Book Description
The second volume of this modern account of Kaehlerian geometry and Hodge theory starts with the topology of families of algebraic varieties. The main results are the generalized Noether-Lefschetz theorems, the generic triviality of the Abel-Jacobi maps, and most importantly, Nori's connectivity theorem, which generalizes the above. The last part deals with the relationships between Hodge theory and algebraic cycles. The text is complemented by exercises offering useful results in complex algebraic geometry. Also available: Volume I 0-521-80260-1 Hardback $60.00 CDownload Description
The second volume of this modern account of Kaehlerian geometry and Hodge theory starts with the topology of families of algebraic varieties. Proofs of the Lefschetz theorem on hyperplane sections, the Picard-Lefschetz study of Lefschetz pencils, and Deligne theorems on the degeneration of the Leray spectral sequence and the global invariant cycles follow. The main results of the second part are the generalized Noether-Lefschetz theorems, the generic triviality of the Abel-Jacobi maps, and most importantly Nori's connectivity theorem, which generalizes the above. The last part of the book is devoted to the relationships between Hodge theory and algebraic cycles. The book concludes with the example of cycles on abelian varieties, where some results of Bloch and Beauville, for example, are expounded. The text is complemented by exercises giving useful results in complex algebraic geometry. It will be welcomed by researchers in both algebraic and differential geometry. ... Read more

Book Description
The aim of this book is to describe 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 (5)

very good
excellent book ... can make learning algebraic geometry as easy as bedside reading ... highly recommended.

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!

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.

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.

Very good, but understated prerequisites
On the back of the book, it says "Few algebraic prerequisites are presumed beyond a basic course in linear algebra."This is not, in fact, true.It uses a lot of ring theory (and the review of commutative ring theory in ch. 2 is a bit fast for someone unfamiliar with the subject), and a fair amount of topology.When I first got it, I read the first several pages and found them readable, but when I read more (on the car-ride home) I was confronted with the fact that the Zariski topology is coarser than the standard topology on C^n, and is not even Hausdorff.Several questions came to mind (What's a topology? What does for one to be coarser than another?What is a Hausdorff topology?).Still, after I learned more topology, I found the book a delight.Everything is light and interesting, and does a good job of portraying algebraic geometry without technical details.All mathematics looks nicer when you do that, but it takes away from rigor.Hence this should not be your only text on Algebraic Geometry, but I would suggest it as one of them... ... Read more

Book Description
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)

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.

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.

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.

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.

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

Book Description
This is the first of three volumes on algebraic geometry. The second volume, Algebraic Geometry 2: Sheaves and Cohomology, is available from the AMS as Volume 197 in the Translations of Mathematical Monographs series.

Early in the 20th century, algebraic geometry underwent a significant overhaul, as mathematicians, notably Zariski, introduced a much stronger emphasis on algebra and rigor into the subject. This was followed by another fundamental change in the 1960s with Grothendieck's introduction of schemes. Today, most algebraic geometers are well-versed in the language of schemes, but many newcomers are still initially hesitant about them. Ueno's book provides an inviting introduction to the theory, which should overcome any such impediment to learning this rich subject.

The book begins with a description of the standard theory of algebraic varieties. Then, sheaves are introduced and studied, using as few prerequisites as possible. Once sheaf theory has been well understood, the next step is to see that an affine scheme can be defined in terms of a sheaf over the prime spectrum of a ring. By studying algebraic varieties over a field, Ueno demonstrates how the notion of schemes is necessary in algebraic geometry.

This first volume gives a definition of schemes and describes some of their elementary properties. It is then possible, with only a little additional work, to discover their usefulness. Further properties of schemes will be discussed in the second volume. ... Read more

Customer Reviews (1)

A Good Book Overall
A nice book with details worked out but quite a few typos. ... Read more

Book Description

Book Description
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 Gröbner bases and resultants. In order to do this, the authors provide an introduction to some algebraic objects and techniques which are more advanced than one typically encounters in a first course, but nonetheless of great utility. The book is written for nonspecialists and for readers with a diverse range of backgrounds. It assumes knowledge of the material covered in a standard undergraduate course in abstract algebra, and it would help to have some previous exposure to Gröbner bases. The book does not assume the reader is familiar with more advanced concepts such as modules. For this new edition the authors added two new sections and a new chapter, updated the references and made numerous minor improvements throughout the text. ... Read more

Customer Reviews (2)

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.

