By Rick Miranda
During this e-book, Miranda takes the method that algebraic curves are most sensible encountered for the 1st time over the complicated numbers, the place the reader's classical instinct approximately surfaces, integration, and different strategies may be introduced into play. for this reason, many examples of algebraic curves are provided within the first chapters. during this method, the booklet starts off as a primer on Riemann surfaces, with advanced charts and meromorphic services 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 provided in an algebraic demeanour, through an variation of the adelic facts, expressed thoroughly by way of 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 heritage of a one semester of complicated variable! conception and a yr of summary algebra, this is often a very good graduate textbook for a second-semester direction in advanced variables or a year-long direction in algebraic geometry.
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