addANodeAndExtend2Torsion -- add a node to a canonical curve and extend a 2-Torsion divisor to the new curve

Synopsis

Usage:
(J,E,Pts1)=addANodeAndExtend2Torsion(I,D,Pts)
Inputs:
- I, an ideal, of a curve C in its canonical embedding over a finite field kk
- D, a list, a pair {D₀,D₁}of two effective divisors represented by their ideal
- Pts, a list, a list of ideals of kk-rational points on C
Outputs:
- J, an ideal, of canonical curve C’ obtained by identifying two points from the list to anode
- E, a list, a pair of divisors {E₀,E₁}such that E₀-E₁ is two torsion on C’
- Pts1, a list, of ideals of the remaining points on C’

Description

Given a canonical possibly nodal curve C and a two Torsion divisor D₀-D₁, we compute a curve C’ obtained by identifying two points p, q on C to a node, and a 2-Torsion line bundle whose pullback under C-> C’ is the given 2 Torsion line bundle O_C(D₀-D₁). Note that C’ together E₀-E₁ is defined over the given ground field if and only if the values of the rational function f with (f)=2D₀-2D₁ at the two points p, q differ by a square, i.e. f(p)/f(q) ∈kk². We use a probabilistic approach to over come this difficulty, by trying several pairs of points if necessary.

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

i3 : time (J,D,Pts)=getCanonicalCurveOfGenus8With2Torsion(R,20);
     -- used 1.24501 seconds

i4 : betti J, dim J, codim J, degree J, genus J, apply(D,d->degree d)

             0  1
o4 = (total: 1 15, 2, 6, 14, 8, {8, 8})
          0: 1  .
          1: . 15

o4 : Sequence

i5 : time (I,D,Pts)=addANodeAndExtend2Torsion(J,D,Pts);
     -- used 8.94068 seconds

i6 : betti I, dim I, codim I, degree I, genus I, apply(D,e->degree e)

             0  1
o6 = (total: 1 21, 2, 7, 16, 9, {22, 22})
          0: 1  .
          1: . 21

o6 : Sequence

i7 : (E,Pts1)=simplifyDivisor(I,D,Pts);

i8 : apply(E,e->degree e)

o8 = {9, 9}

o8 : List

i9 : #Pts,#Pts1

o9 = (18, 5)

o9 : Sequence

i10 : betti(J=prymCanonicalEmbedding(I,E)), dim J,codim J,degree J,genus J

              0  1
o10 = (total: 1 12, 2, 6, 16, 9)
           0: 1  .
           1: . 12

o10 : Sequence

The key idea of the algoritm to compute a rational function g such that f(p)/f(q)*(g(p)/g(q))² = 1. The function g can be obtained from any pencil of homogeneous forms via a suitable fractional transformation. We compute suitable linear forms g₀,g₁ such that g=g₀/g₁ has the desired property and replace D_i by E_i+g_i.C for i=0,1. The curve C’ is simply the image of C under |K_C+p+q|. The divisor representative E₀,E₁ of the divisor class E₀-E₁ can be simplified by imposing common base points in both |E_i|. This can be carried out with the function "simplifyDivisor", but it is not in the current routine "addANodeAndExtend2Torsion".

Ways to use `addANodeAndExtend2Torsion` :

addANodeAndExtend2Torsion(Ideal,List,List)

addANodeAndExtend2Torsion -- add a node to a canonical curve and extend a 2-Torsion divisor to the new curve

Synopsis

Description

Ways to use addANodeAndExtend2Torsion :

Ways to use `addANodeAndExtend2Torsion` :