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学习代数几何的四条途径

2010-12-23 4页 doc 32KB 15阅读

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学习代数几何的四条途径学习代数几何的四条途径 学习代数几何的四条途径                                        In educating mathematicians about a new area, it is worthwhile to distinguish between the courtship period and what comes after. Some books are written to make you fall in love with a subject; others are...
学习代数几何的四条途径
学习代数几何的四条途径 学习代数几何的四条途径                                        In educating mathematicians about a new area, it is worthwhile to distinguish between the courtship period and what comes after. Some books are written to make you fall in love with a subject; others are written to tell you what you need to know in order to live with it.   I vividly remember how I fell in love with algebraic geometry. I was a first-year graduate student at Princeton, heading for winter vacation. Thus far at Princeton, I had seen some lovely but quite abstract mathematics, but some good soul put a copy of Robert J. Walker's Algebraic Curves [10], a little Dover paperback, into my hands just before I left. When I reached Noether's theorem and its applications to the proof of Pascal's theorem and the group law for a plane cubic, something indefinable happened to me-admittedly, I was on a beach in the Virgin Islands, which may have had an effect-and somewhere in my brain the feeling that every author hopes to evoke was aroused: "I want to know more."  Algebraic geometry is not an easy subject either to teach or to learn. A list of worthwhile topics to know before undertaking the study of algebraic geometry could include commutative algebra, complex analysis, algebraic and differential topology, number theory, elliptic operator theory, K-theory, differential geometry, category theory, Lie groups and representation theory, several complex variables, and lattice polyhedra, with the recent essential addition of elementary particle theory. At the time, I had a decent understanding of perhaps two of these. Where to start? I did not know it then, but the issue of how to help students get started learning algebraic geometry is a question that almost every practicing algebraic geometer would answer differently. Furthermore, at that time, books had not yet been written for many of the paths to algebraic geometry that are commonly trodden now. With the benefit of hindsight, I would distinguish four basic approaches:   1. Schemes. Known to its adherents as "doing it right the first time," the machinery of schemes was developed by Grothendieck in the 1960s, building on earlier work of Serre and Weil. A digestible first taste of the philosophy of schemes was provided by a set of lecture notes by Mumford distributed by the Harvard mathematics department and now available commercially [7]. (Its nickname, the "Little Red Book," alluded not only to its crimson cover but to the famous book of quotations of Mao Zedong.) Hartshorne [5] has endured as the best one-volume treatment of this essential set of tools. Everyone in algebraic geometry eventually studies this book. Some with a strong algebraic background begin with it; others come to grips with it only after dipping into one of the three other approaches. My own recommendation to the student would be to experience at least one of the other approaches first, in order to provide motivation for the questions to which the very powerful techniques of scheme theory can be applied. 2. One dimension at a time. The idea here is to start by learning about algebraic curves. This allows a student to reach beautiful results rather quickly, because the technical issues are much easier in one dimension than higher up. Done well, it begins the long process of developing a good geometric intuition.   An important choice here is whether or not to consider the transcendental part of algebraic geometry. Ever since Abel's work on elliptic integrals, integration and the use of transcendental functions have provided important techniques, beautiful theorems, and fascinating problems. For algebraic curves, this question takes the form of whether to set as a goal the Riemann-Roch theorem (an algebraic result, though originally proved analytically) and the results I mentioned earlier that attracted me to the subject, or whether to incorporate the Abel-Jacobi theorem and perhaps the geometry of the On -divisor. The algebraic approach to algebraic curves is well embodied in Fulton [10]. Weyl [12] is the first exemplar in the modern literature of the transcendental treatment, with a host of excellent successors covering a similar body of material on Riemann surfaces. Siegel [9] is perhaps the best for developing an intuitive feeling for how to use complex analysis in algebraic geometry, while Gunning [3] provides a gentle introduction to sheaf theory as a way to develop Riemann surface theory.  The algebraic/transcendental dichotomy is another form of the basic divide as to whether, for pedagogical purposes, one wants to consider algebraic geometry as a natural continuation of commutative algebra or of complex analysis, that is, whether a prospective student should go through the algebraic or the analytic/geometric "door" to the subject. Historically, as the name chosen indicates, mathematicians approached Riemann surfaces through the analytic/geometric door before the algebraic door had been constructed. My own view of this is that both gateways are necessary, and that the student should enter both of them multiple times. The algebraic door unquestionably leads to some of the most powerful techniques available to modern algebraic geometers. The analytic/geometric door has provided an ever-renewed source of new methods and has fueled major advances in the subject by allowing access to the full fecundity of the geometric imagination. That said, there are mathematicians who have made imperishable contributions to the subject while sticking to one approach or the other.   3. Geometric examples and constructions. An abiding feature of algebraic geometry, inherited from the amazing fertility of nineteenth-century geometry, is an extraordinary array of intricate and beautiful geometric constructions: the cross-ratio, the twisted cubic, the Segre and Veronese embeddings, the Grassmannian, Schubert cycles and the Plucker embedding, and so forth. What better way to get to know the subject than to feast on these morsels and, in doing so, to develop a strong geometric intuition? One of the first introductory books to marry this approach with algebraic techniques was Hodge and Pedoe [61, to which Harris [4] is a fitting successor. One does not come away with from these books immediately armed for battle with a large arsenal of techniques, but they furnish an extremely valuable component of the education of an algebraic geometer.   4. Several complex variables. This is the modern approach to transcendental algebraic geometry, providing a mix of geometric insight and analytic technique, notably harmonic forms and Kahler geometry. Weil [11] blazed a trail here, and the remarkable text of Griffiths and Harris [2] has endured as a highly inspiring introduction to transcendental geometry. There is no single book or course that prepares a student to read the much-feared chapter 0 of [2]; what is needed, rather, is a certain general level of geometric sophistication. The student should be prepared to dip into books on topology, manifolds, and differential geometry in order to pick up whatever is missing in his or her background. Some books combine features from different realms. For example, Shafarevich [81 touches upon all four approaches。 REFERENCES 1. W. Fulton, Algebraic Curves, Benjamin, New York, 1969. 2. P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley, New York, 1978. 3. R. C. Gunning, Lectures on Riemann Surfaces, Princeton University Press, Princeton, 1966. 4. J. Harris, Algebraic Geometry: a First Course, Springer-Verlag, New York, 1992. 5. R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977. 6. W. V D. Hodge and D. Pedoe, Methods of Algebraic Geometry, Cambridge University Press, Cambridge, 1947. 7. D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Mathematics, vol. 1358, SpringerVerlag, New York, 1988. 8. I. R. Shafarevich, Basic Algebraic Geometry, 2nd ed., Springer-Verlag, New York, 1994. 9. C. L. Siegel, Topics in Complex Function Theory, Wiley-Interscience, New York, 1969-73. 10. R. J. Walker, Algebraic Curves, Princeton University Press, Princeton, 1950; reprinted by Dover, New York, 1962, and by Springer-Verlag, New York, 1978. 11. A. Weil, Introduction a l'Etude des Varietes Kahleriennes, Hermann, Paris, 1958. 12. H. Weyl, Die Idee der Riemannschen Flache, Teubner, Leipzig, 1913; translated as The Concept of a Riemann Surface, Addison-Wesley, Reading, MA, 1964.
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