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NumericalGodeaux :: precomputedCoxModel

precomputedCoxModel -- load the equation of the 5-dimensional hypersurface in a Cox ring of a toric variety

Synopsis

Description

$G$ is the ideal of a birational model of the 5-dimensional quotient $F(Q)//(G_m)^3$ as an anti-canonical hypersurface in a toric variety. The toric variety is a $\mathbb{P}^3$-bundle over a $\mathbb{P}^2$-bundle over $\mathbb{P}^1$. The fibers over the $\mathbb{P}^2$-bundle over $\mathbb{P}^1$ are K3-surfaces

i1 : G= precomputedCoxModel(QQ)

                  3      2     3             2              2      
o1 = ideal(s s t r r  - s t t r r  + 2s s t r r r  - s s t r r r  -
            0 2 0 0 2    2 0 1 0 2     0 2 0 0 1 2    0 2 1 0 1 2  
     ------------------------------------------------------------------------
       2     2        2 2 2        2   2            2              2    
     2s t t r r r  + s t r r r  + s r r r  - s s r r r  + s s t r r r  -
       2 0 1 0 1 2    2 1 0 1 2    0 0 1 2    0 1 0 1 2    0 2 0 0 1 2  
     ------------------------------------------------------------------------
               2              2      2       2       2 2   2      2 3    
     3s s t r r r  + s s t r r r  - s t t r r r  + 2s t r r r  + s r r  -
       0 2 1 0 1 2    1 2 1 0 1 2    2 0 1 0 1 2     2 1 0 1 2    0 1 2  
     ------------------------------------------------------------------------
          3             3            3      2 2 3            2 2    2 2 2 2  
     s s r r  - 2s s t r r  + s s t r r  + s t r r  - s s t r r  + s t r r  +
      0 1 1 2     0 2 1 1 2    1 2 1 1 2    2 1 1 2    0 2 0 0 2    2 0 0 2  
     ------------------------------------------------------------------------
      2     2 2              2    2 2     2              2    2 2     2  
     s t t r r  - s s t r r r  + s t r r r  + s s t r r r  - s t r r r  -
      2 0 1 0 2    0 2 0 0 1 2    2 0 0 1 2    0 2 1 0 1 2    2 1 0 1 2  
     ------------------------------------------------------------------------
      2 2 2        2 2          2 2          2 2           2 2          2 2  
     s r r  + s s r r  + s s t r r  - s s t r r  + 2s s t r r  - s s t r r  -
      0 1 2    0 1 1 2    0 2 0 1 2    1 2 0 1 2     0 2 1 1 2    1 2 1 1 2  
     ------------------------------------------------------------------------
      2     2 2    2 2 2 2    2       3                3        2   2 3    
     s t t r r  - s t r r  - s s t t r r  + s s s t t r r  + s s t t r r  -
      2 0 1 1 2    2 1 1 2    0 2 0 1 0 3    0 1 2 0 1 0 3    0 2 0 1 0 3  
     ------------------------------------------------------------------------
        2   2 3      3   2         2     2          2   2      
     s s t t r r  + s t r r r  - 2s s t r r r  + s s t r r r  -
      1 2 0 1 0 3    0 0 0 1 3     0 1 0 0 1 3    0 1 0 0 1 3  
     ------------------------------------------------------------------------
       2       2                   2        2       2        2   2 2      
     2s s t t r r r  + 3s s s t t r r r  - s s t t r r r  + s s t r r r  -
       0 2 0 1 0 1 3     0 1 2 0 1 0 1 3    1 2 0 1 0 1 3    0 2 1 0 1 3  
     ------------------------------------------------------------------------
            2 2          2   2 2          2   2 2          2 3 2      
     s s s t r r r  + s s t t r r r  - s s t t r r r  - s s t r r r  +
