Singularities of plane curves / Eduardo Casas-Alvero.

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Projective spaces -- Power series -- Surfaces, local coordinates -- Morphisms -- Local rings -- Tangent and cotangent spaces -- Curves -- Germs of curves -- Multiplicity and tangent cone -- Smooth germs -- Examples of singular germs -- Newton--Puiseux algorithm -- Newton polygon -- Fractionary power series -- Search for y-roots of f(x, y) -- The Newton-Puiseux algorithm -- Puiseux theorem -- Separation of y-roots -- The case of convergent series -- Algebraic properties of C{x, y} -- First local properties of plane curves -- The branches of a germ -- The Puiseux series of a germ -- Points on curves around O -- Local rings of germs -- Parameterizing branches -- Intersection multiplicity -- Pencils and linear systems -- Infinitely near points -- Blowing up -- Transforming curves and germs -- Infinitely near points -- Enriques' definition of infinitely near points -- Proximity -- Free and satellite points -- Resolution of singularities -- Equisingularity -- Enriques diagrams -- The ring in the first neighbourhood -- The rings in the successive neighbourhoods -- Artin theorem for plane curves -- Virtual multiplicities -- Curves through a weighted cluster -- When virtual multiplicities are effective -- Blowing up all points in a cluster -- Exceptional divisors and dual graphs -- The totla transform of a curve -- Unloading -- The number of conditions -- Adjoint germs and curves -- Noether's Af + B[phi] theorem -- Analysis of branches -- Characteristic exponents -- The first characteristic exponent.

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number of important classical results for which there was formerly no suitable reference, and includes new, previously unpublished results as well as applications to algebra and algebraic geometry.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).