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Algebraic Curves and Surfaces

Laurent Busé , Fabrizio Catanese , Elisa Postinghel
Publication Date: 
Number of Pages: 
SISSA Springer Series
[Reviewed by
Felipe Zaldivar
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The geometry of algebraic curves and surfaces is a wide and venerable subject, and there are many monographs and textbooks aimed to graduate students and experienced researchers. The novelties of the book under review include a historical overview of the classification theorem of algebraic surfaces, from the classical results of the Italian school of Castelnuovo and Enriques to the methods of the French school of Grothendieck and Serre in the hands of Mumford and Bombieri and the Russian school of Shafarevich and his students.
The first chapter is a focused approach to the classification of algebraic surfaces according to the value of the twelfth plurigenus (a more nuanced classification than the usual one using the Kodaira dimension). The results are formulated and proved in the modern language, allowing the author a few paragraphs for the conjectural analogues in higher dimensions.
The second chapter’s topic is the dimension of the linear systems of plane curves and hypersurfaces in complex projective spaces. Again, a topic studied initially by B. Segre and G. Castelnuovo and still wide open.
The third and last chapter considers equations that parametrize the image of a curve or surface, a topic with roots in classical elimination theory and now studied with the methods of homological algebra, and with a renewed interest for its applications in geometric modeling of 3D shapes.
The book can be used as a gentle introduction to the theory of algebraic surfaces in conjunction with more systematic textbooks such as C. Ciliberto’s Classification of Complex Algebraic Surfaces.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is