      0 1 2 1 0 1 3    0 2 0 1 0 1 3    1 2 0 1 0 1 3    0 2 1 0 1 3  
     ------------------------------------------------------------------------
        2 3 2        3     2       2       2        2     2       2   2   2  
     s s t r r r  - s t r r r  + 2s s t r r r  - s s t r r r  + 2s s t r r r 
      1 2 1 0 1 3    0 1 0 1 3     0 1 1 0 1 3    0 1 1 0 1 3     0 2 1 0 1 3
     ------------------------------------------------------------------------
               2   2      2   2   2        2 3   2        2 3   2    
     - 3s s s t r r r  + s s t r r r  - s s t r r r  + s s t r r r  -
         0 1 2 1 0 1 3    1 2 1 0 1 3    0 2 1 0 1 3    1 2 1 0 1 3  
     ------------------------------------------------------------------------
            2 2        2       2                   2          2 2   2      
     s s s t r r r  + s s t t r r r  - 2s s s t t r r r  + s s t t r r r  -
      0 1 2 0 0 2 3    0 2 0 1 0 2 3     0 1 2 0 1 0 2 3    1 2 0 1 0 2 3  
     ------------------------------------------------------------------------
        2   2 2           2   2 2         3               2              
     s s t t r r r  + 2s s t t r r r  - 2s t r r r r  + 3s s t r r r r  -
      0 2 0 1 0 2 3     1 2 0 1 0 2 3     0 0 0 1 2 3     0 1 0 0 1 2 3  
     ------------------------------------------------------------------------
        2                    2             2                
     s s t r r r r  - s s s t r r r r  + 3s s t t r r r r  -
      0 1 0 0 1 2 3    0 1 2 0 0 1 2 3     0 2 0 1 0 1 2 3  
     ------------------------------------------------------------------------
                            2                    2 2            
     4s s s t t r r r r  + s s t t r r r r  + s s t t r r r r  -
       0 1 2 0 1 0 1 2 3    1 2 0 1 0 1 2 3    1 2 0 1 0 1 2 3  
     ------------------------------------------------------------------------
      2   2                   2              2   2              2   2        
     s s t r r r r  + 2s s s t r r r r  - s s t t r r r r  + s s t t r r r r 
      0 2 1 0 1 2 3     0 1 2 1 0 1 2 3    0 2 0 1 0 1 2 3    1 2 0 1 0 1 2 3
     ------------------------------------------------------------------------
          2 3               2 3             2     2           2   2      
     + s s t r r r r  - 2s s t r r r r  - 2s s t r r r  + 2s s t r r r  +
        0 2 1 0 1 2 3     1 2 1 0 1 2 3     0 1 0 1 2 3     0 1 0 1 2 3  
     ------------------------------------------------------------------------
      3   2         2     2           2   2                   2      
     s t r r r  - 3s s t r r r  + 2s s t r r r  + 3s s s t t r r r  -
      0 1 1 2 3     0 1 1 1 2 3     0 1 1 1 2 3     0 1 2 0 1 1 2 3  
     ------------------------------------------------------------------------
       2       2         2   2 2               2 2         2   2 2      
     2s s t t r r r  - 2s s t r r r  + 5s s s t r r r  - 2s s t r r r  -
       1 2 0 1 1 2 3     0 2 1 1 2 3     0 1 2 1 1 2 3     1 2 1 1 2 3  
     ------------------------------------------------------------------------
        2   2 2          2 3 2           2 3 2        3     2    
     s s t t r r r  + s s t r r r  - 2s s t r r r  + s t r r r  -
      1 2 0 1 1 2 3    0 2 1 1 2 3     1 2 1 1 2 3    0 0 1 2 3  
     ------------------------------------------------------------------------
