randomLineTorsZ2 -- compute a line for a numerical Godeaux surface with a cyclic torsion group of order 2

Synopsis

Usage:

randomLineTorsZ2(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 => ..., default value 19, optional argument for the random computations
- PrecomputedParametrization => ..., default value true, optional argument for using a precomputed unirational parametrization
Outputs:
- randLine, a matrix, a 12x2-matrix spanning a line in P11 for a Z/2Z-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 $\mathbb{P}^{11}$. A numerical Godeaux surface $X$ with torsion group $\mathbb{Z}/2$ has a special bicanonical curve of the form $2D_2$, where $D_2 \in |K_X + t_2|$ with a torsion element $t_2$ of order 2. To construct a surface with such a curve, the associated line $l$ must intersect a particular loci in $Q$ in exactly one point.

i1 : kk = QQ;

i2 : s = "22";

i3 : (relLin,relPfaf,d1',d2,Ms) = setupGodeaux(kk,s);

i4 : (randLine,subsLine) = randomLineTorsZ2(d1',relPfaf);

i5 : transpose randLine

o5 = | 2 0    0    -8 0     0     -2 0   0      -1   0  0     |
     | 0 1344 3072 -5 -5364 12288 4  192 -32256 -445 40 16128 |

                                                                                                                                                                                                                                                                                                                                2                                                                                                                                                                                                                                                                                                                          12
o5 : Matrix (QQ[a     , a     , a     , a     , a     , a     , a     , a     , a     , a     , a     , a     , c   , c   , c   , c   ..c   , c   , c   , c   , c   ..c   , c   ..c   , c   , c   ..c   , c   , o     , o     , o     , o     , o     , o     , o     , o     , o     , o     , o     , o     , x ..x , y ..y ])  <--- (QQ[a     , a     , a     , a     , a     , a     , a     , a     , a     , a     , a     , a     , c   , c   , c   , c   ..c   , c   , c   , c   , c   ..c   , c   ..c   , c   , c   ..c   , c   , o     , o     , o     , o     , o     , o     , o     , o     , o     , o     , o     , o     , x ..x , y ..y ])
                 2,1,3   2,0,3   2,2,2   3,1,3   3,0,3   3,2,2   0,1,3   0,1,2   0,0,0   1,1,3   1,1,2   1,0,0   0,0   0,2   0,4   0,6   0,7   1,0   1,2   1,4   1,6   1,7   2,0   2,3   2,5   3,0   3,3   3,5   1,0,1   2,0,1   3,1,3   3,2,3   4,0,1   4,3,3   5,0,1   5,1,1   5,1,3   5,2,1   5,3,3   5,4,3   0   1   0   3              2,1,3   2,0,3   2,2,2   3,1,3   3,0,3   3,2,2   0,1,3   0,1,2   0,0,0   1,1,3   1,1,2   1,0,0   0,0   0,2   0,4   0,6   0,7   1,0   1,2   1,4   1,6   1,7   2,0   2,3   2,5   3,0   3,3   3,5   1,0,1   2,0,1   3,1,3   3,2,3   4,0,1   4,3,3   5,0,1   5,1,1   5,1,3   5,2,1   5,3,3   5,4,3   0   1   0   3

Ways to use `randomLineTorsZ2` :

"randomLineTorsZ2(Matrix,Matrix)"

For the programmer

The object randomLineTorsZ2 is a method function with options.

randomLineTorsZ2 -- compute a line for a numerical Godeaux surface with a cyclic torsion group of order 2

Synopsis

Description

Ways to use randomLineTorsZ2 :

For the programmer

Ways to use `randomLineTorsZ2` :