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# RationalMap || Ideal -- restriction of a rational map

- Operator: ||
- Usage:
    `Phi || J`
- Inputs:
  - Phi, a rational map, $\phi:X \dashrightarrow Y$
  - J, an ideal, a homogeneous ideal of a subvariety $Z\subset Y$
- Outputs:
  - a rational map, the restriction of $\phi$ to ${\phi}^{(-1)} Z$, ${{\phi}|}_{{\phi}^{(-1)} Z}: {\phi}^{(-1)} Z \dashrightarrow Z$

## Description

    i1 : P5 = ZZ/190181[x_0..x_5]

    o1 = P5

    o1 : PolynomialRing

    i2 : Phi = rationalMap {x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3}

    o2 = -- rational map --
                        ZZ
         source: Proj(------[x , x , x , x , x , x ])
                      190181  0   1   2   3   4   5
                        ZZ
         target: Proj(------[x , x , x , x , x , x ])
                      190181  0   1   2   3   4   5
         defining forms: {
                           2
                          x  - x x ,
                           4    3 5

                          x x  - x x ,
                           2 4    1 5

                          x x  - x x ,
                           2 3    1 4

                           2
                          x  - x x ,
                           2    0 5

                          x x  - x x ,
                           1 2    0 4

                           2
                          x  - x x
                           1    0 3
                         }

    o2 : RationalMap (quadratic rational map from PP^5 to PP^5)

    i3 : J = ideal random(1,P5);

    o3 : Ideal of P5

    i4 : Phi' = Phi||J

    o4 = -- rational map --
                                      ZZ
         source: subvariety of Proj(------[x , x , x , x , x , x ]) defined by
                                    190181  0   1   2   3   4   5
                 {
                   2                    2                                                               2
                  x  + 9702x x  - 94294x  - x x  + 68094x x  - 9702x x  - 68094x x  + 93593x x  - 53251x  + 94294x x  - 93593x x  + 53251x x
                   1        1 2         2    0 3         2 3        0 4         1 4         2 4         4         0 5         1 5         3 5
                 }
                                      ZZ
         target: subvariety of Proj(------[x , x , x , x , x , x ]) defined by
                                    190181  0   1   2   3   4   5
                 {
                  x  + 16566x  - 70158x  - 38148x  - 77864x  - 71321x
                   0         1         2         3         4         5
                 }
         defining forms: {
                           2
                          x  - x x ,
                           4    3 5

                          x x  - x x ,
                           2 4    1 5

                          x x  - x x ,
                           2 3    1 4

                           2
                          x  - x x ,
                           2    0 5

                          x x  - x x ,
                           1 2    0 4

                                             2                                                        2
                          - 9702x x  + 94294x  - 68094x x  + 9702x x  + 68094x x  - 93593x x  + 53251x  - 94294x x  + 93593x x  - 53251x x
                                 1 2         2         2 3        0 4         1 4         2 4         4         0 5         1 5         3 5
                         }

    o4 : RationalMap (quadratic rational map from hypersurface in PP^5 to hypersurface in PP^5)

    i5 : describe Phi

    o5 = rational map defined by forms of degree 2
         source variety: PP^5
         target variety: PP^5
         coefficient ring: ZZ/190181

    i6 : describe Phi'

    o6 = rational map defined by forms of degree 2
         source variety: smooth quadric hypersurface in PP^5
         target variety: hyperplane in PP^5
         coefficient ring: ZZ/190181

## See also

- RationalMap | Ideal -- restriction of a rational map

## Ways to use this method:

- RationalMap || Ideal -- restriction of a rational map
- `RationalMap || Ring`
- `RationalMap || RingElement`

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Cremona/documentation.m2:776:0.