount of commutative algebra and algebraic geometry so as to make it as self-contained as possible. The aim of the book is to assemble in a single volume the algebraic aspects of the theory, so as to present the foundations of the theory in characteristic zero. Detailed proofs are included, and some recent results are discussed in the final chapters.

... Read more

5. Introduction to Algebraic Geometry
by Brendan Hassett
Paperback: 264 Pages (2007-05-21)
list price: US$38.99 -- used & new: US$24.56
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Asin: 0521691419
Average Customer Review: 3.5 out of 5 stars
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Algebraic geometry, central to pure mathematics, has important applications in such fields as engineering, computer science, statistics and computational biology, which exploit the computational algorithms that the theory provides. Users get the full benefit, however, when they know something of the underlying theory, as well as basic procedures and facts. This book is a systematic introduction to the central concepts of algebraic geometry most useful for computation. Written for advanced undergraduate and graduate students in mathematics and researchers in application areas, it focuses on specific examples and restricts development of formalism to what is needed to address these examples. In particular, it introduces the notion of Gröbner bases early on and develops algorithms for almost everything covered. It is based on courses given over the past five years in a large interdisciplinary programme in computational algebraic geometry at Rice University, spanning mathematics, computer science, biomathematics and bioinformatics. ... Read more

Customer Reviews (3)

4-0 out of 5 stars Nice
There are many introductions to algebraic geometry focusing on similar topics to this one (varieties affine and projective). However this one is especially friendly to newcomers and is the only such book that I think is accessible to an average undergraduate (junior or senior year). It is very well written with excellent examples. If you are interested in algebraic geometry you will definately need to proceed beyond this book, but it is a very nice first step.

2-0 out of 5 stars Wait for 2nd Edition
Hassett tries to give a more concrete approach to classical algebraic geometry by teaching Grobner Basis early and focus on computational algorithms every step of the way. Although this pedagogical idea is good, the book is not. It is rushed and published way too early; it feels very much like lecture notes. It is not surprising then that the book is plagued by many serious problems.

The Good:
1. Introducing Grobner Basis technique early is a good idea. GB provides a powerful tool for experimenting with many of the later concepts in the book.

2. The book offers ample examples. All difficult concepts are immediately followed by a relatively detailed example.

3. There are many exercises, varying in difficulty. Many of the exercises are computational and I highly recommend that the reader use a computer commutative algebra program to help solve them.

The Bad:

1. Mistakes
Hassett's book is riddled with false statements. Some are blatant and minor, some are subtle and occur in statement of major theorems. Luckily, a careful reader with effort can catch all of the mistakes so the book is still read-able.

2. Leaps in Logic
Hassett's proofs and exposition often lack detail. He is especially hand-wavy on the algebra aspect; many k-algebra isomorphisms are simply asserted and not justified in any degree.

3. Lacking in Geometric Intuition
This is specifically referring to chapter 3 and 4. Hassett presents the definition of morphism and rational and related theorems without giving the reader any idea how they relate to the geometry of polynomial maps.

4. Lack of Graphics
This relates to the lack in geometric intuition. The amount of visual aid in this book is very sparse; what little graphics present are almost all useless.

Overall, I felt that this book did a poor job of teaching a mathematics student how to rigorously think about algebraic geometry. Hassett's most grave problem is that it glosses over steps of critical proofs early on and leaves reader confused about how to actually go about proving even rudimentary propositions on their own.

To use an analogy: an introductory algebra student might not know how to prove Lagrange's theorem on their own, but when presented when a possible proof, they should at least be able to tell whether the proof is rigorous and correct. For me, Hassett's text fails in this respect; I have trouble after reading the book to tell whether my proofs are rigorously enough or not. (this is not due to my own lack of mathematical maturity; I have taken a year of algebra)

This book has great potential to be a classic in algebraic geometry but as of now, it falls far far short. I would recommend that readers Wait for the second edition of Hassett's book and use the introductory algebraic geometry text by Joe Harris in the mean time.

