NetNews Usenet Archive 1992 #27/NN_1992_27.iso/spool/sci/math/research/564

Text File | 1992-11-20 | 1.7 KB | 47 lines

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From: [e-mail] (Emil Volcheck)
Subject: reference sought for plane birational transformation (TeX)
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Sender: Daniel Grayson <[e-mail]>
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Organization: UCLA Mathematics Department
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Date: Thu, 19 Nov 1992 01:25:51 GMT
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I have been investigating a plane birational transformation specified
in a paper by Max Noether in {\it Mathematische Annalen.}  Noether claims
several properties for it which relate to resolving the singularities
of a curve into ordinary multiple points.  It is an extension of the
standard method of quadratic transformations described by Fulton, for instance.

Noether neither justified his claims nor gave any reference to other works
which describe this transformation.  I hope that someone might be able to
tell me a reference that might relate to this particular transformation.
Here's the definition in homogeneous coordinates as he gave it. The notation
might seem a little awkward.  It transforms a curve $f(x_1: x_2: x_3)=0$,
and the new coordinate system is in coordinates $\xi_i$.  Subscripts
denote degree.

$$
\xi_1:\xi_2:\xi_3 = x_1 \cdot S_{s-1} : x_2 \cdot S_{s-1} : S_s,
$$
where
$$
S_{s-1}  =  x_3 U_{s-2}(x_1,x_2) + U_{s-1}(x_1,x_2),
$$
and
$$
S_s      =  x_3 V_{s-1}(x_1,x_2) + V_s(x_1,x_2),
$$
with $U_h,$ $V_h$ homogeneous polynomials of degree $h$.

Thanks in advance!
\bye
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