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 - 2494x , x + 918x , x - 2923x , x + 2833x , x + 4964x , 6 7 5 7 4 7 3 7 2 7 ------------------------------------------------------------------------ x - 239x , x - 3639x ), ideal (x - 2011x , x - 3964x , x + 3166x , 1 7 0 7 6 7 5 7 4 7 ------------------------------------------------------------------------ x + 3170x , x + 1121x , x + 608x , x + 4705x ), ideal (x - 2725x , 3 7 2 7 1 7 0 7 6 7 ------------------------------------------------------------------------ x - 4130x , x - 76x , x + 2541x , x + 4225x , x - 333x , x - 5 7 4 7 3 7 2 7 1 7 0 ------------------------------------------------------------------------ 3916x ), ideal (x - 3954x , x - 4336x , x + 850x , x + 3202x , x - 7 6 7 5 7 4 7 3 7 2 ------------------------------------------------------------------------ 1852x , x - 4758x , x + 4340x ), ideal (x - 3511x , x - 4007x , x - 7 1 7 0 7 6 7 5 7 4 ------------------------------------------------------------------------ 3504x , x + 4670x , x + 1301x , x - 2654x , x - 4557x ), ideal (x - 7 3 7 2 7 1 7 0 7 6 ------------------------------------------------------------------------ 36x , x - 1889x , x - 1087x , x - 633x , x - 4495x , x + 579x , x 7 5 7 4 7 3 7 2 7 1 7 0 ------------------------------------------------------------------------ + 1903x ), ideal (x - 421x , x + 828x , x + 1096x , x - 1260x , x - 7 6 7 5 7 4 7 3 7 2 ------------------------------------------------------------------------ 1780x , x + 1502x , x - 1057x ), ideal (x + 3867x , x + 4990x , x - 7 1 7 0 7 6 7 5 7 4 ------------------------------------------------------------------------ 4374x , x + 4871x , x - 3786x , x + 2300x , x + 3418x )} 7 3 7 2 7 1 7 0 7 o5 : List |