4-0 out of 5 stars The bookat least cracks the door open?
This book seems like a breath of fresh air in a very stale
area of mathematics. He provides an opening that my other book
on the area didn't:Algebraic Geometry.
If it wasn't that he uses terms like 'Ideal' without any real
definition, I would have given the book five stars.
I shopped really hard before buying this book
and read the index and table of contents online.
For once I'm not disappointed. I have four pages turned down from my first read through and it may take some more time and study to
get more out of it, but it does seem to be an honest teaching effort
in print. I say that it well done. ... Read more

6. Using Algebraic Geometry (Graduate Texts in Mathematics)
by David A. Cox, John Little, Donal O'Shea
Paperback: 558 Pages (2005-03-17)
list price: US$54.95 -- used & new: US$38.81
(price subject to change: see help)
Asin: 0387207333
Average Customer Review: 3.0 out of 5 stars
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In recent years, the discovery of new algorithms for dealing with polynomial equations, coupled with their implementation on fast inexpensive computers, has sparked a minor revolution in the study and practice of algebraic geometry. These algorithmic methods have also given rise to some exciting new applications of algebraic geometry. This book illustrates the many uses of algebraic geometry, highlighting some of the more recent applications of Gr ... Read more

Customer Reviews (2)

4-0 out of 5 stars Good introduction
Once thought to be high-brow estoeric mathematics, algebraic geometry is now finding applications in a myriad of different areas, such as cryptography, coding algorithms, and computer graphics. This book gives an overview of some of the techniques involved when applying algebraic geometry. The authors gear the discussion to those who are attempting to write computer code to solve polynomial equations and thus the first few chapters cover the algebraic structure of ideals in polynomial rings and Grobner basis algorithms. The reader is expected to have a fairly good background in undergraduate algebra in order to read this book, but the authors do give an introduction to algebra in the first chapter. Many exercises permeate the text, some of which are quite useful in testing the reader's understanding. The Maple symbolic programming language is used to illustrate the main algorithms, and I think effectively so. The authors do mention other packages such as Axiom, Mathematica, Macauley, and REDUCE to do the calculations. The chapter on local rings is the most well-written in the book, as the idea of a local ring is made very concrete in their discussion and in the examples. The strategy of studying properties of a variety via the study of functions on the variety is illustrated nicely with an example of a circle of radius one. Later, in a chapter on free resolutions, the authors discuss the Hilbert function and give a very instructive example of its calculation, that of a twisted cubic in three-dimensional space. They mention the conjecture on graded resolutions of ideals of canonical curves and refer the reader to the literature for more information. Particularly interesting is the chapter on polytopes, where toric varieties are introduced. The authors motivate nicely how some of the more abstract constructions in this subject, such as the Chow ring and the Veronese map, arise. The important subject of homotopy continuation methods is discussed, and this is helpful since these methods have taken on major applications in recent years. In optimization theory, they serve as a kind of generalization of the gradient methods, but do not have the convergence to local minima problems so characteristic of these methods. In addition, one can use homotopy continuation methods to get a computational handle on the Schubert calculus, namely, the problem of finding explicity the number of m-planes that meet a set of linear subspaces in general position. There are some software packages developed in the academic environment that deal with homotopy continuation, such as "Continuum", which is a projective approach based on Bezout's theorem; and "PHC", which is based on Bernstein's theorem, the latter of which the authors treat in detail in the book. My primary reason for purchasing the book was mainly the last chapter on algebraic coding theory. The authors do give an effective presentation of the concepts, including error-correcting codes, but I was disappointed in not finding a treatment of the soft-decision problem in Reed-Solomon codes.

In general this is a good book and worth reading, if one needs an introduction to the areas covered. Students could definitely benefit from its perusal.

2-0 out of 5 stars Don't bother
I just completed a course that used this book as a...reference.Granted, it is a first edition, but it reads like a rough draft.The presence of three authors is all too obvious in the inconsistent writing of proofs, paragraphs, and even exercises.Some proofs are just plain wrong, and manyhave gaping holes in them.Notation is confusing, and changes withoutwarning or explanation. I will say this much in its favor: many importantresults are presented, although the proofs are absent.It makes a goodsource for named theorems, but that's about it. ... Read more

7. Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics, 6)
by Qing Liu
Hardcover: 592 Pages (2002-07-18)
list price: US$134.99 -- used & new: US$94.49
(price subject to change: see help)
Asin: 0198502842
Average Customer Review: 5.0 out of 5 stars
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This book is a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularization (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the fundamental theorem of stable reduction of Deligne-Mumford. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises. ... Read more

Customer Reviews (2)

5-0 out of 5 stars Algebraic Geometry and Arithmetic
This book together with Matsumura on Commutative Algebra and Hartschone on Algebraic Geometry is an excellent book to learn the subject. I am really enjoying it.

5-0 out of 5 stars Very good exposition
Liu's book has two distinct parts to it. The first 7 chapters combine to give a wonderful exposition of the language of schemes; the other chapters are of a specialised nature and concentrate on arithmetic curves. I will talk about the former. (So when I say "this book", I am only referring to the first 7 chapters)

The book starts off with a chapter on some topics in basic commutative algebra - localisation, flatness and completion. Once this is done, the stage is set to introduce schemes in the next chapter and prove their basic properties. Chapter 3 talks about morphisms of schemes and base change. Chapter 4 continues with a discussion of morphisms and also presents some results about some special types of schemes (normal, regular). It culminates with a proof of Zariski's main theorem.The next chapter takes up sheaf cohomology and is followed up with a chapter on differential calculus on schemes (Kahler differentials, duality theory). Lastly, chapter 7 takes up divisors, proves the Riemann Roch theorem and culminates with some applications to curves.

At a first glance, this would basically look like Hartshorne - the most popular book for an introduction to schemes. However, there are few differences which I will point out. Firstly, Hartshorne emphasizes geometric applications and, as such, uses algebraically closed fields freely. Liu, on the other hand, does not hesistate to give arithmetic applications whenever possible and, therefore, tries to relax the hypotheses on the base field whenever possible. Secondly, Liu is much more readable than Hartshorne which, in its supreme elegance, is a tad dense for a first reading. Unlike Hartshorne, a majority of important results are not presented in the exercises (though many are!). Moreover, unlike Harshorne, this book develops all the necessary commutative algebra along the way (chapter 1,2 of Atiyah-Macdonald should be good enough to read this book). Coming back to the geometry, Hartshorne's chapter 4,5 form an excellent resource for classical geometric applications for theory of schemes. Moreover, chapter 1 presents a very readable and scheme-free account of classical algebraic geometry (pre-Grothendieck) in the language of varieties. Liu's book, however, does not emphasize classical or geometric applications and is not the best place to start if one wishes to learn about varieties.

In the current literature on algebraic geometry, there is a noticeable void. Namely, on one hand, we have Grothendieck's "Elements" (EGA) which present all results about schemes and sheaf cohomology in utmost generality, prove everything with excruciating detail, and are almost unreadable as texts (they're a great references). On the other hand, we have Hartshorne which is basically a beautiful summary of EGA along with geometric applications, but is quite hard to read for an introduction. The book under review is not as concise as Hartshorne's book, presents arithmetic applications and is more readable in a reasonable amount of time than EGA.

In conclusion, this book should be an invaluable resource to anyone who wishes to learn about schemes, especially with arithmetic applications in mind. For those inclined towards geometry, an account of schemes from this book coupled with applications from another book (like Hartshorne) would be a good combination. ... Read more

8. Fourier-Mukai Transforms in Algebraic Geometry (Oxford Mathematical Monographs)
by Daniel Huybrechts
Hardcover: 280 Pages (2006-06-29)
list price: US$125.00 -- used & new: US$107.85
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Asin: 0199296863
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This seminal text on Fourier-Mukai Transforms in Algebraic Geometry by a leading researcher and expositor is based on a course given at the Institut de Mathematiques de Jussieu in 2004 and 2005.Aimed at postgraduate students with a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety.Including notions from other areas, e.g. singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs are given and exercises aid the reader throughout. ... Read more

