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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  - 4642x , x  - 3546x , x  + 3037x , x  - 3107x , x  - 2118x ,
              6        7   5        7   4        7   3        7   2        7 
     ------------------------------------------------------------------------
     x  - 1352x , x  + 264x ), ideal (x  + 819x , x  + 4195x , x  - 1955x ,
      1        7   0       7           6       7   5        7   4        7 
     ------------------------------------------------------------------------
     x  - 3491x , x  - 105x , x  - 4792x , x  + 1768x ), ideal (x  + 4437x ,
      3        7   2       7   1        7   0        7           6        7 
     ------------------------------------------------------------------------
     x  + 640x , x  - 4763x , x  + 3022x , x  + 4533x , x  - 1826x , x  -
      5       7   4        7   3        7   2        7   1        7   0  
     ------------------------------------------------------------------------
     2052x ), ideal (x  + 1270x , x  + 4687x , x  - 2115x , x  - 1314x , x  -
          7           6        7   5        7   4        7   3        7   2  
     ------------------------------------------------------------------------
     4497x , x  + 2488x , x  + 3531x ), ideal (x  - 553x , x  - 1971x , x  +
          7   1        7   0        7           6       7   5        7   4  
     ------------------------------------------------------------------------
     599x , x  + 3215x , x  + 4961x , x  - 2839x , x  - 415x ), ideal (x  +
         7   3        7   2        7   1        7   0       7           6  
     ------------------------------------------------------------------------
     1072x , x  - 1523x , x  - 3140x , x  - 4466x , x  - 4788x , x  - 3266x ,
          7   5        7   4        7   3        7   2        7   1        7 
     ------------------------------------------------------------------------
     x  - 2651x ), ideal (x  + 1313x , x  - 713x , x  + 4861x , x  + 1676x ,
      0        7           6        7   5       7   4        7   3        7 
     ------------------------------------------------------------------------
     x  + 2939x , x  + 4736x , x  - 4974x ), ideal (x  - 2149x , x  - 2354x ,
      2        7   1        7   0        7           6        7   5        7 
     ------------------------------------------------------------------------
     x  + 3549x , x  - 177x , x  + 644x , x  + 3449x , x  - 658x )}
      4        7   3       7   2       7   1        7   0       7

o5 : List

Ways to use randomCanonicalCurveGenus8with8Points :

  • randomCanonicalCurveGenus8with8Points(PolynomialRing)