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 + 2856x , x - 4887x , x + 2520x , x + 1050x , x + 66x , 6 7 5 7 4 7 3 7 2 7 ------------------------------------------------------------------------ x - 251x , x - 4488x ), ideal (x - 4812x , x - 601x , x - 4778x , 1 7 0 7 6 7 5 7 4 7 ------------------------------------------------------------------------ x + 4609x , x + 4664x , x - 4575x , x - 1221x ), ideal (x + 4804x , 3 7 2 7 1 7 0 7 6 7 ------------------------------------------------------------------------ x - 3748x , x + 4000x , x - 3872x , x - 3435x , x - 1818x , x - 5 7 4 7 3 7 2 7 1 7 0 ------------------------------------------------------------------------ 4695x ), ideal (x + 685x , x + 1214x , x - 4520x , x + 4250x , x + 7 6 7 5 7 4 7 3 7 2 ------------------------------------------------------------------------ 4431x , x + 493x , x - 4265x ), ideal (x - 4901x , x - 2403x , x - 7 1 7 0 7 6 7 5 7 4 ------------------------------------------------------------------------ 3009x , x - 285x , x - 316x , x + 1054x , x + 4620x ), ideal (x - 7 3 7 2 7 1 7 0 7 6 ------------------------------------------------------------------------ 3769x , x + 2030x , x - 4155x , x + 3627x , x + 4119x , x - 2793x , 7 5 7 4 7 3 7 2 7 1 7 ------------------------------------------------------------------------ x + 1117x ), ideal (x + 4139x , x + 2321x , x + 4579x , x - 2818x , 0 7 6 7 5 7 4 7 3 7 ------------------------------------------------------------------------ x + 305x , x + 2784x , x - 2153x ), ideal (x + 3029x , x + 2079x , 2 7 1 7 0 7 6 7 5 7 ------------------------------------------------------------------------ x - 551x , x - 3827x , x - 1230x , x + 3945x , x - 2882x )} 4 7 3 7 2 7 1 7 0 7 o5 : List |