RationalMap | Ideal -- restriction of a rational map

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 : I = ideal(random(2,P5),random(3,P5));

o3 : Ideal of P5

i4 : Phi' = Phi|I

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                                           2                                                        2                                                                    2
              x  + 16566x x  - 47468x  - 70158x x  + 90626x x  - 24212x  - 38148x x  - 31779x x  + 19605x x  - 769x  - 77864x x  + 10660x x  + 7410x x  - 66565x x  - 49192x  - 71321x x  + 18433x x  - 46460x x  + 4698x x  - 76890x x  - 55574x ,
               0         0 1         1         0 2         1 2         2         0 3         1 3         2 3       3         0 4         1 4        2 4         3 4         4         0 5         1 5         2 5        3 5         4 5         5
              
                 2         3                       2            2           2        3                       2                                       2             2           2           2         3                       2                                     2                                                     2             2           2           2           2         3                       2                                       2                                                     2                                                                   2             2           2          2           2          2         3
              x x  - 51610x  - 80517x x x  + 21123x x  - 3492x x  + 85424x x  + 3605x  + 12124x x x  - 30548x x  - 57743x x x  - 12590x x x  - 19357x x  - 25505x x  + 11508x x  + 35026x x  - 79088x  + 26615x x x  - 10036x x  + 39457x x x  + 879x x x  + 78560x x  + 83997x x x  - 42552x x x  + 69752x x x  - 78713x x  + 52698x x  - 46597x x  - 42666x x  + 34140x x  - 29206x  - 88405x x x  - 15770x x  + 18059x x x  - 14362x x x  + 87935x x  + 67371x x x  - 60584x x x  - 81481x x x  - 61286x x  + 63878x x x  - 33047x x x  + 53673x x x  + 88603x x x  - 71230x x  - 45859x x  + 10167x x  + 8993x x  - 77222x x  - 4109x x  - 16028x
               0 1         1         0 1 2         1 2        0 2         1 2        2         0 1 3         1 3         0 2 3         1 2 3         2 3         0 3         1 3         2 3         3         0 1 4         1 4         0 2 4       1 2 4         2 4         0 3 4         1 3 4         2 3 4         3 4         0 4         1 4         2 4         3 4         4         0 1 5         1 5         0 2 5         1 2 5         2 5         0 3 5         1 3 5         2 3 5         3 5         0 4 5         1 4 5         2 4 5         3 4 5         4 5         0 5         1 5        2 5         3 5        4 5         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
                     }

o4 : RationalMap (quadratic rational map from threefold in PP^5 to 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: complete intersection of type (2,3) in PP^5
     target variety: PP^5
     coefficient ring: ZZ/190181