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

Book Description

Tropical geometry is algebraic geometry over the semifield of tropical numbers, i.e., the real numbers and negative infinity enhanced with the (max,+)-arithmetics. Geometrically, tropical varieties are much simpler than their classical counterparts. Yet they carry information about complex and real varieties.

Book Description
Algebraic geometry is, essentially, the study of the solution of equations and occupies a central position in pure mathematics. With the minimum of prerequisites, Dr. Reid introduces the reader to the basic concepts of algebraic geometry, including: plane conics, cubics and the group law, affine and projective varieties, and nonsingularity and dimension. He stresses the connections the subject has with commutative algebra as well as its relation to topology, differential geometry, and number theory. The book contains numerous examples and exercises illustrating the theory. ... Read more

Customer Reviews (1)

baked just right for the first timers !
There are many good books on the subject of algebraic geometry, so what was the use of one more - asks the author in the preface to this book.But there are none -at the UG level- which for the first time reveal to theyounger mathematicians the secrets of this vast and growing subject. Thebook treats every new concept with the rigour that keeps in mind the levelit is meant for, and yet maintains its mathematical "beauty" -setting firmly the basics for those who would want to take up this courseat an advanced level as well as keeping the more casual mathematics readerinterested. ... Read more

Book Description
Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris.After receiving his Ph.D. from Princeton in 1963, Hartshorne became a Junior Fellow at Harvard, then taught there for several years.In 1972 he moved to California where he is now Professor at the University of California at Berkeley.He is the author of "Residues and Duality" (1966), "Foundations of Projective Geometry (1968), "Ample Subvarieties of Algebraic Varieties" (1970), and numerous research titles.His current research interest is the geometry of projective varieties and vector bundles. He has been a visiting professor at the College de France and at Kyoto University, where he gave lectures in French and in Japanese, respectively.Professor Hartshorne is married to Edie Churchill, educator and psychotherapist, and has two sons.He has travelled widely, speaks several foreign languages, and is an experienced mountain climber. He is also an accomplished amateur musician: he has played the flute for many years, and during his last visit to Kyoto he began studying the shakuhachi. ... Read more

Customer Reviews (5)

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.

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.

Be prepared...
This book is one of the most used in graduate courses in algebraic geometry and one that causes most beginning students the most trouble. But it is a subject that is now a "must-learn" for those interested in its many applications, such as cryptography, coding theory, physics, computer graphics, and engineering. That algebraic geometry has so many applications is quite amazing, since it was not too long ago that it was thought of as a highly abstract, esoteric topic. That being said, most of the books on the subject, including this one, are written from a very formal point of view. Those interested in applications will have to face up to this when attempting to learn the subject. To read this book productively one should gain a thorough knowledge of commutative algebra, a good start being Eisenbud's book on this subject. Also, it is important to dig into the original literature on algebraic geometry, with the goal of gaining insight into the constructions and problems involved. The author of this book does not make an attempt to motivate the subject with historical examples, and so such a perusal of the literature is mandatory for a deeper appreciation of algebraic geometry. The study of algebraic geometry is well worth the time however, since it is one that is marked by brilliant developments, and one that will no doubt find even more applications in this century.

Varieties, both affine and projective, are introduced in chapter 1. The discussion is purely formal, with the examples given unfortunately in the exercises. The Zariski topology is introduced by first defining algebraic sets, which are zero sets of collections of polynomials. The algebraic sets are closed under intersection and under finite unions. Therefore their complements form a topology which is the Zariski topology. The properties of varieties are discussed, along with morphisms between them. "Functionals" on varieties, called regular functions in algebraic geometry, are introduced to define these morphisms. Rational and birational maps, so important in "classical" algebraic geometry are introduced here also. Blowing up is discussed as an example of a birational map. A very interesting way, due to Zariski, of defining a nonsingular variety intrinsically in terms of local rings is given. The more specialized case of nonsingular curves is treated, and the reader gets a small taste of elliptic curves in the exercises. A very condensed treatment of intersection theory in projective space is given. The discussion is primarily from an algebraic point of view. It would have been nice if the author would have given more motivation of why graded modules are necessary in the definition of intersection multiplicity.

