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Branch Curves of Multiple Planes and Continuous Systems of Plane Algebraic Curves

  • Sheeram Shankar Abhyankar
  • David Mumford
Part of the Classics in Mathematics book series (CLASSICS, volume 61)

Abstract

Let z be a k-valued algebraic function of two complex variables x and y, defined by an irreducible algebraic equation,
$$F(x,y,z){\rm{ }} = {\rm{ }}0$$
(1)
. The branch curve f,
$$f(x,y){\rm{ }} = {\rm{ }}0$$
(2)
, of the function z is found by eliminating z between F = 0 and \(\partial F/\partial z = 0\) and by neglecting in the resultant certain factors which correspond to multiple curves of the surface F = 0 (apparent branch curves) The definition of f may be rendered exact by assuming that: (a) the polynomial f contains no multiple factors; (b) the curve f is the locus of the effective branch points (x1, y), (x2, y),…,(xn, y) of the function z = z(x,y), for y fixed, and of the lines y = c = const. such that y = c is an effective branch point of z if x is fixed and generic. It may be necessary to include the line at infinity of the projective plane (x,y) in the branch curve. However, we may always choose the coördinates x and y in such a manner that the line at infinity does not belong to the branch curve.

Keywords

Modulus Space Fundamental Group Irreducible Component Algebraic Variety Plane Curf 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Sheeram Shankar Abhyankar
  • David Mumford

There are no affiliations available

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