Algebraic Geometry 1: From Algebraic Varieties to Schemes Kenji Ueno Publication Year: ISBN ISBN Kenji Ueno is a Japanese mathematician, specializing in algebraic geometry. He was in the s at the University of Tokyo and was from to a. Algebraic geometry is built upon two fundamental notions: schemes and sheaves . The theory of schemes was explained in Algebraic Geometry 1: From.

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By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. I think almost everyone agrees that Hartshorne’s Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc. One suggestion per answer please.

Algebraic Geometry by Kenji Ueno

Also, please include an explanation of why you like the book, or what makes it unique or useful. I think Algebraic Geometry is too broad a subject to choose only one book. Maybe if one is a beginner then a clear introductory book is enough or if algebraic geometry is not ones major field of study then a self-contained reference dealing with the important topics thoroughly is enough. But Algebraic Geometry nowadays has grown into such a deep and ample field of study that a graduate student has to focus heavily on one or two topics whereas at the same time must be able to use the fundamental results of other close subfields.

That is why I have collected what in my humble opinion are the best books for each stage and topic of study, my personal choices for the best books are then: Very complete proves Riemann-Roch for curves in an easy language and concrete in classic constructions needed to understand the reasons about why things are done the way they are in advanced purely algebraic books.

There are very few books like this and they should be a must to start learning the subject. Check out Dolgachev’s review. Shafarevich – “Basic Algebraic Geometry” vol. They may be the most complete on foundations for varieties up to introducing schemes and complex geometry, so they are very useful before more abstract studies. But the problems are hard for many beginners. They do not prove Riemann-Roch which is done classically without cohomology in the previous recommendation so a modern more orthodox course would be Perrin’s “Algebraic Geometry, An Introduction”, which in fact introduce cohomology and prove RR.

This new title is wonderful: The second half then jumps into a categorical introduction to schemes, bits of cohomology and even glimpses of intersection theory. Gathmann – “Algebraic Geometry” which can be found here. Just amazing notes; short but very complete, dealing even with schemes and cohomology and proving Riemann-Roch and even hinting Hirzebruch-R-R. It is the best free course in my opinion, to get enough algebraic geometry background to understand the other more advanced and abstract titles.

For an abstract algebraic approach, a freely available online course is available by the nicely done new long notes by R. It is a very complete book even introducing some needed commutative algebra and preparing the reader to learn arithmetic geometry like Mordell’s conjecture, Faltings’ or even Fermat-Wiles Theorem.


Griffiths; Harris – “Principles of Algebraic Geometry”. By far the best for a complex-geometry-oriented mind. Also useful coming from studies on several complex variables or differential geometry. It develops a lot of algebraic geometry without so much advanced commutative and homological algebra as the modern books tend to emphasize.

It does a great job complementing Hartshorne’s treatment of schemes, above all because of the more solvable exercises.

It is a classic and although the flavor is clearly of typed concise notes, it is by far the shortest but thorough book on curves, which serves as a very nice introduction to the whole subject. It does everything that is needed to prove Riemann-Roch for curves and introduces many concepts useful to motivate more advanced courses.

This one is focused on the reader, therefore many results are stated to be worked out. So some people find it the best way to really master the subject. Beauville – “Complex Algebraic Surfaces”. I have not found a quicker and simpler way to learn and clasify algebraic surfaces. The background needed is minimum compared to other titles. Badescu – “Algebraic Surfaces”. Excellent complete and advanced kenju for surfaces. Very well done and indispensable for those needing a companion, but above all an expansion, to Hartshorne’s chapter.

The 11 volume can serve almost as an introduction to complex geometry and the second to its topology. They are becoming more and more the standard reference on these topics, fitting nicely between abstract algebraic geometry and complex differential geometry. Mukai – An Introduction to Invariants and Moduli. Excellent but extremely expensive hardcover book. When geomstry cheaper paperback edition is released by Cambridge Press any serious student of algebraic geometry should own a copy since, again, it is one of those titles that help geometey and give conceptual insights needed to make any sense of abstract monographs like the next ones.

