Geometry of projective algebraic curves. (English) Zbl 0556.14012

Namba, Makoto

Pure and Applied Mathematics, Vol. 88. New York - Basel: Marcel Dekker, Inc. X, 409 p. $ 69.75 (U.S. & Canada); $ 83.50; SFr. 195.00 (all other countries) (1984).

This intriguing and unorthodox text on algebraic curves consists of two approximately equal parts. The first, on extrinsic geometry, is a richly illustrated and geometrically oriented treatment of projective curves covering the projective geometry of conics, Bezout’s theorem, linear systems, dual curves and Plücker’s formula followed by an extended discussion of singular plane quartics, plane quintics, and space curves. - The second part treats complex manifolds, compact Riemann surfaces including a section on elliptic functions, and culminates with an introduction to Jacobian varieties which leads into current research on linear systems.
The author has adopted a ”dialectic” construction where results from the second part is extensively used in the first part. Thus, although the text contains a large number of problems, it would be difficult to use for a conventional course. It is, however a rich and interesting source of results and examples which could be very useful in the preparation of a course and warmly recommended for collateral reading and browsing. It is unfortunate that the publisher’s pricing policy puts the book out of reach of most individuals.

Reviewer: H.H.Martens

MSC:

14H45	Special algebraic curves and curves of low genus
14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry
30Fxx	Riemann surfaces
14Nxx	Projective and enumerative algebraic geometry
30-02	Research exposition (monographs, survey articles) pertaining to functions of a complex variable

Keywords:

conics; linear systems; dual curves; compact Riemann surfaces