Algebraic Curves and Riemann Surfaces
The book was easy to understand, with many examples. The exercises were well chosen, and served to give further examples and developments of the theory. --William Goldman, University of Maryland 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 center stage. But the main examples come from projective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Duality 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 and cohomology are introduced as a unifying device in the latter chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one semester of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-semester course in complex variables or a year-long course in algebraic geometry.
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Chapter II Functions and Maps
Chapter III More Examples of Riemann Surfaces
Chapter IV Integration on Riemann Surfaces
Chapter V Divisors and Meromorphic Functions
Chapter VI Algebraic Curves and the RiemannRoch Theorem
Chapter VII Applications of RiemannRoch
Chapter VIII Abels Theorem
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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