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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  + 4343x , x  + 3780x , x  - 4269x , x  - 369x , x  - 4049x ,
              6        7   5        7   4        7   3       7   2        7 
     ------------------------------------------------------------------------
     x  + 3326x , x  + x ), ideal (x  - 3755x , x  + 4705x , x  - 862x , x  +
      1        7   0    7           6        7   5        7   4       7   3  
     ------------------------------------------------------------------------
     2736x , x  - 328x , x  - 4669x , x  + 1787x ), ideal (x  - 593x , x  -
          7   2       7   1        7   0        7           6       7   5  
     ------------------------------------------------------------------------
     3336x , x  + 2310x , x  - 230x , x  - 809x , x  - 3492x , x  - 3928x ),
          7   4        7   3       7   2       7   1        7   0        7  
     ------------------------------------------------------------------------
     ideal (x  - 350x , x  + 2226x , x  - 3821x , x  + 675x , x  + 4965x , x 
             6       7   5        7   4        7   3       7   2        7   1
     ------------------------------------------------------------------------
     + 2302x , x  + 1537x ), ideal (x  - 1191x , x  - 2066x , x  - 3554x , x 
            7   0        7           6        7   5        7   4        7   3
     ------------------------------------------------------------------------
     - 1197x , x  + 500x , x  + 528x , x  - 613x ), ideal (x  - 2621x , x  -
            7   2       7   1       7   0       7           6        7   5  
     ------------------------------------------------------------------------
     3923x , x  + 3995x , x  + 705x , x  - 3477x , x  - 3199x , x  + 2634x ),
          7   4        7   3       7   2        7   1        7   0        7  
     ------------------------------------------------------------------------
     ideal (x  - 1778x , x  - 2454x , x  - 2324x , x  - 234x , x  + 2583x ,
             6        7   5        7   4        7   3       7   2        7 
     ------------------------------------------------------------------------
     x  + 1108x , x  - 1446x ), ideal (x  - 972x , x  - 937x , x  + 1938x ,
      1        7   0        7           6       7   5       7   4        7 
     ------------------------------------------------------------------------
     x  + 2153x , x  - 4170x , x  + 4805x , x  + 4386x )}
      3        7   2        7   1        7   0        7

o5 : List

Ways to use randomCanonicalCurveGenus8with8Points :

  • randomCanonicalCurveGenus8with8Points(PolynomialRing)