9. Computational Methods in Commutative Algebra and Algebraic Geometry (Algorithms and Computation in Mathematics)
by Wolmer Vasconcelos
Paperback: 408 Pages (2004-07-12)
list price: US$89.95 -- used & new: US$11.04
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Asin: 3540213112
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From the reviews:

... Read more

10. Hodge Theory and Complex Algebraic Geometry I: Volume 1 (Cambridge Studies in Advanced Mathematics) (v. 1)
by Claire Voisin
Paperback: 332 Pages (2008-02-04)
list price: US$46.99 -- used & new: US$37.92
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Asin: 0521718015
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This is a modern introduction to Kaehlerian geometry and Hodge structure. Coverage begins with variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory (with the latter being treated in a more theoretical way than is usual in geometry). The book culminates with the Hodge decomposition theorem. In between, the author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem.The second part of the book investigates the meaning of these results in several directions. ... Read more

11. An Invitation to Algebraic Geometry (Universitext)
by Karen E. Smith, Lauri Kahanpää, Pekka Kekäläinen, William Traves
Paperback: 184 Pages (2010-11-02)
list price: US$59.95 -- used & new: US$48.31
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Asin: 1441931953
Average Customer Review: 5.0 out of 5 stars
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This is a description of the underlying principles of algebraic geometry, some of its important developments in the twentieth century, and some of the problems that occupy its practitioners today. It is intended for the working or the aspiring mathematician who is unfamiliar with algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites. Few algebraic prerequisites are presumed beyond a basic course in linear algebra. ... Read more

Customer Reviews (4)

5-0 out of 5 stars very good
excellent book ... can make learning algebraic geometry as easy as bedside reading ... highly recommended.

5-0 out of 5 stars Splendid introduction
For people just starting on Algebraic Geometry, Robin Hartshorne's book, is very daunting--but it is the ULTIMATE book for professional and advanced readers. But for starters, Karen Smith's "An Invitation to Algebraic Geometry" is simply a SPLENDID way to start working on the basic ideas. The author has some stunning graphs and pictures to help understand material. I loved the book the minute I opened it. BUY it NOW!

5-0 out of 5 stars enjoyable guidance
i'm not a math student, but this book is very readable. it's short(150 pages) but many illustrative examples and exercises cover chief topics and facts, i assume. at first, i tried Eisenbud's "geometry of schemes" but it was too hard and Hartshorne's was somewhat alien to me. then comes this book. it helped me through the Eisenbud's, and convinced me algebraic geometry is an intriguing discipline.

5-0 out of 5 stars Wow!
This could be your only book on algebraic geometry if you just want a sound idea of what algebraic geometry can do. If you actually want to know the field, and you do not already have a lot of expert friends telling you about it, then the advanced books will go much more easily with this expert around. It is a terrific guide to the key ideas--what they mean, how they work, how they look.

The only book like this one in brevity and scope is Reid UNDERGRADUATE ALGEBRAIC GEOMETRY--with its highly informed, highly polemical, final chapter on the state of the art. Both are very good. This one is more advanced. Beyond what Reid covers, Smith sketches Hilbert polynomials, Hironaka's (and very briefly even De Jong's) approach to removing singularities, and ample line bundles. You do need a bit of topology and analysis to follow it. Smith has very many fewer concrete examples than Reid. They are beautifully chosen classics, like Veronese maps and Segre maps, so they teach a lot. And the more you know to start with, the more you will see in each.

The book does geometry over the complex numbers. It is good old conservative material, with terrific graphics of curves and surfaces. The proofs and partial proofs are very clear, intuitive and to the point. But, in fact, just because the proofs are so clear and to the point they usually work in a much broader setting. Long stretches of the book apply just as well over any field or any algebraically complete field. This generality is only mentioned a few times, in passing, but is there if you want it. Smith describes schemes very briefly, and mentions them at each point where they naturally arise. You will not know what schemes "are" at the end of this book. You will know some things they DO. She has no time for fights between "concretely complex" and "abstractly scheming" approaches--for her it is all geometry. ... Read more

