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Algebraic geometry ii mumford12/9/2023 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 and introduces many concepts useful to motivate more advanced courses.Īrbarello Cornalba Griffiths Harris - "Geometry of Algebraic Curves" vol 1 and 2. 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 a great job complementing Hartshorne's treatment of schemes, above all because of the more solvable exercises.įulton - "Algebraic Curves, an Introduction to Algebraic Geometry" which can be found here. Tons of stuff on schemes more complete than Mumford's Red Book (For an online free alternative check Mumfords' Algebraic Geometry II unpublished notes on schemes.). Görtz Wedhorn - Algebraic Geometry I, Schemes with Examples and Exercises. It develops a lot of algebraic geometry without so much advanced commutative and homological algebra as the modern books tend to emphasize. Also useful coming from studies on several complex variables or differential geometry. 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: (A link to all versions the latest is of 2017.) It is the best free course in my opinion, to get enough algebraic geometry background to understand the other more advanced and abstract titles.įor an abstract algebraic approach, the nice, long notes by Ravi Vakil is found here. The latest is of 2019.) Just amazing notes short but very complete, dealing even with schemes and cohomology and proving Riemann-Roch and even hinting Hirzebruch-R-R. Gathmann - "Algebraic Geometry" (All versions are found here. The second half then jumps into a categorical introduction to schemes, bits of cohomology and even glimpses of intersection theory. This new title is wonderful: it starts by introducing algebraic affine and projective curves and varieties and builds the theory up in the first half of the book as the perfect introduction to Hartshorne's chapter I. Holme - "A Royal Road to Algebraic Geometry". 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. But the problems are hard for many beginners. 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. 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" which starts from the very beginning with a classical geometric style. 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:īeltrametti-Carletti-Gallarati-Monti. Therefore I find the attempt to reduce his/her study to just one book (besides Hartshorne's) too hard and unpractical. 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. 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. The (penultimate) draft version of their book contains an updated version of Mumford's original drawing on page 120.I think Algebraic Geometry is too broad a subject to choose only one book. This is a collection of "charts" for the affine scheme $\Spec\mathbb$ is a modified version of Mumford's 1967 map and is discussed by Lieven Le Bruyn Manin's geometric axis.Ģ015, David Mumford and Tadao Oda, Algebraic Geometry II
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