randomLineTorsZ3 -- compute a line for a numerical Godeaux surface with a cyclic torsion group of order 3

Synopsis

Usage:
(randLine,subsLine) = randomLineTorsZ3(d1',relPfaf)
Inputs:
- d1', a matrix, the first syzygy matrix in normal form with the first row omitted
- relPfaf, a matrix, a row matrix containing the quadratic (Pfaffian) relations
Optional inputs:
- Height => ...,
- PrecomputedParametrization => ...,
Outputs:
- randLine, a matrix, a 12x2-matrix spanning a line P11 leading to a Z/3Z-Godeaux surface
- subsLine, a matrix, row matrix containing the variables of the normalized set-up with the a-variables being replaced by the assignments from the chosen line

Description

The order of the torsion group does only depend on the choice of the line in the complete intersection of the quadratic relations Q in ℙ¹¹. A numerical Godeaux surface X with torsion group ℤ/3 has a special reducible bicanonical curve of the form D₁+D₂, where D_i ∈|K_x + t_i| with a torsion element t_i of order i. To construct a surface with such a curve, the associated line l must intersect a particular loci in Q in exactly one point.

Ways to use `randomLineTorsZ3` :

randomLineTorsZ3(Matrix,Matrix)

randomLineTorsZ3 -- compute a line for a numerical Godeaux surface with a cyclic torsion group of order 3

Synopsis

Description

Ways to use randomLineTorsZ3 :

Ways to use `randomLineTorsZ3` :