12. Real Algebraic Geometry (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics)
by Jacek Bochnak, Michel Coste, Marie-Francoise Roy
Paperback: 430 Pages (2010-11-02)
list price: US$199.00 -- used & new: US$199.00
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Asin: 364208429X
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This book is a systematic treatment of real algebraic geometry, a subject that has strong interrelation with other areas of mathematics: singularity theory, differential topology, quadratic forms, commutative algebra, model theory, complexity theory etc. The careful and clearly written account covers both basic concepts and up-to-date research topics. It may be used as text for a graduate course. The present edition is a substantially revised and expanded English version of the book "Géometrie algébrique réelle" originally published in French, in 1987, as Volume 12 of ERGEBNISSE. Since the publication of the French version the theory has made advances in several directions. Many of these are included in this English version. Thus the English book may be regarded as a completely new treatment of the subject. ... Read more

13. Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series (Ergebnisse der Mathematik Und Ihrer Grenzgebiete) (Volume 0)
by R.K. Lazarsfeld
Paperback: 387 Pages (2004-08-24)
list price: US$34.95 -- used & new: US$28.56
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Asin: 3540225285
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This two volume work on ... Read more

14. Complex Analysis and Algebraic Geometry: A Collection of Papers Dedicated to K. Kodaira
Paperback: 416 Pages (2009-04-13)
list price: US$50.00 -- used & new: US$43.19
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Asin: 0521093341
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The articles in this volume cover some developments in complex analysis and algebraic geometry. The book is divided into three parts. Part I includes topics in the theory of algebraic surfaces and analytical surfaces. Part II covers topics in moduli and classification problems, as well as structure theory of certain complex manifolds. Part III is devoted to various topics in algebraic geometry analysis and arithmetic. A survey article by Ueno serves as an introduction to the general background of the subject matter of the volume. The volume was written for Kunihiko Kodaira, on the occasion of his sixtieth birthday, by his friends and students. Professor Kodaira was one of the world's leading mathematicians in algebraic geometry and complex manifold theory: and the contributions reflect those concerns. ... Read more

15. Rudiments of algebraic geometry (University texts in the mathematical sciences)
by William Elliott Jenner
 Paperback: 104 Pages (1963)

Asin: B0006AYF2I
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16. Algebraic and Analytic Geometry (London Mathematical Society Lecture Note Series)
by Amnon Neeman
Paperback: 434 Pages (2007-09-24)
list price: US$85.00 -- used & new: US$68.70
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Asin: 0521709830
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This textbook, for an undergraduate course in modern algebraic geometry, recognizes that the typical undergraduate curriculum contains a great deal of analysis and, by contrast, little algebra. Because of this imbalance, it seems most natural to present algebraic geometry by highlighting the way it connects algebra and analysis; the average student will probably be more familiar and more comfortable with the analytic component. The book therefore focuses on Serre's GAGA theorem, which perhaps best encapsulates the link between algebra and analysis. GAGA provides the unifying theme of the book: we develop enough of the modern machinery of algebraic geometry to be able to give an essentially complete proof, at a level accessible to undergraduates throughout. The book is based on a course which the author has taught, twice, at the Australian National University. ... Read more

17. Principles of Algebraic Geometry
by Phillip Griffiths, Joseph Harris
 Paperback: 832 Pages (1994-08-02)
list price: US$137.00 -- used & new: US$93.86
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Asin: 0471050598
Average Customer Review: 5.0 out of 5 stars
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A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special topics in complex manifolds. ... Read more

Customer Reviews (8)

5-0 out of 5 stars high five
I agree with most earlier commentators that this is a very nice introduction to the subject. That said, depending on your background, you may find that cover to cover may not be the most efficient way of reading this book. Also it differs from 'modern' treatments of the subject. All in all, it's an indispensible reference for most beginners and 'advanced beginners' if not more readers.

4-0 out of 5 stars A review from a graduate student
If you are a graduate student in mathematics or related fields and you are interested in learning algebraic geometry in the Griffiths-Harris way, then I suggest before buying this book to have a good background in the following:

1. Complex Analysis
2. Differential Geometry and calculus on manifolds
3. Homology-Cohomology Theory
4. Undergraduate Algebraic Geometry

Do not expect chapter 0, "Foundational Material", to be the place where you are supposed to build your "foundation". You can try the books of Michael Spivak, David A. Cox, Fangyang Zheng, among other books for foundational material but not chapter 0.

