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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  - 2735x , x  - 2225x , x  + 1961x , x  + 505x , x  + 2153x ,
              6        7   5        7   4        7   3       7   2        7 
     ------------------------------------------------------------------------
     x  + 729x , x  + 50x ), ideal (x  - 4917x , x  - 3517x , x  + 1118x , x 
      1       7   0      7           6        7   5        7   4        7   3
     ------------------------------------------------------------------------
     + 2762x , x  + 823x , x  - 1308x , x  - 2539x ), ideal (x  - 2060x , x 
            7   2       7   1        7   0        7           6        7   5
     ------------------------------------------------------------------------
     - 4683x , x  - 4563x , x  + 1795x , x  + 4252x , x  - 948x , x  -
            7   4        7   3        7   2        7   1       7   0  
     ------------------------------------------------------------------------
     2016x ), ideal (x  - 3770x , x  + 4777x , x  + 3280x , x  + 3531x , x  +
          7           6        7   5        7   4        7   3        7   2  
     ------------------------------------------------------------------------
     1525x , x  - 4104x , x  + 4982x ), ideal (x  - 4619x , x  - 4648x , x  -
          7   1        7   0        7           6        7   5        7   4  
     ------------------------------------------------------------------------
     1148x , x  - 434x , x  + 519x , x  + 2402x , x  + 4005x ), ideal (x  +
          7   3       7   2       7   1        7   0        7           6  
     ------------------------------------------------------------------------
     4085x , x  - 2587x , x  - 900x , x  - 1112x , x  - 3518x , x  + 1867x ,
          7   5        7   4       7   3        7   2        7   1        7 
     ------------------------------------------------------------------------
     x  - 4489x ), ideal (x  - 1234x , x  + 554x , x  + 4744x , x  + 1589x ,
      0        7           6        7   5       7   4        7   3        7 
     ------------------------------------------------------------------------
     x  + 2331x , x  - 3884x , x  - 840x ), ideal (x  - 3570x , x  - 2104x ,
      2        7   1        7   0       7           6        7   5        7 
     ------------------------------------------------------------------------
     x  - 4797x , x  - 598x , x  - 4697x , x  - 487x , x  - 218x )}
      4        7   3       7   2        7   1       7   0       7

o5 : List

Ways to use randomCanonicalCurveGenus8with8Points :

  • randomCanonicalCurveGenus8with8Points(PolynomialRing)