The theory of schemes follows in chapter 2, and to that end sheaf theory is developed very quickly and with no motivation (such as could be obtained from a discussion of analytic continuation in complex analysis). Needless to say scheme theory is very abstract and requires much dedication on the reader's part to gain an in-depth understanding. I have found the best way to learn this material is via many examples: try to experiment and invent some of your own. The author's discussion on divisors in this chapter is fairly concrete however.

The reader is introduced to the cohomology of sheaves in chapter 3, and the reader should review a book on homological algebra before taking on this chapter. Derived functors are used to construct sheaf cohomology which is then applied to a Noetherian affine scheme, and shown to be the same as the Cech cohomology for Noetherian separated schemes. A very detailed discussion is given of the Serre duality theorem.

Things get much more concrete in the next chapter on curves. After a short proof o the Riemann-Roch theorem, the author studies morphisms of curves via Hurwitz's theorem. The author then treats embeddings in projective space, and shows that any curve can be embedded in P(3), and that any curve can be mapped birationally into P(2) if one allows nodes as singularities in the image. And then the author treats the most fascinating objects in all of mathematics: elliptic curves. Although short, the author does a fine job of introducing most important results.

This is followed in the next chapter by a discussion of algebraic surfaces in the last chapter of the book. The treatment is again much more concrete than the earlier chapters of the book, and the author details modern formulations of classical constructions in algebraic geometry. Ruled surfaces, and nonsingular cubic surfaces in P(3) are discussed, as well as intersection theory. A short overview of the classification of surfaces is given. The reader interested in more of the details of algebraic surfaces should consult some of the early works on the subject, particularly ones dealing with Riemann surfaces. It was the study of algebraic functions of one variable that led to the introduction of Riemann surfaces, and the later to a consideration of algebraic functions of two variables. A perusal of the works of some of the Italian geometers could also be of benefit as it will give a greater appreciation of the methods of modern algebraic geometry to put their results on a rigorous foundation.

Terrific, if you want it.
This book hardly needs a review on Amazon, because if you have as much math background as it needs, then you must already know it is indispensible for learning about schemes in algebraic geometry. The book is clear, concise, very well organized, and very long. If you do not already know the Noether normalization theorem, and the Hilbert Nullstellensatz, then you do not want this book yet--you want an introduction to commutative algebra.

Indispensable!
Excelent and useful text, indispensable for graduate students and research,athematicians working on algebraic geometry. Hartshorne walks the fineline between commutative algebra and their geometrical counterparts withelegance. The book is also rich in references, providing many directionsfor further study. ... Read more

Book Description
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

Book Description
The second volume of Shafarevich's introductory book on algebraic varieties and complex manifolds. As with Volume 1, the author has revised the text and added new material, e.g. as a section on real algebraic curves. 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 Volume 1 and is suitable for beginning graduate students in mathematics as well as those in theoretical physics. ... Read more

Customer Reviews (2)

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.

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

Book Description

Aimed primarily at graduate students and beginning researchers, this book provides an introduction to algebraic geometry that is particularly suitable for those with no previous contact with the subject and assumes only the standard background of undergraduate algebra. It is developed from a masters course given at the Université Paris-Sud, Orsay, and focusses on projective algebraic geometry over an algebraically closed base field.