However, if you have most of the above-mentioned foundational material, then this book is good in presenting complex manifolds for example in chapter 0 section 2 and also in presenting (complex) holomorphic vector bundles, as well as many other things.

So, in summary, I would say a good book but not for students trying to learn the basics in algebraic geometry.

5-0 out of 5 stars algebraic geometry: the real stuff
The book is beautifully written and easy to read, with emphasis on geometric picture instead of abstract nonsense. By far the best introduction to algebraic geometry for string theorists.

5-0 out of 5 stars Work of Art
This is an amazing book with an amazing subject (complex algebraic geometry).Every section presents something interesting and wonderful.I've only read chapters 0 (Complex manifolds, Hodge theory), 1 (Divisors & line bundles, vanishing theorems, embeddings), and 2 (Riemann surfaces).I had had a bad experience with alg geom before this book.Required reading for mathematicians in complex manifolds, algebraic geometry, or string theorists.There are some very trivial typos scattered, but nothing problematic in the least (like capital lambda instead of a big wedge, or indices).If you read the book carefully you will get a lot out of it.

5-0 out of 5 stars Absolutely indispensable
This book is fabulous - it is an indispensable reference for complex algebraic geometry. It is very clearly written and ideas are always motivated by examples and problems. Moreover, if you want to learn modern algebraic geometry, it's imperative to learn the classical case (over the complexes - which in practice is easier to work in) in order to understand the generalisations a la Grothendieck. ... Read more

18. The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra (Graduate Texts in Mathematics)
by David Eisenbud
Paperback: 248 Pages (2005-02-01)
list price: US$44.95 -- used & new: US$27.99
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Asin: 0387222324
Average Customer Review: 5.0 out of 5 stars
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Algebraic Geometry often seems very abstract, but in fact it is full of concrete examples and problems. This side of the subject can be approached through the equations of a variety, and the syzygies of these equations are a necessary part of the study. This book is the first textbook-level account of basic examples and techniques in this area. It illustrates the use of syzygies in many concrete geometric considerations, from interpolation to the study of canonical curves. The text has served as a basis for graduate courses by the author at Berkeley, Brandeis, and in Paris. It is also suitable for self-study by a reader who knows a little commutative algebra and algebraic geometry already. As an aid to the reader, an appendix provides a summary of commutative algebra, tying together examples and major results from a wide range of topics.

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Customer Reviews (1)

5-0 out of 5 stars Great book
Great book. It is really worthy the money, especially if you want to better understand geometry from an algebraic perspectives. For pure geometers it might be a little bit too algebraic. ... Read more

19. Algebraic Geometry: A First Course (Graduate Texts in Mathematics)
by Joe Harris
Paperback: 328 Pages (2010-11-02)
list price: US$74.95 -- used & new: US$60.14
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Asin: 144193099X
Average Customer Review: 3.0 out of 5 stars
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"This book succeeds brilliantly by concentrating on a number of core topics...and by treating them in a hugely rich and varied way. The author ensures that the reader will learn a large amount of classical material and perhaps more importantly, will also learn that there is no one approach to the subject. The essence lies in the range and interplay of possible approaches. The author is to be congratulated on a work of deep and enthusiastic scholarship." --MATHEMATICAL REVIEWS

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Customer Reviews (4)

1-0 out of 5 stars Not for a beginner
This book is not a good first text for a person trying to learn algebraic geometry.Prof. Harris gives very limited explanations and few clear examples.Try Shafarevich, Basic Algebraic Geometry vols I & II or Miles Reid, Undergraduate Algebraic Geometry.For both books, you would need commutative algebra, at least at the level of Miles Reid, Undergraduate Commutative Algebra.

