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 - 1058x , x - 4862x , x + 2201x , x + 419x , x - 3803x , 6 7 5 7 4 7 3 7 2 7 ------------------------------------------------------------------------ x - 1608x , x - 1378x ), ideal (x + 2480x , x - 2154x , x + 3366x , 1 7 0 7 6 7 5 7 4 7 ------------------------------------------------------------------------ x - 2571x , x + 1569x , x - 3599x , x - 1729x ), ideal (x - 3272x , 3 7 2 7 1 7 0 7 6 7 ------------------------------------------------------------------------ x - 1341x , x + 4656x , x + 320x , x - 108x , x - 3838x , x + 5 7 4 7 3 7 2 7 1 7 0 ------------------------------------------------------------------------ 3164x ), ideal (x + 2719x , x - 3665x , x - 2077x , x + 1248x , x + 7 6 7 5 7 4 7 3 7 2 ------------------------------------------------------------------------ 690x , x - 1131x , x + 1693x ), ideal (x - 4521x , x + 1768x , x - 7 1 7 0 7 6 7 5 7 4 ------------------------------------------------------------------------ 1802x , x - 2273x , x - 3982x , x + 3762x , x - 2796x ), ideal (x + 7 3 7 2 7 1 7 0 7 6 ------------------------------------------------------------------------ 4979x , x + 2569x , x + 876x , x + 3026x , x + 4444x , x + 4828x , 7 5 7 4 7 3 7 2 7 1 7 ------------------------------------------------------------------------ x + 3176x ), ideal (x - 2463x , x - 3429x , x - 1197x , x + 1753x , 0 7 6 7 5 7 4 7 3 7 ------------------------------------------------------------------------ x - 878x , x + 730x , x + 3958x ), ideal (x + 2627x , x - 4904x , 2 7 1 7 0 7 6 7 5 7 ------------------------------------------------------------------------ x + 1142x , x + 4168x , x + 2869x , x - 2817x , x - 1007x )} 4 7 3 7 2 7 1 7 0 7 o5 : List |