By Rick Miranda
During this ebook, Miranda takes the procedure 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 innovations should be introduced into play. for this reason, many examples of algebraic curves are provided within the first chapters. during this manner, the publication starts off as a primer on Riemann surfaces, with advanced charts and meromorphic features taking middle degree. however the major examples come from projective curves, and slowly yet without doubt the textual content strikes towards the algebraic class. Proofs of the Riemann-Roch and Serre Duality Theorems are awarded 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 visible. Requiring a historical past of a one semester of advanced variable! idea and a 12 months of summary algebra, this can be a superb graduate textbook for a second-semester path in complicated variables or a year-long direction in algebraic geometry.
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