Singularities of plane curves / Eduardo Casas-Alvero.

Format:

Book

Author/Creator:

Casas-Alvero, E. (Eduardo), 1948- author.

Series:

London Mathematical Society lecture note series ; 276.

London Mathematical Society lecture note series ; 276

Language:

English

Subjects (All):

Curves, Plane.

Singularities (Mathematics).

Physical Description:

1 online resource (xv, 345 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 2000.

Language Note:

English

Summary:

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number of important classical results for which there was formerly no suitable reference, and includes new, previously unpublished results as well as applications to algebra and algebraic geometry.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

Contents:

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

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.

Notes:

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

Includes bibliographical references and index.

ISBN:

1-139-88565-0

1-107-36802-2

1-107-37254-2

1-107-36311-X

1-107-37026-4

1-299-40560-6

1-107-36556-2

0-511-56932-7