In Farkas, Ludwig [2008] it is conjectured that a general smooth Prym canonical curve of even genus g ≥6 has a pure resolution, i.e. C embedded by |KC ⊗L| into Pg-2 has a pure resolution, where L is an n-torsion Line bundle. With this function the conjecture can be verified for curve of small g (and levels n) by computing the syzgies of random g-nodal Prym canonical rational curve over a finite ground field and arguing by with semi-continuity of Betti numbers.
It is interesting that the verification fails along this approach for (g,n)=(8,2) and (16,2), but worked fine in all other cases.
i1 : g=8
o1 = 8
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i2 : syzygiesOfPrymCanonicalNodalCurve(8,3,Printing=>true);
expected syzygies:
0 1 2 3 4 5
total: 1 7 35 56 35 8
0: 1 . . . . .
1: . 7 . . . .
2: . . 35 56 35 8
first strand in the example
0 1
total: 1 7
0: 1 .
1: . 7
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i3 : syzygiesOfPrymCanonicalNodalCurve(8,2,Printing=>true);
warning: clearing value of symbol t to allow access to subscripted variables based on it
: debug with expression debug 5504 or with command line option --debug 5504
expected syzygies:
0 1 2 3 4 5
total: 1 7 35 56 35 8
0: 1 . . . . .
1: . 7 . . . .
2: . . 35 56 35 8
first strand in the example
0 1 2
total: 1 8 1
0: 1 . .
1: . 7 1
2: . 1 .
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Why the case of genus 8 and 2 torsion does not have expected syzygies is mysterious is to us:
i4 : (L,I)=randomPrymCanonicalNodalCurve(8,2);
warning: clearing value of symbol t to allow access to subscripted variables based on it
: debug with expression debug 5504 or with command line option --debug 5504
|
i5 : S=L_7
o5 = S
o5 : PolynomialRing
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i6 : betti(fI=res I)
0 1 2 3 4 5
o6 = total: 1 8 36 56 35 8
0: 1 . . . . .
1: . 7 1 . . .
2: . 1 35 56 35 8
o6 : BettiTally
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i7 : pt=fI.dd_2_{0};
8 1
o7 : Matrix S <--- S
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i8 : CE1=ideal((fI.dd_1)*syz transpose (syz transpose pt)_{0,1});
o8 : Ideal of S
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i9 : CE=saturate(CE1,ideal pt);
o9 : Ideal of S
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i10 : degree CE
o10 = 21
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i11 : betti res CE
0 1 2 3 4 5
o11 = total: 1 7 22 22 7 1
0: 1 . . . . .
1: . 6 1 . . .
2: . 1 21 21 1 .
3: . . . 1 6 .
4: . . . . . 1
o11 : BettiTally
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i12 : E=CE:I;
o12 : Ideal of S
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i13 : betti res E
0 1 2 3 4 5
o13 = total: 1 14 35 35 14 1
0: 1 . . . . .
1: . 14 35 35 14 .
2: . . . . . 1
o13 : BettiTally
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i14 : dim E == 2, degree E == 7, genus E == 1
o14 = (true, true, true)
o14 : Sequence
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i15 : dim ideal pt == 1
o15 = true
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i16 : degree (E+I)
o16 = 14
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At least in the example, the syzygy scheme of the extra syzygy is a reducible half canonical curve of degree 21, whose second component is an elliptc normal curve of degree 7.