randomCanonicalCurveGenus8with8Points -- Compute a random canonical curve of genus 8 with 8 marked point

Synopsis

Usage:

(I,idealsOfPts)=randomCanonicalCurveGenus8with8Points S
Inputs:
- S, a polynomial ring, homogeneous coordinate ring of $\PP^7$
Outputs:
- I, an ideal, a canonical curve C of genus 8
- idealsOfPts, a list, 8 ideals of K-rational points on C

Description

According to Mukai [Mu] any smooth curve of genus 8 and Clifford index 3 is the transversal intersection $C=\PP^7 \cap\ G(2,6) \subset \ \PP^{15}$. In particular this is true for the general curve of genus 8. Picking 8 points in the Grassmannian $G(2,6)$ at random and \PP^7 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  - 1611x , x  + 2414x , x  - 1550x , x  - 4251x , x  + 4315x ,
              6        7   5        7   4        7   3        7   2        7 
     ------------------------------------------------------------------------
     x  + 2701x , x  - 4090x ), ideal (x  + 1668x , x  - 2841x , x  - 2811x ,
      1        7   0        7           6        7   5        7   4        7 
     ------------------------------------------------------------------------
     x  - 2503x , x  - 2868x , x  + 3683x , x  + 4539x ), ideal (x  - 1972x ,
      3        7   2        7   1        7   0        7           6        7 
     ------------------------------------------------------------------------
     x  - 3164x , x  + 2439x , x  - 3731x , x  - 3223x , x  - 4255x , x  -
      5        7   4        7   3        7   2        7   1        7   0  
     ------------------------------------------------------------------------
     3245x ), ideal (x  - 3018x , x  - 4221x , x  + 3745x , x  - 3558x , x  +
          7           6        7   5        7   4        7   3        7   2  
     ------------------------------------------------------------------------
     3717x , x  - 879x , x  - 4286x ), ideal (x  - 3067x , x  - 941x , x  +
          7   1       7   0        7           6        7   5       7   4  
     ------------------------------------------------------------------------
     4350x , x  - 49x , x  + 4179x , x  - 3962x , x  + 2213x ), ideal (x  +
          7   3      7   2        7   1        7   0        7           6  
     ------------------------------------------------------------------------
     4309x , x  + 2155x , x  - 613x , x  - 155x , x  - 542x , x  + 1237x , x 
          7   5        7   4       7   3       7   2       7   1        7   0
     ------------------------------------------------------------------------
     + 2446x ), ideal (x  - 4950x , x  - 597x , x  - 4613x , x  - 1907x , x 
            7           6        7   5       7   4        7   3        7   2
     ------------------------------------------------------------------------
     + 4217x , x  - 886x , x  + 3338x ), ideal (x  + 2599x , x  + 3310x , x 
            7   1       7   0        7           6        7   5        7   4
     ------------------------------------------------------------------------
     + 1584x , x  + 3379x , x  + 1721x , x  - 3470x , x  + 3687x )}
            7   3        7   2        7   1        7   0        7

o6 : List

Ways to use randomCanonicalCurveGenus8with8Points :

randomCanonicalCurveGenus8with8Points(PolynomialRing)

For the programmer

The object randomCanonicalCurveGenus8with8Points is a method function.