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 |
i5 : points o5 = {ideal (x - 1867x , x + 1915x , x - 272x , x - 5001x , x - 3143x , 6 7 5 7 4 7 3 7 2 7 ------------------------------------------------------------------------ x + 4373x , x - 4284x ), ideal (x - 3947x , x - 4792x , x + 4994x , 1 7 0 7 6 7 5 7 4 7 ------------------------------------------------------------------------ x + 4944x , x - 2806x , x - 4652x , x + 41x ), ideal (x + 2632x , 3 7 2 7 1 7 0 7 6 7 ------------------------------------------------------------------------ x + 588x , x + 1777x , x + 563x , x - 3873x , x - 2669x , x + 5 7 4 7 3 7 2 7 1 7 0 ------------------------------------------------------------------------ 2823x ), ideal (x - 4433x , x - 2306x , x - 578x , x - 841x , x + 7 6 7 5 7 4 7 3 7 2 ------------------------------------------------------------------------ 375x , x + 3674x , x - 1428x ), ideal (x + 4791x , x - 3411x , x - 7 1 7 0 7 6 7 5 7 4 ------------------------------------------------------------------------ 519x , x - 3045x , x + 2263x , x - 265x , x - 1078x ), ideal (x + 7 3 7 2 7 1 7 0 7 6 ------------------------------------------------------------------------ 3443x , x + 265x , x - 1521x , x + 2623x , x - 152x , x + 2525x , 7 5 7 4 7 3 7 2 7 1 7 ------------------------------------------------------------------------ x - 219x ), ideal (x - 4382x , x + 3639x , x - 942x , x - 3390x , 0 7 6 7 5 7 4 7 3 7 ------------------------------------------------------------------------ x + 470x , x + 374x , x + 1190x ), ideal (x + 3017x , x + 2844x , 2 7 1 7 0 7 6 7 5 7 ------------------------------------------------------------------------ x - 2251x , x - 2134x , x - 3147x , x + 2312x , x - 2719x )} 4 7 3 7 2 7 1 7 0 7 o5 : List |