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NumericalGodeaux :: randomLineTorsZ5

randomLineTorsZ5 -- compute a line for a numerical Godeaux surface with a cyclic torsion group of order 5

Synopsis

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 $\mathbb{Z}/5$ has two special reducible curves in the bicanonical system. These curves are for the form $D_1+D_4$ and $D_2+D_3$, where $D_i \in |K_X + t_i|$ for a torsion element $t_i$ of order $i$. To construct a surface with such curves, the associated line $l$ must intersect each of two different loci in $Q$ in exactly one point. These loci are computed in the procedure lineConditionsTorsZ5.

i1 : kk = QQ;
 
i2 : s = "1111";
 
i3 : (relLin,relPfaf,d1',d2,Ms) = setupGodeaux(kk,s);
 
i4 : (randLine,subsLine) = randomLineTorsZ5(d1',relPfaf);
 
i5 : transpose randLine

o5 = | 0 0  0 2 0 0 0 0 -8 0 0  0 |
     | 0 -2 0 0 0 0 0 0 0  0 -1 0 |

                                                                                                                                                                                                                                                                                                                                      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   , 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   , c   , o     , o     , o     , o     , o     , o     , o     , o     , o     , o     , o     , o     , x ..x , y ..y ])
                 3,2,3   3,1,3   3,0,3   2,2,3   2,1,2   2,0,2   1,1,3   1,1,2   1,0,1   0,0,3   0,0,2   0,0,1   0,0   0,2   0,4   0,6   0,7   1,1   1,3   1,6   2,1   2,3   2,5   2,7   3,0   3,1   3,3   3,5   3,7   1,0,0   2,0,1   2,1,2   3,0,0   3,1,0   4,0,1   4,2,1   4,3,3   5,1,2   5,2,2   5,3,3   5,4,3   0   1   0   3              3,2,3   3,1,3   3,0,3   2,2,3   2,1,2   2,0,2   1,1,3   1,1,2   1,0,1   0,0,3   0,0,2   0,0,1   0,0   0,2   0,4   0,6   0,7   1,1   1,3   1,6   2,1   2,3   2,5   2,7   3,0   3,1   3,3   3,5   3,7   1,0,0   2,0,1   2,1,2   3,0,0   3,1,0   4,0,1   4,2,1   4,3,3   5,1,2   5,2,2   5,3,3   5,4,3   0   1   0   3

See also

Ways to use randomLineTorsZ5 :

For the programmer

The object randomLineTorsZ5 is a method function with options.