According to Mukai [Mu] any smooth curve of genus 8 and Clifford index 3 is the transversal intersection $C=\PP^7 \cap\ G(2,6) \subset \ \PP^{15}$. In particular this is true for the general curve of genus 8. Picking 8 points in the Grassmannian $G(2,6)$ at random and \PP^7 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 - 1611x , x + 2414x , x - 1550x , x - 4251x , x + 4315x , x + 2701x , x - 4090x ), ideal (x + 1668x , x - 6 7 5 7 4 7 3 7 2 7 1 7 0 7 6 7 5 ---------------------------------------------------------------------------------------------------------------------------- 2841x , x - 2811x , x - 2503x , x - 2868x , x + 3683x , x + 4539x ), ideal (x - 1972x , x - 3164x , x + 2439x , x - 7 4 7 3 7 2 7 1 7 0 7 6 7 5 7 4 7 3 ---------------------------------------------------------------------------------------------------------------------------- 3731x , x - 3223x , x - 4255x , x - 3245x ), ideal (x - 3018x , x - 4221x , x + 3745x , x - 3558x , x + 3717x , x - 7 2 7 1 7 0 7 6 7 5 7 4 7 3 7 2 7 1 ---------------------------------------------------------------------------------------------------------------------------- 879x , x - 4286x ), ideal (x - 3067x , x - 941x , x + 4350x , x - 49x , x + 4179x , x - 3962x , x + 2213x ), ideal 7 0 7 6 7 5 7 4 7 3 7 2 7 1 7 0 7 ---------------------------------------------------------------------------------------------------------------------------- (x + 4309x , x + 2155x , x - 613x , x - 155x , x - 542x , x + 1237x , x + 2446x ), ideal (x - 4950x , x - 597x , x 6 7 5 7 4 7 3 7 2 7 1 7 0 7 6 7 5 7 4 ---------------------------------------------------------------------------------------------------------------------------- - 4613x , x - 1907x , x + 4217x , x - 886x , x + 3338x ), ideal (x + 2599x , x + 3310x , x + 1584x , x + 3379x , x 7 3 7 2 7 1 7 0 7 6 7 5 7 4 7 3 7 2 ---------------------------------------------------------------------------------------------------------------------------- + 1721x , x - 3470x , x + 3687x )} 7 1 7 0 7 o6 : List |
The object randomCanonicalCurveGenus8with8Points is a method function.