By Rick Miranda
During this e-book, Miranda takes the technique that algebraic curves are top encountered for the 1st time over the complicated numbers, the place the reader's classical instinct approximately surfaces, integration, and different strategies will be introduced into play. for that reason, many examples of algebraic curves are offered within the first chapters. during this means, the e-book starts as a primer on Riemann surfaces, with advanced charts and meromorphic features taking middle level. however the major examples come from projective curves, and slowly yet definitely the textual content strikes towards the algebraic classification. Proofs of the Riemann-Roch and Serre Duality Theorems are offered in an algebraic demeanour, through an version of the adelic facts, expressed thoroughly when it comes to fixing a Mittag-Leffler challenge. Sheaves and cohomology are brought as a unifying machine within the latter chapters, in order that their application and naturalness are instantly visible. Requiring a history of a one semester of complicated variable! thought and a yr of summary algebra, this is often an outstanding graduate textbook for a second-semester direction in advanced variables or a year-long direction in algebraic geometry.
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