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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  - 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

Ways to use randomCanonicalCurveGenus8with8Points :

  • randomCanonicalCurveGenus8with8Points(PolynomialRing)