Hartshorne – “Deformation Theory”. Just the perfect complement to Hartshorne’s main book, since it did not deal with these matters, and other books approach the subject from a different point of view e. Simply put, it is still the best and most complete.

Besides, Mumford himself developed the subject. Alternatives are more introductory lectures by Dolgachev. Fulton – “Intersection Theory”. It is the standard reference and is also cheap compared to others. It deals with all the material needed on intersections geometrry a serious student going beyond Ueho appendix; it is algebrwic good reference for the use of the language of characteristic classes in algebraic geometry, proving Hirzebruch-Riemann-Roch and Grothendieck-Riemann-Roch among many interesting results.

Great exposition, useful contents and examples on topics one has to deal with sooner or later. As a fundamental complement check Hauser’s wonderful paper on the Hironaka theorem. Lazarsfeld – Positivity in Algebraic Geometry I: Positivity for Vector Geometrt and Multiplier Ideals.

Amazingly well written and unique on the topic, summarizing and bringing together lots of information, results, and many many examples. Debarre – “Higher Dimensional Algebraic Geometry”.

Kenji Ueno

Considered as harder to learn from by some students, it has become the standard reference on birational geometry. Liu wrote a nice book, which is a bit more oriented to arithmetic geometry.

The last few chapters contain some material which is very pretty but unusual for a basic text, such as reduction of algebraic curves. I’ve found it quite rewarding to to familiarize myself with the contents of EGA.


Many krnji geometry students are able to say with confidence “that’s one of the exercises in Hartshorne, chapter II, section 4. I’ve found this combined table of contents to be useful in this quest. The combined table of contents unfortunately seems to be defunct. Its great for a conceptual introduction that won’t turn people off as fast as Hartshorne. However, it barely even mentions the concept of geomery module of a scheme, and I geometrh it ignores sheaf cohomology entirely.

Shafarevich wrote a very basic introduction, it’s used in undergraduate classes in algebraic geometry sometimes. Basic Algebraic Geometry 1: Varieties in Projective Space.

Ideals, Varieties, and Algorithms: At a lower level then Hartshorne is the fantastic “Algebraic Curves” by Fulton. It’s available on his website. Geeometry Ueno’s three-volume “Algebraic Geometry” is well-written, clear, and has the perfect mix of text and diagrams. It’s undoubtedly a real masterpiece- very user-friendly. I’ve been teaching an introductory course in algebraic geometry this semester and I’ve been looking at geometrh sources.

I’ve found that Milne’s online book jmilne. He gives quite a thorough treatment of the theory of varieties over an algebraic closed field. The book is very complete and everything seems to be done “in the nicest way”. I second Shafarevitch’s two volumes on Basic Algebraic Geometry: Another very nice book is Miranda’s Algebraic Curves which manages to get a long way Riemann-Roch etc without doing sheaves and line gelmetry until kenju end. Of course, by then, you are really wanting sheaves and line bundles!

I liked Mumford’s “Algebraic geometry I: Complex projective varieties” a lot, and also Griffiths’ “Introduction to algebraic curves”. For the record, I hate Hartshorne’s.

Joe Harris’s book Algebraic Geometry might be a good ufno to Hartshorne. Little, Don O’Shea http: I’ve tried learning algebraic geometry several times. I asked around and was told to read Hartshorne. I started reading it several times and each time put it away. I realized that I could work through the sections and solve some of the problems, algebrzic I gained absolutely no intuition for reading Hartshorne. Discussing this with other people, I found that it was a common occurrence for students to read Hartshorne and afterwards have no idea how to do algebraic geometry.

I imagine this was the motivation for asking this question.

While Mumford doesn’t do cohomology, he motivates the definitions of schemes and and many of there basic properties while providing the reader with geometric intuition. This book isn’t easy to read and you have to work out a lot, but the rewards are great. Another great feature of this book is that Mumford bought the rights to the book back from Springer and the book is available for free online.

Another book was supposed to be written that built on the “Red book” including cohomology. After many years, I think this is near completion; see Algebraic Geometry 2. Whlile many of the above books are excellent, it’s a surprise that these books aren’t the standard.