According to Mukai [Mu] any smooth curve of genus 8 and Clifford index 3 is the transversal intersection C=ℙ7 ∩ G(2,6) ⊂ ℙ15. In particular this is true for the general curve of genus 8. Picking 8 points in the Grassmannian G(2,6) at random and ℙ7 as their span gives the result.
i1 : setRandomSeed("alpha");
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i2 : FF=ZZ/10007; |
i3 : S=FF[x_0..x_7]; |
i4 : (I,points)=randomCanonicalCurveGenus8with8Points S; |
i5 : betti res I
0 1 2 3 4 5 6
o5 = total: 1 15 35 42 35 15 1
0: 1 . . . . . .
1: . 15 35 21 . . .
2: . . . 21 35 15 .
3: . . . . . . 1
o5 : BettiTally
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i6 : points
o6 = {ideal (x + 315x , x + 1084x , x - 2032x , x + 2800x , x - 2677x ,
6 7 5 7 4 7 3 7 2 7
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x + 1089x , x + 4275x ), ideal (x + 3175x , x + 4236x , x - 752x ,
1 7 0 7 6 7 5 7 4 7
------------------------------------------------------------------------
x + 1946x , x + 2344x , x + 4872x , x - 2944x ), ideal (x + 2063x ,
3 7 2 7 1 7 0 7 6 7
------------------------------------------------------------------------
x - 694x , x - 114x , x + 930x , x - 3445x , x + 1502x , x -
5 7 4 7 3 7 2 7 1 7 0
------------------------------------------------------------------------
2284x ), ideal (x - 3009x , x - 77x , x - 3932x , x - 199x , x -
7 6 7 5 7 4 7 3 7 2
------------------------------------------------------------------------
4949x , x + 4697x , x - 4337x ), ideal (x - 914x , x - 3912x , x -
7 1 7 0 7 6 7 5 7 4
------------------------------------------------------------------------
2686x , x + 2847x , x + 3526x , x - 1868x , x + 4707x ), ideal (x +
7 3 7 2 7 1 7 0 7 6
------------------------------------------------------------------------
2307x , x - 2134x , x + 259x , x + 3283x , x + 2393x , x - 2654x ,
7 5 7 4 7 3 7 2 7 1 7
------------------------------------------------------------------------
x + 2537x ), ideal (x + 815x , x - 4561x , x - 43x , x + 168x , x
0 7 6 7 5 7 4 7 3 7 2
------------------------------------------------------------------------
- 1494x , x + 4790x , x + 3457x ), ideal (x - 1919x , x + 2630x , x
7 1 7 0 7 6 7 5 7 4
------------------------------------------------------------------------
- 4885x , x + 2450x , x + 2739x , x + 1825x , x - 2642x )}
7 3 7 2 7 1 7 0 7
o6 : List
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