Algebraic curves and compact Riemann surfaces comprise the most
developed and arguably the most beautiful portion of algebraic
geometry. However, the majority of books written on the subject
discuss algebraic curves and compact Riemann surfaces separately,
as parts of distinct general theories. Most texts and university
courses on curve theory generally conclude with the Riemann-Roch
theorem, despite the fact that this theorem is the gateway to some
of the most fascinating results in the theory of algebraic
curves.This book is based on a six-week series of lectures
presented by the author to third- and fourth-year undergraduates
and graduate students at Beijing University in 1982. The lectures
began with minimal technical requirements (a working knowledge of
elementary complex function theory and algebra together with some
exposure to topology of compact surfaces) and proceeded directly to
the Riemann-Roch and Abel theorems. This book differs from a number
of recent books on this subject in that it combines analytic and
geometric methods at the outset, so that the reader can grasp the
basic results of the subject. Although such modern techniques of
sheaf theory, cohomology, and commutative algebra are not covered
here, the book provides a solid foundation to proceed to more
advanced texts in general algebraic geometry, complex manifolds,
and Riemann surfaces, as well as algebraic curves. Containing
numerous exercises and two exams, this book would make an excellent
introductory text.
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