next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
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  - 1867x , x  + 1915x , x  - 272x , x  - 5001x , x  - 3143x ,
              6        7   5        7   4       7   3        7   2        7 
     ------------------------------------------------------------------------
     x  + 4373x , x  - 4284x ), ideal (x  - 3947x , x  - 4792x , x  + 4994x ,
      1        7   0        7           6        7   5        7   4        7 
     ------------------------------------------------------------------------
     x  + 4944x , x  - 2806x , x  - 4652x , x  + 41x ), ideal (x  + 2632x ,
      3        7   2        7   1        7   0      7           6        7 
     ------------------------------------------------------------------------
     x  + 588x , x  + 1777x , x  + 563x , x  - 3873x , x  - 2669x , x  +
      5       7   4        7   3       7   2        7   1        7   0  
     ------------------------------------------------------------------------
     2823x ), ideal (x  - 4433x , x  - 2306x , x  - 578x , x  - 841x , x  +
          7           6        7   5        7   4       7   3       7   2  
     ------------------------------------------------------------------------
     375x , x  + 3674x , x  - 1428x ), ideal (x  + 4791x , x  - 3411x , x  -
         7   1        7   0        7           6        7   5        7   4  
     ------------------------------------------------------------------------
     519x , x  - 3045x , x  + 2263x , x  - 265x , x  - 1078x ), ideal (x  +
         7   3        7   2        7   1       7   0        7           6  
     ------------------------------------------------------------------------
     3443x , x  + 265x , x  - 1521x , x  + 2623x , x  - 152x , x  + 2525x ,
          7   5       7   4        7   3        7   2       7   1        7 
     ------------------------------------------------------------------------
     x  - 219x ), ideal (x  - 4382x , x  + 3639x , x  - 942x , x  - 3390x ,
      0       7           6        7   5        7   4       7   3        7 
     ------------------------------------------------------------------------
     x  + 470x , x  + 374x , x  + 1190x ), ideal (x  + 3017x , x  + 2844x ,
      2       7   1       7   0        7           6        7   5        7 
     ------------------------------------------------------------------------
     x  - 2251x , x  - 2134x , x  - 3147x , x  + 2312x , x  - 2719x )}
      4        7   3        7   2        7   1        7   0        7

o5 : List

Ways to use randomCanonicalCurveGenus8with8Points :

  • randomCanonicalCurveGenus8with8Points(PolynomialRing)