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 + 4343x , x + 3780x , x - 4269x , x - 369x , x - 4049x , 6 7 5 7 4 7 3 7 2 7 ------------------------------------------------------------------------ x + 3326x , x + x ), ideal (x - 3755x , x + 4705x , x - 862x , x + 1 7 0 7 6 7 5 7 4 7 3 ------------------------------------------------------------------------ 2736x , x - 328x , x - 4669x , x + 1787x ), ideal (x - 593x , x - 7 2 7 1 7 0 7 6 7 5 ------------------------------------------------------------------------ 3336x , x + 2310x , x - 230x , x - 809x , x - 3492x , x - 3928x ), 7 4 7 3 7 2 7 1 7 0 7 ------------------------------------------------------------------------ ideal (x - 350x , x + 2226x , x - 3821x , x + 675x , x + 4965x , x 6 7 5 7 4 7 3 7 2 7 1 ------------------------------------------------------------------------ + 2302x , x + 1537x ), ideal (x - 1191x , x - 2066x , x - 3554x , x 7 0 7 6 7 5 7 4 7 3 ------------------------------------------------------------------------ - 1197x , x + 500x , x + 528x , x - 613x ), ideal (x - 2621x , x - 7 2 7 1 7 0 7 6 7 5 ------------------------------------------------------------------------ 3923x , x + 3995x , x + 705x , x - 3477x , x - 3199x , x + 2634x ), 7 4 7 3 7 2 7 1 7 0 7 ------------------------------------------------------------------------ ideal (x - 1778x , x - 2454x , x - 2324x , x - 234x , x + 2583x , 6 7 5 7 4 7 3 7 2 7 ------------------------------------------------------------------------ x + 1108x , x - 1446x ), ideal (x - 972x , x - 937x , x + 1938x , 1 7 0 7 6 7 5 7 4 7 ------------------------------------------------------------------------ x + 2153x , x - 4170x , x + 4805x , x + 4386x )} 3 7 2 7 1 7 0 7 o5 : List |