Singularities of Plane Curves (London Mathematical Society Lecture Note Series, Series Number 276)

This Book Provides A Comprehensive And Self-contained Exposition Of The Algebro-geometric Theory Of Singularities Of Plane Curves, Covering Both Its Classical And Its Modern Aspects. The Book Gives A Unified Treatment, With Complete Proofs, Presenting Modern Results Which Have Only Ever Appeared In Research Papers. It Updates And Correctly Proves A Number Of Important Classical Results For Which There Was Formerly No Suitable Reference, And Includes New, Previously Unpublished Results As Well As Applications To Algebra And Algebraic Geometry. This Book Will Be Useful As A Reference Text For Researchers In The Field. It Is Also Suitable As A Textbook For Postgraduate Courses On Singularities, Or As A Supplementary Text For Courses On Algebraic Geometry (algebraic Curves) Or Commutative Algebra (valuations, Complete Ideals). Eduardo Casas-alvero. Includes Bibliographical References And Index. This comprehensive and self-contained exposition of the algebro-geometric theory of singularities of plane curves covers both the classical and modern aspects of the field. It gives a unified treatment with complete proofs and presents modern results which have only appeared in research papers. It updates and correctly proves a number of important classical results for which there was formerly no suitable reference. With new, previously unpublished results as well as applications to algebra and algebraic geometry, this book will be useful as a reference text for researchers in the field. It is also suitable as a textbook for postgraduate courses on singularities. Cover; Title; Copyright; Dedication; Contents; Preface; 0 Preliminaries; 0.1 Projective spaces; 0.2 Power series; 0.3 Surfaces, local coordinates; 0.4 Morphisms; 0.5 Local rings; 0.6 Tangent and cotangent spaces; 0.7 Curves; 0.8 Germs of curves; 0.9 Multiplicity and tangent cone; 0.10 Smooth germs; 0.11 Examples of singular germs; 0.11.1 Nodes; 0.11.2 Cusps; 0.11.3 Tacnodes; 0.11.4 Ordinary singularities; 0.11.5 Higher-order cusps; 1 Newton-Puiseux algorithm; 1.1 Newton polygon; 1.2 Fractionary power series; 1.3 Search for y-roots of f(x, y); 1.4 The Newton-Puiseux algorithm This chapter is devoted to setting our general assumptions and conventions, to fixing notations and to recalling some basic notions and results in the form to be used throughout this book.