Singularities of plane curves [E-Book] / Eduardo Casas-Alvero.
Casas-Alvero, E., (author)
Cambridge : Cambridge University Press, 2000
1 online resource (xv, 345 pages)
englisch
9780521789592
9780511569326
London Mathematical Society lecture note series ; 276
Full Text
Table of 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
  • 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.