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Geometric Genus of Projective
Curves -
SINGULAR
Example
LIB "normal.lib";
ring r = 32003,(x,y,z,w,u),dp;
ideal i = x2+y2+z2+w2+u2, x3+xy2+z3, z4+w4+u4; // a curve in P^4
genus(i);
To obtain more information on the performed computations, you should
increase the printlevel:
printlevel=3;
genus(i);
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==>
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The ideal of the projective curve:
J[1]=x2+y2+z2+w2+u2
J[2]=xz2-z3+xw2+xu2
J[3]=z4+w4+u4
J[4]=y2z2+y2w2-xzw2+2z2w2-w4+y2u2-xzu2+2z2u2+2w2u2-u4
J[5]=z3w2-2xw4+zw4+z3u2-2xw2u2-2xu4+zu4
J[6]=xzw4+16001z2w4-16001w6+xzw2u2-16001w4u2+xzu4+16001z2u4-16001w2u4-16001u6
J[7]=y2w4+z2w4-16001w6+y2w2u2-xzw2u2+z2w2u2-16000w4u2+y2u4+z2u4-16000w2u4-16001u6
The coefficients of the Hilbert polynomial: -48,24
arithmetic genus: 49
degree: 24
the projected curve:
1901x24+6354x22y2-7492x20y4- ... many terms ... +4y2t22+3540xt23+t24
the arithmetic genus of the plane curve: 253
analyse the singularities
......
many data
......
The projected plane curve has locally:
singularities: 109
branches: 232
nodes: 108
cusps: 0
Tjurina number: 300
Milnor number: 349
delta of the projected curve: 236
delta of the curve: 32
genus: 17
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