Michael Artin Algebraic Geometry
Maral Mende
This book is an introduction to the geometry of complex algebraic varieties. It is intended for students who have learned algebra, analysis, and topology, as taught in standard undergraduate courses. So it is a suitable text for a beginning graduate course or an advanced undergraduate course.
The book begins with a study of plane algebraic curves, then introduces affine and projective varieties, going on to dimension and constructibility. \(\mathcalO\)-modules (quasicoherent sheaves) are defined without reference to sheaf theory, and their cohomology is defined axiomatically. The Riemann-Roch Theorem for curves is proved using projection to the projective line.
Some of the points that aren't always treated in beginning courses are Hensel's Lemma, Chevalley's Finiteness Theorem, and the Birkhoff-Grothendieck Theorem. The book contains extensive discussions of finite group actions, lines in \(\mathbbP^3\), and double planes, and it ends with applications of the Riemann-Roch Theorem.
The expository book under review is an expansion of the lecture notes grown out of a course in algebraic geometry that the author taught at MIT seven times within the last twelve years. That is why the book, which the author likes to refer to as lecture notes, benefits from a fresh living expository style. The output is not that of a plastered grey collection of notions but rather the logbook of an original educational journey in which the author flies like an eagle at high altitude without missing any detail on the ground thanks to its long view, driving his young companions past the marvels of the mathematical landscapes. Or it may look like an ancient workshop, similar to those of the great Italian painters and sculptors, where the master enabled disciples to learn the art by imitation and absorption.
Overall, Artin's text offers an excellent graduate-level introductory course in algebraic geometry. It covers the core topics from varieties to cohomology as a one-semester course that can be taken without a prior class in commutative algebra. Artin shares valuable insights of what is essential to algebraic geometry and where one should focus to appreciate the bigger picture and cautions the reader of technical pitfalls and points of confusion. He provides motivation and well-chosen examples that train the readers' intuition. Moreover, the style is personal and inviting, like a professor talking with students. The book offers the chance to be his student, an experience I enjoyed and learned from by reading 'Algebraic: Notes on a Course.'
The present book under review entitled "Algebraic Geometry - Notes on a Course," by Michael Artin is, according to my very subjective viewpoint, one of the best textbooks devoted to basics on algebraic geometry. I am aware of the fact that this is a rather bold statement, but I will try to justify my claim here. There are many textbooks devoted to the foundations of algebraic geometry, and it seems that there is no room for new ideas or strategies in writing such books. Everything has been tried. This was also my first prediction before receiving this book, and I can honestly say that I am very happy to have been wrong.
In a nutshell, algebraic geometrystudies systems of polynomial equations and the geometry of their solutionsets. It is one of the oldest branches of mathematics, with many connections toother areas such as number theory, complex geometry, combinatorics, ortheoretical physics.
During most weeks, I will be collecting written homework; we will also talkabout some problems in class. For each assignment, please write up yoursolutions nicely and hand them in by the due date. You can either send yourhomework to me by email (in PDF if possible), or hand in a printed copy at thebeginning of Tuesday's class (stapled and with your name on the first page).
And indeed, there are a lot of high quality 'articles', and often you can find alternative approaches to a theory or a problem, which are more suitable for you. In addition, you can actually ask questions (a feature thoroughly missed in e.g. Hartshorne's book).
I've found something extraordinary and of equally extraordinary pedigree online recently. I mentioned it briefly in response to R. Vakil's question about the best way to introduce schemes to students. But this question is really where it belongs and I hope word of it spreads far and wide from here.
Last fall at MIT, Michael Artin taught an introductory course in algebraic geometry that required only a year of basic algebra at the level of his textbook. The official text was William Fulton's Algebraic Curves, but Artin also wrote an extensive set of lecture notes and exercise sets. I found them quite wonderful and very much in the spirit of his classic textbook. (By the way, simply can't wait for the second edition.)
Not only has he posted these notes for download, he's asked anyone working through them to email him any errors found and suggestions for improvements. All the course materials can be found at the MIT webpage. I've also posted the link at MathOnline, of course.
I don't know if most of the hardcore algebraic geometers here would recommend these materials for a beginning course. But for any student not looking to specialize in AG, I can't think of a better source to begin with. That's just my opinion. But it certainly belongs as a possible response to this question. Then again, it may be too softball for the experts, particularly those of the Grothendieck school.
I think it's hard to say which one is the best, but for my own experience, I got into this area pretty much by reading most of this "3264 & all that" book, and completing almost all exercises (this is crucial!). It is said to be on intersection theory, but when I worked through it, I learned many other perspectives and came up with lots of concrete questions as well. I strongly recommend this book (again, the crucial point is doing exercises).
But of course I think it would be good to not just stick on one book. For example, Hartshorne definitely has a very quick and useful intro in cohomology, but for the part "higher direct images" I think 3264 is better. Also Beauville's surface book is of course good intro to surfaces, but for discussion of ruled surfaces, I think Hartshorne is actuallty better...
About Hartshorne and Griffiths, I think a comparison between the two texts is misleading.The first is a introduction to the "Grothendieck yoga" where geometrical classical ideas are "immersed" in the larger but abstract mathematical world of schemes.But also if the complex differential manifold style of Griffiths is "more concrete" is very different from the "Algebraic Geometry" idea, also if it is a deep study of it.
The word "best" is relative. If you have a strong background in commutative algebra and have had considerable exposure to algebraic geometry I would say Hartshorne would suit you. But for an introductory graduate text, I don't think so. We're using Fulton. Organization and exposition is okay, and the discussion is not as "hardcore" as that of Hartshorne. I'm surprised it didn't show up from those of you who posted here.
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