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**Introduction to algebraic curves. Transl. from the Chinese by Kuniko Weltin.**
*(English)*
Zbl 0696.14012

Translations of Mathematical Monographs, 76. Providence, RI: American Mathematical Society (AMS). x, 221 p. $ 93.00 (1989).

The book under review is based on a series of lectures given by the author at Beijing University in the summer of 1982. Assuming only minimal acquaintance with algebra and functions of a complex variable and using analytic and (to a less extent) geometric tools, it provides a quick introduction to the theory of algebraic curves and compact Riemann surfaces.

After reviewing preliminaries and establishing some basic results (like Bezout’s theorem and the Riemann-Hurwitz formula) the author passes to the Riemann-Roch and Abel theorems which occupy a central position in the book. Having proved the Riemann-Roch theorem, he proceeds with giving some applications including canonical and hyperelliptic curves and classification of curves of low genera (up to genus 4). The last chapter is devoted to Jacobian varieties and Abel’s theorem and also contains a brief exposition of the elementary theory of elliptic curves.

Throughout the book there are numerous exercises, and the book ends with test problems for exams. On the whole, this can be the first book for an undergraduate student wishing to specialize in algebraic geometry or complex manifolds, but one should keep in mind that of the three possible approaches to the subject, viz. algebraic, geometric and analytic, preference is clearly given to the last one so that people interested in algebra and arithmetic might prefer a different viewpoint.

After reviewing preliminaries and establishing some basic results (like Bezout’s theorem and the Riemann-Hurwitz formula) the author passes to the Riemann-Roch and Abel theorems which occupy a central position in the book. Having proved the Riemann-Roch theorem, he proceeds with giving some applications including canonical and hyperelliptic curves and classification of curves of low genera (up to genus 4). The last chapter is devoted to Jacobian varieties and Abel’s theorem and also contains a brief exposition of the elementary theory of elliptic curves.

Throughout the book there are numerous exercises, and the book ends with test problems for exams. On the whole, this can be the first book for an undergraduate student wishing to specialize in algebraic geometry or complex manifolds, but one should keep in mind that of the three possible approaches to the subject, viz. algebraic, geometric and analytic, preference is clearly given to the last one so that people interested in algebra and arithmetic might prefer a different viewpoint.

Reviewer: F.L.Zak

### MSC:

14Hxx | Curves in algebraic geometry |

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

30F10 | Compact Riemann surfaces and uniformization |