By Rick Miranda
During this e-book, 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 innovations may be introduced into play. for this reason, many examples of algebraic curves are awarded within the first chapters. during this manner, the ebook starts off as a primer on Riemann surfaces, with advanced charts and meromorphic capabilities taking heart degree. however the major examples come from projective curves, and slowly yet without doubt 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 equipment within the latter chapters, in order that their application and naturalness are instantly noticeable. Requiring a historical past of a one semester of complicated variable! idea and a yr of summary algebra, this can be a very good graduate textbook for a second-semester path in complicated variables or a year-long direction in algebraic geometry.
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