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"); |
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 |
i6 : points o6 = {ideal (x + 315x , x + 1084x , x - 2032x , x + 2800x , x - 2677x , 6 7 5 7 4 7 3 7 2 7 ------------------------------------------------------------------------ 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 |