By Rick Miranda
During this ebook, Miranda takes the strategy that algebraic curves are top encountered for the 1st time over the advanced numbers, the place the reader's classical instinct approximately surfaces, integration, and different ideas might be introduced into play. as a result, many examples of algebraic curves are offered within the first chapters. during this method, the booklet starts as a primer on Riemann surfaces, with advanced charts and meromorphic capabilities 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 provided in an algebraic demeanour, through an edition of the adelic facts, expressed thoroughly when it comes to fixing a Mittag-Leffler challenge. Sheaves and cohomology are brought as a unifying gadget within the latter chapters, in order that their software and naturalness are instantly visible. Requiring a history of a one semester of complicated variable! idea and a yr of summary algebra, this can be an exceptional graduate textbook for a second-semester path in complicated variables or a year-long direction in algebraic geometry.
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