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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  + 2856x , x  - 4887x , x  + 2520x , x  + 1050x , x  + 66x ,
              6        7   5        7   4        7   3        7   2      7 
     ------------------------------------------------------------------------
     x  - 251x , x  - 4488x ), ideal (x  - 4812x , x  - 601x , x  - 4778x ,
      1       7   0        7           6        7   5       7   4        7 
     ------------------------------------------------------------------------
     x  + 4609x , x  + 4664x , x  - 4575x , x  - 1221x ), ideal (x  + 4804x ,
      3        7   2        7   1        7   0        7           6        7 
     ------------------------------------------------------------------------
     x  - 3748x , x  + 4000x , x  - 3872x , x  - 3435x , x  - 1818x , x  -
      5        7   4        7   3        7   2        7   1        7   0  
     ------------------------------------------------------------------------
     4695x ), ideal (x  + 685x , x  + 1214x , x  - 4520x , x  + 4250x , x  +
          7           6       7   5        7   4        7   3        7   2  
     ------------------------------------------------------------------------
     4431x , x  + 493x , x  - 4265x ), ideal (x  - 4901x , x  - 2403x , x  -
          7   1       7   0        7           6        7   5        7   4  
     ------------------------------------------------------------------------
     3009x , x  - 285x , x  - 316x , x  + 1054x , x  + 4620x ), ideal (x  -
          7   3       7   2       7   1        7   0        7           6  
     ------------------------------------------------------------------------
     3769x , x  + 2030x , x  - 4155x , x  + 3627x , x  + 4119x , x  - 2793x ,
          7   5        7   4        7   3        7   2        7   1        7 
     ------------------------------------------------------------------------
     x  + 1117x ), ideal (x  + 4139x , x  + 2321x , x  + 4579x , x  - 2818x ,
      0        7           6        7   5        7   4        7   3        7 
     ------------------------------------------------------------------------
     x  + 305x , x  + 2784x , x  - 2153x ), ideal (x  + 3029x , x  + 2079x ,
      2       7   1        7   0        7           6        7   5        7 
     ------------------------------------------------------------------------
     x  - 551x , x  - 3827x , x  - 1230x , x  + 3945x , x  - 2882x )}
      4       7   3        7   2        7   1        7   0        7

o5 : List

Ways to use randomCanonicalCurveGenus8with8Points :

  • randomCanonicalCurveGenus8with8Points(PolynomialRing)