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Number of Branches: Checking plausibility
LIB "primdec.lib";
ring r=0,(x,y,z),ds;
ideal i=x^4-y*z^2,x*y-z^3,y^2-x^3*z;
qhweight(i);
The isolated space curve singularity is quasihomogeneous.
resolution ires=mres(i,0);
ires;
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==>
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1 3 2
rr <-- rr <-- rr
0 1 2
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It is of Cohen-Macaulay type t=2.
LIB "sing.lib";
T_1(i);
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==>
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// dim T_1 = 13
_[1]=gen(6)+2z*gen(5)
_[2]=gen(4)+3x2*gen(2)
_[3]=gen(3)+gen(1)
_[4]=x*gen(5)-y*gen(2)-z*gen(1)
_[5]=x*gen(1)-z2*gen(2)
_[6]=y*gen(5)+3x2z*gen(2)
_[7]=y*gen(2)-z*gen(1)
_[8]=2y*gen(1)-z2*gen(5)
_[9]=z2*gen(5)
_[10]=z2*gen(1)
_[11]=x3*gen(2)
_[12]=x2z2*gen(2)
_[13]=xz3*gen(2)
_[14]=z4*gen(2)
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The Tjurina number is 13. As the quasihomogeneous space curve singularity is
Cohen-Macaulay of codimension 2, it is unobstructed and hence we can apply the
formula
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Hence the Milnor number is 12, in particular it is
even. By the formula
this contradicts the number of branches (= 2) we computed
before. |
==> We have to decompose the second component further, e.g. using
normalization.
<-- Branches of an isolated space curve singularity
<-- computed via Primary Decomposition
--> computed via Normalization
--> computed via Factorizing Gröbner
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