getCanonicalCurveOfGenus8With2Torsion -- Construct a canonical curve C of genus 8 together with a 2-torsion divisor and n extra points

Synopsis

Usage:
(I,D,Pts)=getCanonicalCurveOfGenus8With2Torsion(S,n)
Inputs:
- S, a polynomial ring, a polynomial ring in 8 variables over a finite ground field kk
- n, an integer, the desired number of extra points
Outputs:
- I, an ideal, of a canonical curve C of genus 8.
- D, a list, a pair of effective divisors {D₀,D₁}specified by ideals of a collection of points on C, such that D₀-D₁ is 2-torsion in Pic C.
- Pts, a list, of ideals of n further kk-rational points on C

Description

We construct a curve C of genus 8 with a g²₇ together with a 2-torsion divisor, via its plane model and transfer the data into canonical space.

i1 : kk=ZZ/10007;R=kk[x_0..x_7];

i3 : time (J,D,Pts)=getCanonicalCurveOfGenus8With2Torsion(R,2);
     -- used 0.600697 seconds

i4 : time betti res J
     -- used 1.87693 seconds

            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 : apply(D,d->(dim d,degree d))

o5 = {(1, 8), (1, 8)}

o5 : List

i6 : tally apply(Pts,pt->betti pt)

                  0 1
o6 = Tally{total: 1 7 => 2}
               0: 1 7

o6 : Tally

Ways to use `getCanonicalCurveOfGenus8With2Torsion` :

getCanonicalCurveOfGenus8With2Torsion(PolynomialRing,ZZ)

getCanonicalCurveOfGenus8With2Torsion -- Construct a canonical curve C of genus 8 together with a 2-torsion divisor and n extra points

Synopsis

Description

Ways to use getCanonicalCurveOfGenus8With2Torsion :

Ways to use `getCanonicalCurveOfGenus8With2Torsion` :