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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  - 2494x , x  + 918x , x  - 2923x , x  + 2833x , x  + 4964x ,
              6        7   5       7   4        7   3        7   2        7 
     ------------------------------------------------------------------------
     x  - 239x , x  - 3639x ), ideal (x  - 2011x , x  - 3964x , x  + 3166x ,
      1       7   0        7           6        7   5        7   4        7 
     ------------------------------------------------------------------------
     x  + 3170x , x  + 1121x , x  + 608x , x  + 4705x ), ideal (x  - 2725x ,
      3        7   2        7   1       7   0        7           6        7 
     ------------------------------------------------------------------------
     x  - 4130x , x  - 76x , x  + 2541x , x  + 4225x , x  - 333x , x  -
      5        7   4      7   3        7   2        7   1       7   0  
     ------------------------------------------------------------------------
     3916x ), ideal (x  - 3954x , x  - 4336x , x  + 850x , x  + 3202x , x  -
          7           6        7   5        7   4       7   3        7   2  
     ------------------------------------------------------------------------
     1852x , x  - 4758x , x  + 4340x ), ideal (x  - 3511x , x  - 4007x , x  -
          7   1        7   0        7           6        7   5        7   4  
     ------------------------------------------------------------------------
     3504x , x  + 4670x , x  + 1301x , x  - 2654x , x  - 4557x ), ideal (x  -
          7   3        7   2        7   1        7   0        7           6  
     ------------------------------------------------------------------------
     36x , x  - 1889x , x  - 1087x , x  - 633x , x  - 4495x , x  + 579x , x 
        7   5        7   4        7   3       7   2        7   1       7   0
     ------------------------------------------------------------------------
     + 1903x ), ideal (x  - 421x , x  + 828x , x  + 1096x , x  - 1260x , x  -
            7           6       7   5       7   4        7   3        7   2  
     ------------------------------------------------------------------------
     1780x , x  + 1502x , x  - 1057x ), ideal (x  + 3867x , x  + 4990x , x  -
          7   1        7   0        7           6        7   5        7   4  
     ------------------------------------------------------------------------
     4374x , x  + 4871x , x  - 3786x , x  + 2300x , x  + 3418x )}
          7   3        7   2        7   1        7   0        7

o5 : List

Ways to use randomCanonicalCurveGenus8with8Points :

  • randomCanonicalCurveGenus8with8Points(PolynomialRing)