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Introduction to plane algebraic curves. Translated from the 1991 German edition by Richard G. Belshoff. (English) Zbl 1078.14041
Boston, MA: Birkhäuser (ISBN 0-8176-4381-8/pbk). xiv, 293 p. (2005).
This English edition is the faithful translation of the German lecture notes “Ebene algebraische Kurven” by Professor E. Kunz [Regensb. Trichter 23 (1991; Zbl 0744.14023)], which appeared in 1991 and were extensively reviewed (and praised) back then. Apart from having been written in German, this booklet was only published in the lecture note series of the Mathematics Department of the University of Regensburg, Germany, and therefore it might have found a rather restricted audience only, very much in contrast to what this excellent elaboration of the author’s multiple courses on the subject deserved. For this reason, it is very gratifying for the international mathematical community that, after nearly 15 years, these very special lecture notes on the fascinating, evergreen topic of plane algebraic curves have been made available in English, in book form, and in modern printing. As a thorough review of the mathematically unaltered text has been given in 1991, shortly after the appearance of the German original, may it suffice here to briefly recall the unique peculiarities of the text. First of all, this introduction to plane algebraic curves stresses the algebraic aspects of the subject, rather than the geometric or topological ones. Working over general ground fields, and emphasizing the local theory of plane algebraic curves, the author actually provides an introduction to commutative algebra along the theory of plane curves, with all the algebraic geometry needed in this special case. The intersection theory of plane curves is largely deduced by means of the theory of filtered algebras, their graded rings and Rees algebras, and by applying algebraic residue calculus. This approach culminates in a new, methodologically adapted version of F. K. Schmidt’s [Math. Z. 41, 415–438 (1936; Zbl 0014.34101)] classical proof of the Riemann-Roch theorem for plane curves, thereby strikingly combining classical ideas and modern algebraic methods. Secondly, this text stands out by the author’s notorious writing style characterized by its systematic representation, didactical perfection, comprehensiveness, mathematical rigor, thematic determination, and striving for self-containedness. Like in most of his other textbooks on algebra and algebraic geometry, Professor Kunz focuses on the inseparable interplay between those two branches of mathematics, and again he presents new viewpoints, methods, and connections, together with numerous examples, illustrations, working problems, and hints for further reading. There is no doubt that the international mathematical community, including students and teachers, will welcome the overdue English edition of this masterly textbook as a very special and useful addition to the great standard texts on plane curves.

14H50 Plane and space curves
14H45 Special algebraic curves and curves of low genus
14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
14H20 Singularities of curves, local rings
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
13A02 Graded rings