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 + 3951x , x - 3080x , x + 3930x , x + 1514x , x - 1732x , 6 7 5 7 4 7 3 7 2 7 ------------------------------------------------------------------------ x - 4667x , x - 3321x ), ideal (x + 900x , x + 1695x , x - 4467x , 1 7 0 7 6 7 5 7 4 7 ------------------------------------------------------------------------ x - 4638x , x + 2601x , x + 4202x , x + 3299x ), ideal (x + 2116x , 3 7 2 7 1 7 0 7 6 7 ------------------------------------------------------------------------ x - 3720x , x + 531x , x - 3928x , x + 2810x , x - 2247x , x - 5 7 4 7 3 7 2 7 1 7 0 ------------------------------------------------------------------------ 36x ), ideal (x + 2525x , x + 3686x , x + 392x , x - 3681x , x + 7 6 7 5 7 4 7 3 7 2 ------------------------------------------------------------------------ 1509x , x + 4453x , x - 583x ), ideal (x - 3732x , x + 1336x , x - 7 1 7 0 7 6 7 5 7 4 ------------------------------------------------------------------------ 3693x , x - 80x , x - 1477x , x + 829x , x + 2369x ), ideal (x - 7 3 7 2 7 1 7 0 7 6 ------------------------------------------------------------------------ 377x , x - 4471x , x - 931x , x + 1701x , x + 1827x , x - 4387x , 7 5 7 4 7 3 7 2 7 1 7 ------------------------------------------------------------------------ x - 4465x ), ideal (x - 64x , x - 4495x , x + 2474x , x - 2396x , 0 7 6 7 5 7 4 7 3 7 ------------------------------------------------------------------------ x + 3261x , x + 4628x , x + 4473x ), ideal (x - 2091x , x + 3211x , 2 7 1 7 0 7 6 7 5 7 ------------------------------------------------------------------------ x + 834x , x + 2189x , x + 3098x , x + 151x , x + 2490x )} 4 7 3 7 2 7 1 7 0 7 o5 : List |