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**Introduction to algebraic curves. Transl. from the Chinese by Kuniko Weltin.
2nd ed.**
*(English)*
Zbl 0873.14030

Translations of Mathematical Monographs. 76. Providence, RI: American Mathematical Society (AMS). x, 225 p. (1996).

The present book is the second edition of an English translation of the original text. The first edition appeared in 1989 (Zbl 0696.14012). The text on its own has been left completely intact and unchanged. However, having put it in a modern type-setting, the editors have taken the occasion to eliminate the numerous (trivial) misprints in the original and, to such a degree, improve the outward aesthetical appearance of this great book. Also, being now available as a relatively inexpensive paperback edition, this very special introductory text on the theory of complex algebraic curves has finally become affordable for less wealthy prospective customers.

Now as before, Griffiths’s elementary crash course on the geometry of complex curves, which was taught at the University of Beijing (China) in the summer of 1982 and first published in Chinese, represents one of the most efficient modern introductions to this fascinating classical subject. Keeping the technical requirements to an absolute minimum, and even relinquishing sheaf theory and cohomological methods, the author leads the beginner to the deep classical theorems of Abel, Jacobi, Riemann-Hurwitz, Riemann-Roch, and others, basically so by laying emphasis on the purely geometric and complex-analytic aspects of the theory. In regard of this particularly beginner-friendly approach, Griffiths’s textbook will certainly maintain its timelessly unique character of being an excellent and thorough guide to the more advanced topics in algebraic curve theory, such as they are comprehensively treated in the two-volume monograph “Geometry of algebraic curves” by E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris [Vol. I: Grundlehren Math. Wiss. 267 (1985; Zbl 0559.14017); Vol. II: to appear]. – As to a related, however more recent and somewhat deeper-going introduction to complex curve theory, the interested reader is referred to R. Miranda’s textbook “Algebraic curves and Riemann surfaces” [Graduate Stud. Math. 5 (1995; Zbl 0820.14022)], whereas the more algebraic and arithmetic aspects of algebraic curve theory are excellently (and complementarily) provided by D. Lorenzini’s “An invitation to arithmetic geometry” [Graduate Stud. Math. 9 (1995; Zbl 0847.14013)].

Now as before, Griffiths’s elementary crash course on the geometry of complex curves, which was taught at the University of Beijing (China) in the summer of 1982 and first published in Chinese, represents one of the most efficient modern introductions to this fascinating classical subject. Keeping the technical requirements to an absolute minimum, and even relinquishing sheaf theory and cohomological methods, the author leads the beginner to the deep classical theorems of Abel, Jacobi, Riemann-Hurwitz, Riemann-Roch, and others, basically so by laying emphasis on the purely geometric and complex-analytic aspects of the theory. In regard of this particularly beginner-friendly approach, Griffiths’s textbook will certainly maintain its timelessly unique character of being an excellent and thorough guide to the more advanced topics in algebraic curve theory, such as they are comprehensively treated in the two-volume monograph “Geometry of algebraic curves” by E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris [Vol. I: Grundlehren Math. Wiss. 267 (1985; Zbl 0559.14017); Vol. II: to appear]. – As to a related, however more recent and somewhat deeper-going introduction to complex curve theory, the interested reader is referred to R. Miranda’s textbook “Algebraic curves and Riemann surfaces” [Graduate Stud. Math. 5 (1995; Zbl 0820.14022)], whereas the more algebraic and arithmetic aspects of algebraic curve theory are excellently (and complementarily) provided by D. Lorenzini’s “An invitation to arithmetic geometry” [Graduate Stud. Math. 9 (1995; Zbl 0847.14013)].

Reviewer: W.Kleinert (Berlin)

### MSC:

14Hxx | Curves in algebraic geometry |

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

30F10 | Compact Riemann surfaces and uniformization |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |