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Cremona :: ideal(RationalMap)

ideal(RationalMap) -- base locus of a rational map

Synopsis

Description

This is generally difficult, but in some cases it is equivalent to ideal matrix phi, which does not perform any computation.

i1 : y = gens(QQ[x_0..x_5]/(x_2^2-x_2*x_3+x_1*x_4-x_0*x_5));
i2 : phi = rationalMap {y_4^2-y_3*y_5,-y_2*y_4+y_3*y_4-y_1*y_5, -y_2*y_3+y_3^2-y_1*y_4, -y_1*y_2+y_1*y_3-y_0*y_4, y_1^2-y_0*y_3}

o2 = -- rational map --
     source: subvariety of Proj(QQ[x , x , x , x , x , x ]) defined by
                                    0   1   2   3   4   5
             {
               2
              x  - x x  + x x  - x x
               2    2 3    1 4    0 5
             }
     target: Proj(QQ[y , y , y , y , y ])
                      0   1   2   3   4
     defining forms: {
                       2
                      x  - x x ,
                       4    3 5
                      
                      - x x  + x x  - x x ,
                         2 4    3 4    1 5
                      
                                2
                      - x x  + x  - x x ,
                         2 3    3    1 4
                      
                      - x x  + x x  - x x ,
                         1 2    1 3    0 4
                      
                       2
                      x  - x x
                       1    0 3
                     }

o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
i3 : time ideal phi
     -- used 0.00311728 seconds

             2                                     2                      
o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
             4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3  
     ------------------------------------------------------------------------
            2
     x x , x  - x x )
      0 4   1    0 3

                     QQ[x ..x ]
                         0   5
o3 : Ideal of -----------------------
               2
              x  - x x  + x x  - x x
               2    2 3    1 4    0 5
i4 : assert(ideal phi == ideal matrix phi)
i5 : phi' = last graph phi

o5 = -- rational map --
     source: subvariety of Proj(QQ[x , x , x , x , x , x ]) x Proj(QQ[y , y , y , y , y ]) defined by
                                    0   1   2   3   4   5              0   1   2   3   4
             {
              x y  - x y  + x y ,
               1 2    3 3    4 4
              
              x y  - x y  - x y  + x y ,
               0 2    1 3    2 4    3 4
              
              x y  - x y  + x y  - x y ,
               2 1    3 1    4 2    5 3
              
              x y  - x y  + x y ,
               1 1    2 2    5 4
              
              x y  - x y  + x y ,
               0 1    2 3    4 4
              
              x y  - x y  + x y  - x y ,
               2 0    3 0    4 1    5 2
              
              x y  - x y  + x y ,
               1 0    3 1    4 2
              
              x y  - x y  + x y ,
               0 0    2 2    4 3
              
               2
              x  - x x  + x x  - x x
               2    2 3    1 4    0 5
             }
     target: Proj(QQ[y , y , y , y , y ])
                      0   1   2   3   4
     defining forms: {
                      y ,
                       0
                      
                      y ,
                       1
                      
                      y ,
                       2
                      
                      y ,
                       3
                      
                      y
                       4
                     }

o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
i6 : time ideal phi'
     -- used 0.14212 seconds

o6 = ideal 1

                                                                                                            QQ[x ..x , y ..y ]
                                                                                                                0   5   0   4
o6 : Ideal of --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                     2
              (x y  - x y  + x y , x y  - x y  - x y  + x y , x y  - x y  + x y  - x y , x y  - x y  + x y , x y  - x y  + x y , x y  - x y  + x y  - x y , x y  - x y  + x y , x y  - x y  + x y , x  - x x  + x x  - x x )
                1 2    3 3    4 4   0 2    1 3    2 4    3 4   2 1    3 1    4 2    5 3   1 1    2 2    5 4   0 1    2 3    4 4   2 0    3 0    4 1    5 2   1 0    3 1    4 2   0 0    2 2    4 3   2    2 3    1 4    0 5
i7 : assert(ideal phi' != ideal matrix phi')

See also