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NumericalGodeaux :: randomSection

randomSection -- choose a point in the solution space defined by the linear relations

Synopsis

Description

After choosing a line in the Pfaffian variety $Q \subset \mathbb{P}^{11}$ and substituting the $a$-variables by the corresponding assignments, we obtain a solution matrix over a $\mathbb{P}^{1}$. Computing (a basis) of all syzygies in the designated degree, we obtain the solution space for the $c$- and $o$-variables. The solution space is a finite dimensional $k$-vector space, and hence isomorphic to a $k^n$. The procedure chooses then a random point in this $k^n$, computes afterwards the corresponding solution for the $c$- and $o$-variables and updates finally the row matrix subsLine with the chosen solutions. After this step, all unknown $a$-, $c$- and $o$-variables have been replaced by a possible assignment and depend only on the variables $x_0,x_1$ and $y_0,\ldots,y_3$. For the configuration "1111", we obtain in the general case (i.e. torsion-free, no hyperelliptic fibers) a 4-dimensional solution space.

i1 : kk = ZZ/29;
 
i2 : s = "1111";
 
i3 : (relLin,relPfaf,d1',d2,Ms) = setupGodeaux(kk,s);
 
i4 : Sa = getP11(relPfaf);
 
i5 : point1 = randomPoint(ideal relPfaf,Sa);

o5 : Ideal of kk[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
 
i6 : (randLine,subsLine) = randomLine(point1,relPfaf,Sa);
 
i7 : (solutionMat,restVars) = solutionMatrix(relLin);
 
i8 : (randPoint,subsPoint) = randomSection(solutionMat,restVars,subsLine);
 
i9 : randPoint

o9 = | -7  |
     | -12 |
     | -12 |
     | 12  |

                                                                                                                                                                                                                                                                                                                                      4                                                                                                                                                                                                                                                                                                                                1
o9 : Matrix (kk[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 ])  <--- (kk[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 randomSection :

For the programmer

The object randomSection is a method function.