Algebraic Curves and Riemann SurfacesAmerican Mathematical Soc., 1995 - 390 pages In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking centre stage. But the main examples come fromprojective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Dualtiy Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves andcohomology are introduced as a unifying device in the later chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one term of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-term course in complex variables or a year-long course in algebraic geometry. |
Table des matières
| 1 | |
Chapter II Functions and Maps | 21 |
Chapter III More Examples of Riemann Surfaces | 57 |
Chapter IV Integration on Riemann Surfaces | 105 |
Chapter V Divisors and Meromorphic Functions | 129 |
Chapter VI Algebraic Curves and the RiemannRoch Theorem | 169 |
Chapter VII Applications of RiemannRoch | 195 |
Chapter VIII Abels Theorem | 247 |
Chapter IX Sheaves and Cech Cohomology | 269 |
Chapter X Algebraic Sheaves | 309 |
Chapter XI Invertible Sheaves Line Bundles and Hsup1 | 323 |
References | 371 |
| 377 | |
Back Cover | 391 |
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Expressions et termes fréquents
affine plane curve algebraic curve atlas automorphisms base point branch points canonical divisor cocycle cohomology groups compact Riemann surface complete linear system complex charts complex torus computation COROLLARY curve of degree curve of genus defined DEFINITION deg(D denoted dim L(D dimension div(f div(H equation exactly EXAMPLE finite formula function f genus g gives global meromorphic function Hence holomorphic 1-form holomorphic functions holomorphic map homogeneous polynomial hyperelliptic hyperplane hyperplane divisor identically zero integral invertible sheaf isomorphism Jac(X kernel Laurent series Laurent tail divisor LEMMA Let F line bundle line bundle charts linearly equivalent map F meromorphic 1-form meromorphic function Moreover multiplicity neighborhood nonconstant nonzero Note open covering open set open subset ordp(f path preimages presheaf projective plane curve projective space PROOF PROPOSITION quotient rational function Riemann Sphere sending sheaf map sheaves Show smooth projective curve subgroup Suppose topology transition functions trivial vanish