random(MultiprojectiveVariety) -- apply a random automorphism of the ambient multi-projective space

Synopsis

Function: random
Usage:

random X
Inputs:
- X, a multi-projective variety, a subvariety of $\mathbb{P}^{k_1}\times\cdots\times\mathbb{P}^{k_n}$
Optional inputs:
- Density => ..., default value 1,
- Height => ..., default value 10,
- MaximalRank => ..., default value false,
- UpperTriangular => ..., default value false,
Outputs:
- a multi-projective variety, the image of $X$ under the action of a random element of $\mathrm{Aut}(\mathbb{P}^{k_1})\times\cdots\times\mathrm{Aut}(\mathbb{P}^{k_n})$

Description

i1 : K = ZZ/65521;

i2 : X = PP_K^({1,1},{2,3});

o2 : ProjectiveVariety, surface in PP^2 x PP^3

i3 : ideal X

              2                              2             2
o3 = ideal (x1  - x1 x1 , x1 x1  - x1 x1 , x1  - x1 x1 , x0  - x0 x0 )
              2     1  3    1  2     0  3    1     0  2    1     0  2

o3 : Ideal of K[x0 ..x0 , x1 ..x1 ]
                  0    2    0    3

i4 : Y = random X;

o4 : ProjectiveVariety, surface in PP^2 x PP^3

i5 : ideal Y

              2                                    2               
o5 = ideal (x1  + 20603x1 x1  - 6053x1 x1  + 6350x1  - 2214x1 x1  -
              1          0  2         1  2         2         0  3  
     ------------------------------------------------------------------------
                                        2                                    
     25885x1 x1  + 10120x1 x1  - 10487x1 , x1 x1  + 32257x1 x1  + 14135x1 x1 
            1  3          2  3          3    0  1          0  2          1  2
     ------------------------------------------------------------------------
             2                                                    2    2  
     + 9059x1  - 14139x1 x1  - 18793x1 x1  + 10289x1 x1  + 21002x1 , x1  -
             2          0  3          1  3          2  3          3    0  
     ------------------------------------------------------------------------
                                      2                             
     14850x1 x1  - 1416x1 x1  - 4590x1  - 29865x1 x1  + 2365x1 x1  +
            0  2         1  2         2          0  3         1  3  
     ------------------------------------------------------------------------
                         2    2                      2                
     2855x1 x1  - 16986x1 , x0  - 9730x0 x0  - 2767x0  - 29185x0 x0  -
           2  3          3    0         0  1         1          0  2  
     ------------------------------------------------------------------------
                          2
     15865x0 x0  + 11692x0 )
            1  2          2

o5 : Ideal of K[x0 ..x0 , x1 ..x1 ]
                  0    2    0    3

Ways to use this method: