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 : FF=ZZ/10007;S=FF[x_0..x_7]; |
i3 : (I,points)=randomCanonicalCurveGenus8with8Points S; |
i4 : betti res I
0 1 2 3 4 5 6
o4 = total: 1 15 35 42 35 15 1
0: 1 . . . . . .
1: . 15 35 21 . . .
2: . . . 21 35 15 .
3: . . . . . . 1
o4 : BettiTally
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i5 : points
o5 = {ideal (x - 4642x , x - 3546x , x + 3037x , x - 3107x , x - 2118x ,
6 7 5 7 4 7 3 7 2 7
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x - 1352x , x + 264x ), ideal (x + 819x , x + 4195x , x - 1955x ,
1 7 0 7 6 7 5 7 4 7
------------------------------------------------------------------------
x - 3491x , x - 105x , x - 4792x , x + 1768x ), ideal (x + 4437x ,
3 7 2 7 1 7 0 7 6 7
------------------------------------------------------------------------
x + 640x , x - 4763x , x + 3022x , x + 4533x , x - 1826x , x -
5 7 4 7 3 7 2 7 1 7 0
------------------------------------------------------------------------
2052x ), ideal (x + 1270x , x + 4687x , x - 2115x , x - 1314x , x -
7 6 7 5 7 4 7 3 7 2
------------------------------------------------------------------------
4497x , x + 2488x , x + 3531x ), ideal (x - 553x , x - 1971x , x +
7 1 7 0 7 6 7 5 7 4
------------------------------------------------------------------------
599x , x + 3215x , x + 4961x , x - 2839x , x - 415x ), ideal (x +
7 3 7 2 7 1 7 0 7 6
------------------------------------------------------------------------
1072x , x - 1523x , x - 3140x , x - 4466x , x - 4788x , x - 3266x ,
7 5 7 4 7 3 7 2 7 1 7
------------------------------------------------------------------------
x - 2651x ), ideal (x + 1313x , x - 713x , x + 4861x , x + 1676x ,
0 7 6 7 5 7 4 7 3 7
------------------------------------------------------------------------
x + 2939x , x + 4736x , x - 4974x ), ideal (x - 2149x , x - 2354x ,
2 7 1 7 0 7 6 7 5 7
------------------------------------------------------------------------
x + 3549x , x - 177x , x + 644x , x + 3449x , x - 658x )}
4 7 3 7 2 7 1 7 0 7
o5 : List
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