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 - 2735x , x - 2225x , x + 1961x , x + 505x , x + 2153x , 6 7 5 7 4 7 3 7 2 7 ------------------------------------------------------------------------ x + 729x , x + 50x ), ideal (x - 4917x , x - 3517x , x + 1118x , x 1 7 0 7 6 7 5 7 4 7 3 ------------------------------------------------------------------------ + 2762x , x + 823x , x - 1308x , x - 2539x ), ideal (x - 2060x , x 7 2 7 1 7 0 7 6 7 5 ------------------------------------------------------------------------ - 4683x , x - 4563x , x + 1795x , x + 4252x , x - 948x , x - 7 4 7 3 7 2 7 1 7 0 ------------------------------------------------------------------------ 2016x ), ideal (x - 3770x , x + 4777x , x + 3280x , x + 3531x , x + 7 6 7 5 7 4 7 3 7 2 ------------------------------------------------------------------------ 1525x , x - 4104x , x + 4982x ), ideal (x - 4619x , x - 4648x , x - 7 1 7 0 7 6 7 5 7 4 ------------------------------------------------------------------------ 1148x , x - 434x , x + 519x , x + 2402x , x + 4005x ), ideal (x + 7 3 7 2 7 1 7 0 7 6 ------------------------------------------------------------------------ 4085x , x - 2587x , x - 900x , x - 1112x , x - 3518x , x + 1867x , 7 5 7 4 7 3 7 2 7 1 7 ------------------------------------------------------------------------ x - 4489x ), ideal (x - 1234x , x + 554x , x + 4744x , x + 1589x , 0 7 6 7 5 7 4 7 3 7 ------------------------------------------------------------------------ x + 2331x , x - 3884x , x - 840x ), ideal (x - 3570x , x - 2104x , 2 7 1 7 0 7 6 7 5 7 ------------------------------------------------------------------------ x - 4797x , x - 598x , x - 4697x , x - 487x , x - 218x )} 4 7 3 7 2 7 1 7 0 7 o5 : List |