      2       2            2   2      2   2   2      2         2    
     s s t r r r  - s s s t r r r  + s s t r r r  - s s t t r r r  +
      0 1 0 1 2 3    0 1 2 0 1 2 3    1 2 0 1 2 3    0 2 0 1 1 2 3  
     ------------------------------------------------------------------------
                  2      2       2 2 2      2     2 2 2        2   3 2 2  
     s s s t t r r r  + s s s t t r r  - s s s t t r r  - s s s t t r r  +
      0 1 2 0 1 1 2 3    0 1 2 0 1 0 3    0 1 2 0 1 0 3    0 1 2 0 1 0 3  
     ------------------------------------------------------------------------
      2 2   3 2 2    3           2     2 2         2      3         2  
     s s t t r r  - s s t t r r r  + 2s s t t r r r  - s s t t r r r  +
      1 2 0 1 0 3    0 1 0 1 0 1 3     0 1 0 1 0 1 3    0 1 0 1 0 1 3  
     ------------------------------------------------------------------------
       2       2     2       2     2     2    3     2     2    2     3     2
     2s s s t t r r r  - 3s s s t t r r r  + s s t t r r r  - s s s t r r r 
       0 1 2 0 1 0 1 3     0 1 2 0 1 0 1 3    1 2 0 1 0 1 3    0 1 2 1 0 1 3
     ------------------------------------------------------------------------
          2   3     2        2   3     2    2 2   3     2        2 4     2  
     + s s s t r r r  - s s s t t r r r  + s s t t r r r  + s s s t r r r  -
        0 1 2 1 0 1 3    0 1 2 0 1 0 1 3    1 2 0 1 0 1 3    0 1 2 1 0 1 3  
     ------------------------------------------------------------------------
      2 2 4     2    3   2 2 2     2 2 2 2 2      3 2 2 2     2     3 2 2  
     s s t r r r  + s s t r r  - 2s s t r r  + s s t r r  - 2s s s t r r  +
      1 2 1 0 1 3    0 1 1 1 3     0 1 1 1 3    0 1 1 1 3     0 1 2 1 1 3  
     ------------------------------------------------------------------------
         2   3 2 2    3   3 2 2        2 4 2 2    2 2 4 2 2    2 2 2     2  
     3s s s t r r  - s s t r r  + s s s t r r  - s s t r r  + s s t r r r  -
       0 1 2 1 1 3    1 2 1 1 3    0 1 2 1 1 3    1 2 1 1 3    0 1 0 1 2 3  
     ------------------------------------------------------------------------
        3 2     2    3           2    2 2         2      2   2       2  
     s s t r r r  + s s t t r r r  - s s t t r r r  - s s s t t r r r  +
      0 1 0 1 2 3    0 1 0 1 1 2 3    0 1 0 1 1 2 3    0 1 2 0 1 1 2 3  
     ------------------------------------------------------------------------
      3   2       2    2       2     2      2     2     2
     s s t t r r r  - s s s t t r r r  + s s s t t r r r )
      1 2 0 1 1 2 3    0 1 2 0 1 1 2 3    0 1 2 0 1 1 2 3

o1 : Ideal of QQ[s ..s , t ..t , r ..r ]
                  0   2   0   1   0   3
 
i2 : coxRing = ring G

o2 = coxRing

o2 : PolynomialRing
 
i3 : 5==dim G-3

o3 = true
 
i4 : vars coxRing

o4 = | s_0 s_1 s_2 t_0 t_1 r_0 r_1 r_2 r_3 |

                   1             9
o4 : Matrix coxRing  <--- coxRing
 
i5 : degrees coxRing

o5 = {{1, 1, 0}, {1, 1, 0}, {1, 0, 0}, {0, 1, 0}, {0, 1, 0}, {1, 2, 1}, {1,
     ------------------------------------------------------------------------
     2, 1}, {1, 2, 1}, {0, 0, 1}}

o5 : List
 
i6 : degrees source gens G

o6 = {{6, 10, 4}}

o6 : List
 
i7 : sum degrees coxRing

o7 = {6, 10, 4}

o7 : List

See also

Ways to use precomputedCoxModel :

For the programmer

The object precomputedCoxModel is a method function.