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 : setRandomSeed("alpha");
i2 : FF=ZZ/10007;
i3 : S=FF[x_0..x_7];
i4 : (I,points)=randomCanonicalCurveGenus8with8Points S;
i5 : betti res I

            0  1  2  3  4  5 6
o5 = total: 1 15 35 42 35 15 1
         0: 1  .  .  .  .  . .
         1: . 15 35 21  .  . .
         2: .  .  . 21 35 15 .
         3: .  .  .  .  .  . 1

o5 : BettiTally
i6 : points

o6 = {ideal (x  + 315x , x  + 1084x , x  - 2032x , x  + 2800x , x  - 2677x ,
              6       7   5        7   4        7   3        7   2        7 
     ------------------------------------------------------------------------
     x  + 1089x , x  + 4275x ), ideal (x  + 3175x , x  + 4236x , x  - 752x ,
      1        7   0        7           6        7   5        7   4       7 
     ------------------------------------------------------------------------
     x  + 1946x , x  + 2344x , x  + 4872x , x  - 2944x ), ideal (x  + 2063x ,
      3        7   2        7   1        7   0        7           6        7 
     ------------------------------------------------------------------------
     x  - 694x , x  - 114x , x  + 930x , x  - 3445x , x  + 1502x , x  -
      5       7   4       7   3       7   2        7   1        7   0  
     ------------------------------------------------------------------------
     2284x ), ideal (x  - 3009x , x  - 77x , x  - 3932x , x  - 199x , x  -
          7           6        7   5      7   4        7   3       7   2  
     ------------------------------------------------------------------------
     4949x , x  + 4697x , x  - 4337x ), ideal (x  - 914x , x  - 3912x , x  -
          7   1        7   0        7           6       7   5        7   4  
     ------------------------------------------------------------------------
     2686x , x  + 2847x , x  + 3526x , x  - 1868x , x  + 4707x ), ideal (x  +
          7   3        7   2        7   1        7   0        7           6  
     ------------------------------------------------------------------------
     2307x , x  - 2134x , x  + 259x , x  + 3283x , x  + 2393x , x  - 2654x ,
          7   5        7   4       7   3        7   2        7   1        7 
     ------------------------------------------------------------------------
     x  + 2537x ), ideal (x  + 815x , x  - 4561x , x  - 43x , x  + 168x , x 
      0        7           6       7   5        7   4      7   3       7   2
     ------------------------------------------------------------------------
     - 1494x , x  + 4790x , x  + 3457x ), ideal (x  - 1919x , x  + 2630x , x 
            7   1        7   0        7           6        7   5        7   4
     ------------------------------------------------------------------------
     - 4885x , x  + 2450x , x  + 2739x , x  + 1825x , x  - 2642x )}
            7   3        7   2        7   1        7   0        7

o6 : List

Ways to use randomCanonicalCurveGenus8with8Points :

  • randomCanonicalCurveGenus8with8Points(PolynomialRing)