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 - 347x , x + 3004x , x - 2213x , x + 4946x , x - 3583x , 6 7 5 7 4 7 3 7 2 7 ------------------------------------------------------------------------ x - 2629x , x - 3869x ), ideal (x + 4391x , x + 2554x , x + 635x , 1 7 0 7 6 7 5 7 4 7 ------------------------------------------------------------------------ x - 350x , x - 2371x , x + 1854x , x + 4836x ), ideal (x + 2262x , 3 7 2 7 1 7 0 7 6 7 ------------------------------------------------------------------------ x - 3267x , x + 1166x , x - 4543x , x - 1408x , x - 2951x , x - 5 7 4 7 3 7 2 7 1 7 0 ------------------------------------------------------------------------ 748x ), ideal (x - 2837x , x - 2710x , x - 4981x , x - 2428x , x + 7 6 7 5 7 4 7 3 7 2 ------------------------------------------------------------------------ 1835x , x - 163x , x - 2525x ), ideal (x + 4001x , x - 2195x , x - 7 1 7 0 7 6 7 5 7 4 ------------------------------------------------------------------------ 957x , x + 450x , x - 4901x , x - 1557x , x + 2331x ), ideal (x - 7 3 7 2 7 1 7 0 7 6 ------------------------------------------------------------------------ 2986x , x + 107x , x + 1659x , x - 3678x , x - 2717x , x + 694x , 7 5 7 4 7 3 7 2 7 1 7 ------------------------------------------------------------------------ x + 2683x ), ideal (x + 4334x , x + 1258x , x + 1486x , x - 288x , 0 7 6 7 5 7 4 7 3 7 ------------------------------------------------------------------------ x - 4522x , x - 1797x , x - 3188x ), ideal (x - 3719x , x - 3707x , 2 7 1 7 0 7 6 7 5 7 ------------------------------------------------------------------------ x + 788x , x - 417x , x - 2702x , x + 605x , x - 2235x )} 4 7 3 7 2 7 1 7 0 7 o5 : List |