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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  - 347x , x  + 3004x , x  - 2213x , x  + 4946x , x  - 3583x ,
              6       7   5        7   4        7   3        7   2        7 
     ------------------------------------------------------------------------
     x  - 2629x , x  - 3869x ), ideal (x  + 4391x , x  + 2554x , x  + 635x ,
      1        7   0        7           6        7   5        7   4       7 
     ------------------------------------------------------------------------
     x  - 350x , x  - 2371x , x  + 1854x , x  + 4836x ), ideal (x  + 2262x ,
      3       7   2        7   1        7   0        7           6        7 
     ------------------------------------------------------------------------
     x  - 3267x , x  + 1166x , x  - 4543x , x  - 1408x , x  - 2951x , x  -
      5        7   4        7   3        7   2        7   1        7   0  
     ------------------------------------------------------------------------
     748x ), ideal (x  - 2837x , x  - 2710x , x  - 4981x , x  - 2428x , x  +
         7           6        7   5        7   4        7   3        7   2  
     ------------------------------------------------------------------------
     1835x , x  - 163x , x  - 2525x ), ideal (x  + 4001x , x  - 2195x , x  -
          7   1       7   0        7           6        7   5        7   4  
     ------------------------------------------------------------------------
     957x , x  + 450x , x  - 4901x , x  - 1557x , x  + 2331x ), ideal (x  -
         7   3       7   2        7   1        7   0        7           6  
     ------------------------------------------------------------------------
     2986x , x  + 107x , x  + 1659x , x  - 3678x , x  - 2717x , x  + 694x ,
          7   5       7   4        7   3        7   2        7   1       7 
     ------------------------------------------------------------------------
     x  + 2683x ), ideal (x  + 4334x , x  + 1258x , x  + 1486x , x  - 288x ,
      0        7           6        7   5        7   4        7   3       7 
     ------------------------------------------------------------------------
     x  - 4522x , x  - 1797x , x  - 3188x ), ideal (x  - 3719x , x  - 3707x ,
      2        7   1        7   0        7           6        7   5        7 
     ------------------------------------------------------------------------
     x  + 788x , x  - 417x , x  - 2702x , x  + 605x , x  - 2235x )}
      4       7   3       7   2        7   1       7   0        7

o5 : List

Ways to use randomCanonicalCurveGenus8with8Points :

  • randomCanonicalCurveGenus8with8Points(PolynomialRing)