Algebraic Surfaces pp 207-247 | Cite as

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

Chapter

## Abstract

Let . , of the function

*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)

*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 (*x*_{1},*y*), (*x*_{2},*y*),…,(*x*_{n},*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

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1995