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Singularities of plane curves - An
Example
ring s = 0,(x,y),ds;
poly f = y2-2x28y-4x21y17+4x14y33-8x7y49+x56+20y65+4x49y16;
LIB "sing.lib";
LIB "classify.lib";
classify(f);
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==>
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The singularity is R-equivalent to A[2260]: y2+x2261
Milnor number(f) = 2260
modality(f) = 0
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ring r = 0,(x,y),dp;
poly f = fetch(s,f);
tjurina(f);
Hence, Tglobal(f) = Tlocal(f)
(= local Milnor number for
Ak-sing.).
ring sh = 0,(x,y,z),dp;
poly f = fetch(s,f);
poly F = homog(f,z); //
homogeneous polynomial defining C
ring r1 = 0,(y,z),dp;
map phi = sh,1,y,z;
poly g = phi(F); // F in
affine chart (x=1)
tjurina(g);
ring r2 = 0,(y,z),ds; //
local ring at (1:0:0)
poly g = fetch(r1,g);
tjurina(g);
We conclude: there is (precisely) 1 singularity of C
at infinity.
(We have considered all points at
infinity except (0:1:0) which is obviously not on C.)
Topological type of the singularity at (1:0:0) : x9-y16 = 0.
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