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 + 4335x , x - 3568x , x - 2321x , x + 1745x , x - 643x , 6 7 5 7 4 7 3 7 2 7 ------------------------------------------------------------------------ x - 2978x , x - 4767x ), ideal (x - 3448x , x + 2716x , x + 1560x , 1 7 0 7 6 7 5 7 4 7 ------------------------------------------------------------------------ x - 3274x , x + 4925x , x - 4687x , x - 1898x ), ideal (x - 3738x , 3 7 2 7 1 7 0 7 6 7 ------------------------------------------------------------------------ x - 1861x , x + 628x , x - 2045x , x + 1782x , x - 1363x , x - 5 7 4 7 3 7 2 7 1 7 0 ------------------------------------------------------------------------ 1510x ), ideal (x + 2192x , x - 4635x , x + 1448x , x + 4283x , x - 7 6 7 5 7 4 7 3 7 2 ------------------------------------------------------------------------ 2858x , x + 2385x , x - 1948x ), ideal (x - 2999x , x - 2000x , x + 7 1 7 0 7 6 7 5 7 4 ------------------------------------------------------------------------ 2676x , x - 494x , x + 1462x , x - 4215x , x - 3304x ), ideal (x - 7 3 7 2 7 1 7 0 7 6 ------------------------------------------------------------------------ 4457x , x - 292x , x - 3635x , x + 1098x , x + 2261x , x + 509x , 7 5 7 4 7 3 7 2 7 1 7 ------------------------------------------------------------------------ x + 4858x ), ideal (x - 1070x , x + 2278x , x + 3019x , x + 370x , 0 7 6 7 5 7 4 7 3 7 ------------------------------------------------------------------------ x + 2033x , x - 195x , x - 4698x ), ideal (x + 3194x , x + 1643x , 2 7 1 7 0 7 6 7 5 7 ------------------------------------------------------------------------ x - 3785x , x + 4897x , x + 3343x , x - 4082x , x + 2319x )} 4 7 3 7 2 7 1 7 0 7 o5 : List |