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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  + 4335x , x  - 3568x , x  - 2321x , x  + 1745x , x  - 643x ,
              6        7   5        7   4        7   3        7   2       7 
     ------------------------------------------------------------------------
     x  - 2978x , x  - 4767x ), ideal (x  - 3448x , x  + 2716x , x  + 1560x ,
      1        7   0        7           6        7   5        7   4        7 
     ------------------------------------------------------------------------
     x  - 3274x , x  + 4925x , x  - 4687x , x  - 1898x ), ideal (x  - 3738x ,
      3        7   2        7   1        7   0        7           6        7 
     ------------------------------------------------------------------------
     x  - 1861x , x  + 628x , x  - 2045x , x  + 1782x , x  - 1363x , x  -
      5        7   4       7   3        7   2        7   1        7   0  
     ------------------------------------------------------------------------
     1510x ), ideal (x  + 2192x , x  - 4635x , x  + 1448x , x  + 4283x , x  -
          7           6        7   5        7   4        7   3        7   2  
     ------------------------------------------------------------------------
     2858x , x  + 2385x , x  - 1948x ), ideal (x  - 2999x , x  - 2000x , x  +
          7   1        7   0        7           6        7   5        7   4  
     ------------------------------------------------------------------------
     2676x , x  - 494x , x  + 1462x , x  - 4215x , x  - 3304x ), ideal (x  -
          7   3       7   2        7   1        7   0        7           6  
     ------------------------------------------------------------------------
     4457x , x  - 292x , x  - 3635x , x  + 1098x , x  + 2261x , x  + 509x ,
          7   5       7   4        7   3        7   2        7   1       7 
     ------------------------------------------------------------------------
     x  + 4858x ), ideal (x  - 1070x , x  + 2278x , x  + 3019x , x  + 370x ,
      0        7           6        7   5        7   4        7   3       7 
     ------------------------------------------------------------------------
     x  + 2033x , x  - 195x , x  - 4698x ), ideal (x  + 3194x , x  + 1643x ,
      2        7   1       7   0        7           6        7   5        7 
     ------------------------------------------------------------------------
     x  - 3785x , x  + 4897x , x  + 3343x , x  - 4082x , x  + 2319x )}
      4        7   3        7   2        7   1        7   0        7

o5 : List

Ways to use randomCanonicalCurveGenus8with8Points :

  • randomCanonicalCurveGenus8with8Points(PolynomialRing)