5-0 out of 5 stars 22 lectures = 5 stars
Last year we used this title as a main reference for the first two quarters of a year-long introductory sequence on algebraic geometry, at the beginning graduate student level (2nd year). Our professor --who was himself a former student of Harris, and a specialist in the Mori program-- backed up the presentation with personal lecture notes, moving mostly in parallel to the book's topics. In the third quarter of the sequence, we moved up to cover the theory of sheaves and schemes from Robin Hartshorne's advanced treatise. Prior to this point, my only exposure to the subject was from the corresponding chapter in Dummit and Foote's algebra, and also from a recent introductory text, "An Invitation to Algebraic Geometry" by Karen Smith et al. At this stage of my studies I was mainly testing the waters; my ineterest on the one hand was driven by a curiosity for the subject itself, which has a reputation for being difficult and hard to grasp, and on the other hand, from its pivotal role in the formulation of new physical theories of mirror symmetry and string theory (for more in this direction, see the 1999 AMS title by A. Cox and S. Katz).

Back to the present text, as the editorial notes correctly point out, this Harris book emphasizes the classical algebraic geometry from the 19th & early 20th centuries prior to the introduction of highly abstract machinery, due to the work of A. Groethendieck in the 50's and 60's. Therefore it's quite natural to base the treatment mostly on the examples and concrete constructions, which were the guiding principles of the abstract development in the first place. This approach also makes the subject accessible for the newcomers who may not have an advanced background in commutative algebra or category theory, and who may not be intending to specialize in the area but merely wish to gain a general understanding of the ideas involved. I found my experience with the subject very fulfilling, and enjoyed many aspects of the presentation. I only suspect many of you could have a similar rewarding experience embarking upon this journey! For the application-oriented readers, I should recommend the Springer-Verlag title "Ideals, Varieties and Algorithms" by Cox et al. which has separate chapters on the Groebner bases, invariant theory, and the robotics.

1-0 out of 5 stars too many examples
I was confused by the many examples. Most of the times I did not see how they fit together in the theory. Perhaps I would have appreciate it more, had I known some algebraic geometry first.

5-0 out of 5 stars Definitely a good start in algebraic geometry
If one is planning to do work in coding theory, cryptography, computer graphics, digitial watermarking, or are hoping to become a mathematician specializing in algebraic geometry, this book will be of an enormous help. The author does a first class job in introducing the reader to the field of algebraic geometry, using a wealth of examples and with the goal of building intuition and understanding. It is great that a mathematician of the author's caliber would take the time to write these lectures here put into book form. It is rare to find a book on algebraic geometry that attempts to make the subject concrete and understandable, and yet points the way to more modern "scheme-theoretic" formulations.

In lecture 1, the author introduces affine and projective varieties over algebraically closed fields. Linear subspaces of n-dimensional projective space P(n) are shown to be varieties, along with any finite subset of P(n). He delays giving rigorous definitions of degree and dimension, emphasizing instead concrete examples of varieties. The twisted cubic is given as the first example of a concrete variety that is not a hypersurface, along with their generalizations, the rational normal curves.

The Zariski topology, considered by the newcomer to the subject as being a rather "strange" topology, is introduced in lecture 2. The author does a great job though explaining its properties, and introduces the regular functions on affine and projective varieties. The Nullstellensatz theorem, needed to prove that the ring of regular functions is the coordinate ring, is deferred to a later lecture. Rational normal curves are further generalized to Veronese maps in this lecture, and the properties of the corresponding Veronese varieties discussed in some detail. Also, the very interesting Segre varieties are discussed here. With these two examples of varieties, the reader already can develop a good geometric intution of the behavior of typical varieties. The Veronese and Segre maps are then combined to give another example of a variety: the rational normal scroll. More concrete examples of varieties are given in the next two lectures, including cones, quadrics, and projections. A "fiber bundle" approach to forming families of varieties parametrized by a given variety is outlined here also.

The author finally gets down to more algebraic matters in lecture 5, with the Nullstellensatz proven in great detail. He also discusses the origins of schemes in algebraic geometry, giving the reader a better appreciation of just where these objects arise, namely the association to an arbitrary ideal, instead of merely a radical ideal.

