randomLineTors0 -- compute a line for a torsion-free numerical Godeaux surface

Synopsis

Usage:

randomLineTors0(d1',relPfaf)

randomLineTors0(d1',relPfaf,Ms,h);
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
- Ms, a list, containing the four skew-symmetric matrices describing the Pfaffian relations
- h, an integer, the number of hyperelliptic fibers in the bicanonical fibration
Outputs:
- randLine, a matrix, a 12x2-matrix spanning a line in P11 for a torsion-free Godeaux surface
- subsLine, a matrix, a row matrix containing the variables of the normalized set-up with the a-variables being replaced by the assginments 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 the $\mathbb{P}^{11}$ of $a$-variables. The associated line to a numerical Godeaux surface with a trivial torsion group and no hyperelliptic bicanonical fibers does not intersect any of the special loci in $Q$. We call such a line a general line in $Q$. To obtain surfaces with one respectively two hyperelliptic fibers, the corresponding line must intersect the hyperelliptic locus in $Q$ in one respectively two points. The number of hyperelliptic fibers must be between 0 and 2.

Ways to use `randomLineTors0` :

"randomLineTors0(Matrix,Matrix)"
"randomLineTors0(Matrix,Matrix,List,ZZ)"

For the programmer

The object randomLineTors0 is a method function.

randomLineTors0 -- compute a line for a torsion-free numerical Godeaux surface

Synopsis

Description

Ways to use randomLineTors0 :

For the programmer

Ways to use `randomLineTors0` :