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 + 2776x , x - 1893x , x - 1851x , x + 917x , x - 4266x ,
6 7 5 7 4 7 3 7 2 7
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x + 1148x , x + 1523x ), ideal (x + 1549x , x - 831x , x - 781x ,
1 7 0 7 6 7 5 7 4 7
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x - 3305x , x + 69x , x - 2610x , x + 732x ), ideal (x - 2145x , x
3 7 2 7 1 7 0 7 6 7 5
------------------------------------------------------------------------
- 2833x , x + 1040x , x + 1470x , x + 591x , x + 2183x , x -
7 4 7 3 7 2 7 1 7 0
------------------------------------------------------------------------
4856x ), ideal (x + 1900x , x - 2964x , x - 747x , x + 129x , x -
7 6 7 5 7 4 7 3 7 2
------------------------------------------------------------------------
3749x , x + 3747x , x + 385x ), ideal (x - 4122x , x - 2329x , x -
7 1 7 0 7 6 7 5 7 4
------------------------------------------------------------------------
3768x , x + 446x , x - 596x , x - 511x , x - 105x ), ideal (x -
7 3 7 2 7 1 7 0 7 6
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3504x , x + 807x , x - 1269x , x - 11x , x - 477x , x + 2179x , x
7 5 7 4 7 3 7 2 7 1 7 0
------------------------------------------------------------------------
- 3206x ), ideal (x + 3353x , x + 1385x , x + 3198x , x - 4928x , x
7 6 7 5 7 4 7 3 7 2
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- 4609x , x - 3236x , x - 2724x ), ideal (x + 475x , x + 2585x , x
7 1 7 0 7 6 7 5 7 4
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- 2359x , x - 4050x , x - 4366x , x + 2113x , x + 1037x )}
7 3 7 2 7 1 7 0 7
o5 : List
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