Book Description
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

Book Description
This text provides an introduction to the theory of convex polytopes and polyhedral sets, to algebraic geometry and to the fascinating connections between these fields: the theory of toric varieties (or torus embeddings).
The fist part of the book contains an introduction to the theory of polytopes - one of the most important parts of classical geometry in n-dimensional Euclidean space. Since the discussion here is independent of any applications to algebraic geometry, it would also be suitable for a course in geometry. This part also provides large parts of the mathematical background of linear optimization and of the geometrical aspects in Computer Science. The second part introduces toric varieties in an elementary way, building on the concepts of combinatorial geometry introduced in the first part. Many of the general concepts of algebraic geometry arise in this treatment and can be dealt with concretely. This part of the book can thus serve for a one-semester introduction to algebraic geometry, with the first part serving as a reference for combinatorial geometry. ... Read more

Customer Reviews (1)

An excellent way to begin a study of algebraic geometry
This book is a very organized introduction to the study of constructions that really go back to Isaac Newton, one of these now being called a Newton polygon. In the context of modern algebraic geometry, the constructions take place when dealing with the resolution of singularities of varieties. Given a variety X, this procedure asks for a map from a nonsingular variety Y to X, such that the map is an isomorphism over the nonsingular locus of X. It was the case of a plane curve singularity that was essentially solved by Newton. His techniques were generalized considerably beginning in the 1970's, and resulted in the theory of toric varieties, which is the main subject of this book.

Loosely speaking, a toric variety is a complex algebraic variety which is the partial compactification of an algebraic torus. The algebraic torus acts on a point in the toric variety such that the orbit of the point is an embedded copy of the algebraic torus. Toric varieties are excellent concrete examples of algebraic varieties since they are characterized entirely by a combinatorial object called its fan, which is a collection of convex cones.

This book is an fine introduction to toric varieties. The author does a thorough job of detailing the relevant background in the first half of the book, which deals mostly with convexity and the geometry of lattice polytopes. A very interesting discussion of the Picard group is given in the last few sections of this part. This is one of the best discussions I have seen in the literature on this subject as it gives the reader a very intuitive and concrete view of this group.

The second half covers toric varieties in detail with systems of rational functions on a toric variety studied via sheaf theory. The reader familiar with sheaf theory from general algebraic geometry will see it take on a beautifully concrete form in this book. Readers new to algebraic geometry will appreciate the more abstract approach to sheaf theory if they move on to these more advanced treatments. The author gives many examples of the constructions involved with toric varieties. The cohomology of toric varieties is also treated very nicely, and here again, a reader with a modest background in combinatorial topology will follow the presentation. The physicist reader doing research into mirror symmetry will appreciate this book, as toric varieties serve as a good starting point for the constructions in that area. ... Read more

Book Description
Basic Algebraic Geometry, Volume I, is a revised and expanded new edition of the first four chapters of Shafarevich's well-known introductory book on algebraic geometry. The author has added plenty of new, mostly concrete geometrical material such as Grassmannian varieties, plane cubic curves, the cubic surface, degenerations of quadrics and elliptic curves, the Bertini theorems, normal surface singularities. There are also some new number-theoretical applications. Shafarevich succeeds in making algebraic geometry accessible to non-specialists and beginners and his two-volume book will remain one of the most popular introductions to this field. The book is suitable for third-year undergraduates in mathematics and also for students of theoretical physics. ... Read more

Customer Reviews (2)

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!!!

After Hartshorne!!!
This book is very good for the secondary course after learning with Harshorne's Algebraic geometry. ... Read more

Book Description
This book will be particularly valuable to the American student because it covers material that is not available in any other textbooks or monographs. The subject of the book is not restricted to commutative algebra developed as a pure discipline for its own sake, nor is it aimed only at algebraic geometry where the intrinsic geometry of a general n-dimensional variety plays the central role. Instead, this book is developed around the vital theme that certain areas of both subjects are best understood together. This link between the two subjects, forged in the nineteenth century, built further by Krull and Zariski, remains as active as ever.In this book, the reader will find as the same time a leisurely and clear exposition of the basic definitions and results in both algebra and geometry, as well as an cxposition of the important recent progress fue to Quillen-Suslin, Evans-Eisenbud, Szpiro, Mohan Kumar and others. The ample exercises are another excellent feature. Professor Kunz has filled a longstanding need for an introduction to commutative algebra and algebraic geometry that emphasizes the concrete elementary nature of objects with which both subjects began. ... Read more