Grassmannian varieties are then introduced in lecture 6, along with some of its subvarieties, such as the Fano varieties. The join operation, widely used in geometric topology, is here defined for two varieties.

More connections with the modern viewpoint are made in lecture 7, where rational functions and rational maps are discussed. The author takes great care in explaining in what sense rational maps can be thought of as maps in the "ordinary" sense, namely they must be thought of as equivalence classes of pairs, instead of acting on points. The very important concept of a birational isomorphism is discussed also, along with blow-ups and blow-downs of varieties.

Many more concrete examples of varieties are given in lectures 8 and 9, such as secant varieties, flag manifolds, and determinantal varieties. In addition, algebraic groups on varieties are discussed in lecture 10, allowing one to discuss a kind of glueing operation on varieties, just as in geometric topology, namely by taking the quotient of varieties via finite groups.

The author then moves on to giving a more rigorous formulation of dimension, giving several different definitions, all of these conforming to intuitive ideas on what the dimension of an algebraic variety should be, and also one compatible with a purely algebraic context. Again, several concrete examples are given to illustrate the actual calculation of the dimension of a variety, both in this lecture and the next one.

The next lecture is very interesting and discusses an important problem in algebraic geometry, namely the determination of how many hypersurfaces of each degree contain a projective variety in P(n). The solution is given in terms of the famous Hilbert polynomial, which is determined for rational normal curves, Veronese varieties, and plane curves in this lecture. The author also explains the utility of using graded modules in the determination of the Hilbert polynomial, something that is usually glossed over in most books on this topic. This discussion leads to the Hilbert syzygy theorem.

Some analogs of basic contructions in differential geometry are defined for varieties in the next four lectures, based on an appropriate notion of smoothness. The tangent spaces, the Gauss map, and duals discussed here.

Then in lecture 18 the author makes good on his promise in earlier lectures of making the notion of the degree of a projective variety more rigorous. The well-known Bezout's theorem is proven, after introducting a notion of transversal intersection for varieties. As usual in the book, several examples are given for the calculation of the degree, including Veronese and Segre varieties, in this lecture and the next.

The behavior of a variety at a singular point is studied in lecture 20 using tangent cones. The author proves the resolution of singularities for curves here also.

Lecture 21 is very important, especially for the physicist reader working in string and M-theories, as the author introduces the concept of a moduli space. Most results are left unproven, but the intuition gained from reading this lecture is invaluable. The all-important Chow and Hilbert varieties are discussed here. The book the ends with a fairly lengthy overview of quadric hypersurfaces. ... Read more

20. Basic Algebraic Geometry 2: Schemes and Complex Manifolds (Volume 0)
by Igor R. Shafarevich
Paperback: 269 Pages (1994-08-08)
list price: US$74.95 -- used & new: US$51.99
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Asin: 3540575545
Average Customer Review: 4.0 out of 5 stars
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The second volume of Shafarevich's introductory book on algebraic geometry focuses on schemes, complex algebraic varieties and complex manifolds. As with first volume the author has revised the text and added new material. Although the material is more advanced than in Volume 1 the algebraic apparatus is kept to a minimum making the book accessible to non-specialists. It can be read independently of the first volume and is suitable for beginning graduate students.

... Read more

Customer Reviews (2)

5-0 out of 5 stars Are you looking for literary criticism? It's a freaking math book!
The first book in this two volume introduction to algebraic geometry was used as the primary textbook for my algebraic geometry class.It was amazing.Easily the most readable (oh, Hartshorne why are you so heartless?) of all the algebraic algebraic geometry I own (which is quite a few).I finally managed to secure the second volume after several fruitless searches and order cancellations (Amazon didn't have it in stock for months).Though I'm only starting to get into this one and am basing this recommendation mostly on my experience with the first volume, I still think that Shafarevich writes the most accessible introduction to schemes that I've ever read.

3-0 out of 5 stars Was the book modified without modifying the index?
Shafarevich explains mathematics well, but I find the index for this book extremely frustrating (I bought my copy in the 2005 Springer sale).I'm almost convinced that they forgot to modify the index after making modifications to the first edition. ... Read more

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