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RandomGenus14Curves :: randomCanonicalCurveGenus8with8Points

randomCanonicalCurveGenus8with8Points -- Compute a random canonical curve of genus 8 with 8 marked point

Synopsis

Description

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  + 3951x , x  - 3080x , x  + 3930x , x  + 1514x , x  - 1732x ,
              6        7   5        7   4        7   3        7   2        7 
     ------------------------------------------------------------------------
     x  - 4667x , x  - 3321x ), ideal (x  + 900x , x  + 1695x , x  - 4467x ,
      1        7   0        7           6       7   5        7   4        7 
     ------------------------------------------------------------------------
     x  - 4638x , x  + 2601x , x  + 4202x , x  + 3299x ), ideal (x  + 2116x ,
      3        7   2        7   1        7   0        7           6        7 
     ------------------------------------------------------------------------
     x  - 3720x , x  + 531x , x  - 3928x , x  + 2810x , x  - 2247x , x  -
      5        7   4       7   3        7   2        7   1        7   0  
     ------------------------------------------------------------------------
     36x ), ideal (x  + 2525x , x  + 3686x , x  + 392x , x  - 3681x , x  +
        7           6        7   5        7   4       7   3        7   2  
     ------------------------------------------------------------------------
     1509x , x  + 4453x , x  - 583x ), ideal (x  - 3732x , x  + 1336x , x  -
          7   1        7   0       7           6        7   5        7   4  
     ------------------------------------------------------------------------
     3693x , x  - 80x , x  - 1477x , x  + 829x , x  + 2369x ), ideal (x  -
          7   3      7   2        7   1       7   0        7           6  
     ------------------------------------------------------------------------
     377x , x  - 4471x , x  - 931x , x  + 1701x , x  + 1827x , x  - 4387x ,
         7   5        7   4       7   3        7   2        7   1        7 
     ------------------------------------------------------------------------
     x  - 4465x ), ideal (x  - 64x , x  - 4495x , x  + 2474x , x  - 2396x ,
      0        7           6      7   5        7   4        7   3        7 
     ------------------------------------------------------------------------
     x  + 3261x , x  + 4628x , x  + 4473x ), ideal (x  - 2091x , x  + 3211x ,
      2        7   1        7   0        7           6        7   5        7 
     ------------------------------------------------------------------------
     x  + 834x , x  + 2189x , x  + 3098x , x  + 151x , x  + 2490x )}
      4       7   3        7   2        7   1       7   0        7

o5 : List

Ways to use randomCanonicalCurveGenus8with8Points :

  • randomCanonicalCurveGenus8with8Points(PolynomialRing)