I haven't looked through all of them but these notes seem to cover as much as Hartshorne does (if not more). I think these notes might be made into a full on textbook someday. My last suggestion would be Ravi Vakil's online notes on the foundations of algebraic geometry. This is a very active frontier of research, with new fundamental results proved in Hacon/Kovács' "Classification of Higher-Dimensional Varieties". The main alternative to this title is Kollar/Mori "Birational Geometry of Algebraic Varieties" but is regarded as much harder to understand by many students. The best alternative to this and the previous title, but more on the introductory side, is Mukai's book on moduli and invariants.ĭebarre - "Higher Dimensional Algebraic Geometry". Besides, Mumford himself developed the subject. Simply put, it is the original reference. Mumford Fogarty Kirwan - "Geometric Invariant Theory". The alternative and easier title is Sernesi's book on deformation of algebraic schemes oriented for complex geometry or for physicists) than what a student of AG from Hartshorne's book may like to learn the subject. 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.g. The only main alternative is Cutkosky's book. Small but fundamental book on singularities, methods of resolution for curves and surfaces and proof of the cornerstone Hironaka's theorem. Kollár - Lectures on Resolution of Singularities. By far the best and most complete book on the subject, from general Bézout's theorem to Grothendieck-Riemann-Roch theorem. Done with more advanced tools than Beauville.įulton - Intersection Theory. For those needing a companion and expansion to Hartshorne's chapter. The background already needed is minimum compared to other titles.īadescu - "Algebraic Surfaces". I have not found a quicker and simpler way to learn and clasify algebraic surfaces. 2 has finally appeared making the two huge volumes a complete reference on the subject.īeauville - Complex Algebraic Surfaces. So some people find it the best way to really master the subject. This one is focused on the reader, therefore many results are stated to be worked out. It does everything that is needed to prove Riemann-Roch for curves.Īrbarello Cornalba Griffiths Harris - "Geometry of Algebraic Curves" vol 1 and vol 2. It is a classic and although the flavor is clearly of typed notes, it is by far the shortest and manageable book on curves, which serves as a very nice introduction to the whole subject. A second volume is on its way on cohomology.įulton - "Algebraic Curves, an Introduction to Algebraic Geometry" which can be found here. It does a great job complementing Hartshorne's treatment of schemes, above all because of the more solvable exercises. Tons of stuff on schemes more complete than Mumford's Red Book. Görtz Wedhorn - Algebraic Geometry I, Schemes with Examples and Exercises. It is the best free book you need to get enough algebraic geometry to understand the other titles. Just amazing notes short but very complete, dealing even with schemes and cohomology and proving Riemann-Roch. Gathmann - "Algebraic Geometry" which can be found here. Also useful coming from studies on several complex variables. By far the best for a complex-geometry-oriented mind. Griffiths Harris - "Principles of Algebraic Geometry". 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. Liu Qing - "Algebraic Geometry and Arithmetic Curves". GRADUATE FOR ALGEBRISTS AND NUMBER THEORISTS: They are the most complete on foundations and introductory into Schemes so they are very useful before more abstract studies. Shafarevich - "Basic Algebraic Geometry" vol. There are very few books like this and they should be a must to start learning the subject. 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. "Lectures on Curves, Surfaces and Projective Varieties" ( errata) which starts from the very beginning with a classical geometric style. But my personal choices for the BEST BOOKS areīeltrametti et al. I think Algebraic Geometry is too broad a subject to choose only one book.
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