substitute(RationalMap,PolynomialRing,PolynomialRing) -- substitute the ambient projective spaces of source and target

Synopsis

Function: substitute
Usage:

sub(phi,R,S)
Inputs:
- phi, a rational map, $\phi:X\subseteq\mathbb{P}^n\dashrightarrow Y\subseteq\mathbb{P}^m$
- R, a polynomial ring, the coordinate ring of $\mathbb{P}^n$
- S, a polynomial ring, the coordinate ring of $\mathbb{P}^m$
Outputs:
- a rational map, a rational map isomorphic to the original map such that the ambient of the source is $R$ and the ambient of the target is $S$

Description

i1 : ZZ/3331[vars(0..5)];

i2 : phi = rationalMap {e^2-d*f, c*e-b*f, c*d-b*e, c^2-a*f, b*c-a*e, b^2-a*d}

o2 = -- rational map --
                   ZZ
     source: Proj(----[a, b, c, d, e, f])
                  3331
                   ZZ
     target: Proj(----[a, b, c, d, e, f])
                  3331
     defining forms: {
                       2
                      e  - d*f,
                      
                      c*e - b*f,
                      
                      c*d - b*e,
                      
                       2
                      c  - a*f,
                      
                      b*c - a*e,
                      
                       2
                      b  - a*d
                     }

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

i3 : R = ZZ/3331[x_0..x_5], S = ZZ/3331[y_0..y_5];

i4 : sub(phi,R,S)

o4 = -- rational map --
                   ZZ
     source: Proj(----[x , x , x , x , x , x ])
                  3331  0   1   2   3   4   5
                   ZZ
     target: Proj(----[y , y , y , y , y , y ])
                  3331  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 PP^5 to PP^5)

Ways to use this method: