The function decomposes the syzygy scheme of the extra syzygy into its component. The six quadrics involved in the syzygy vanish in the union of C with an elliptic normal curve E and a point pt.
i1 : p=10007, R=ZZ/p[x_0..x_6];
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i2 : time I = randomOneNodalPrymCanonicalCurveOfGenus8 R;
-- used 10.4711 seconds
o2 : Ideal of R
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i3 : betti res I
0 1 2 3 4 5
o3 = total: 1 8 36 56 35 8
0: 1 . . . . .
1: . 7 1 . . .
2: . 1 35 56 35 8
o3 : BettiTally
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i4 : (dim I ,degree I, genus I)==(2,14,8)
o4 = true
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i5 : (pt,E)=extraSyzygyInGenus8 I ;
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i6 : (dim E, degree E, genus E)==(2,7,1)
o6 = true
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i7 : betti res E
0 1 2 3 4 5
o7 = total: 1 14 35 35 14 1
0: 1 . . . . .
1: . 14 35 35 14 .
2: . . . . . 1
o7 : BettiTally
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i8 : (dim pt, degree pt)==(1,1)
o8 = true
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i9 : halfCan=intersect(E,I);
o9 : Ideal of R
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i10 : betti res halfCan
0 1 2 3 4 5
o10 = total: 1 7 22 22 7 1
0: 1 . . . . .
1: . 6 1 . . .
2: . 1 21 21 1 .
3: . . . 1 6 .
4: . . . . . 1
o10 : BettiTally
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i11 : (dim halfCan, degree halfCan, genus halfCan)==(2,21,22)
o11 = true
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i12 : betti res intersect(E,I,pt)
0 1 2 3 4 5 6
o12 = total: 1 6 16 27 22 7 1
0: 1 . . . . . .
1: . 6 1 . . . .
2: . . 15 6 1 . .
3: . . . 21 21 6 1
4: . . . . . 1 .
o12 : BettiTally
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i13 : (dim(I+E),degree(I+E))==(1,14)
o13 = true
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