RAT List -- define a multi-rational map

Synopsis

Operator: SPACE
Usage:

H {F1,F2,...}

check H {F1,F2,...}
Inputs:
- H, a hom-set, the hom-set of rational maps between two multi-projective varieties X and Y
- F, a list, a list of homogeneous row matrices over the ring of X of lengths compatible with the shape of Y (these matrices can be replaced by the lists of their entries)
Outputs:
- a multi-rational map, the map from X to Y defined on coordinates by sending a point p to the point (F1(p),F2(p),...)

Description

This is a more controlled way of constructing rational maps than just using the multirationalMap function.

i1 : R = ring PP_(ZZ/3)^{1,2};

i2 : F = {apply(2,i -> random({1,1},R)), apply(4,i -> random({0,1},R)), apply(3,i -> random({1,0},R))}

o2 = {{- x0 x1  + x0 x1 , x0 x1  + x0 x1  + x0 x1  + x0 x1 }, {- x1  - x1  -
           1  0     0  1    0  0     1  0     0  2     1  2        0     1  
     ------------------------------------------------------------------------
     x1 , - x1  - x1 , - x1  + x1  - x1 , x1  - x1  - x1 }, {-x0 , - x0  -
       2      0     2      0     1     2    0     1     2       0      0  
     ------------------------------------------------------------------------
     x0 , x0  + x0 }}
       1    0     1

o2 : List

i3 : H = Hom(PP_(ZZ/3)^{1,2},PP_(ZZ/3)^{1,3,2})

o3 = H

o3 : Hom(PP^1 x PP^2,PP^1 x PP^3 x PP^2)

i4 : f = H F;

o4 : MultirationalMap (rational map from PP^1 x PP^2 to PP^1 x PP^3 x PP^2)

i5 : show f

o5 = -- multi-rational map --
                  ZZ                   ZZ
     source: Proj(--[x0 , x0 ]) x Proj(--[x1 , x1 , x1 ])
                   3   0    1           3   0    1    2
                  ZZ                   ZZ                             ZZ
     target: Proj(--[x0 , x0 ]) x Proj(--[x1 , x1 , x1 , x1 ]) x Proj(--[x2 , x2 , x2 ])
                   3   0    1           3   0    1    2    3           3   0    1    2
     -- rational map 1/3 -- 
     map 1/3, one of its representatives:
     {
      - x0 x1  + x0 x1 ,
          1  0     0  1
      
      x0 x1  + x0 x1  + x0 x1  + x0 x1
        0  0     1  0     0  2     1  2
     }
     -- rational map 2/3 -- 
     map 2/3, one of its representatives:
     {
      - x1  - x1  - x1 ,
          0     1     2
      
      - x1  - x1 ,
          0     2
      
      - x1  + x1  - x1 ,
          0     1     2
      
      x1  - x1  - x1
        0     1     2
     }
     -- rational map 3/3 -- 
     map 3/3, one of its representatives:
     {
      -x0 ,
         0
      
      - x0  - x0 ,
          0     1
      
      x0  + x0
        0     1
     }

The following equality is satisfied for every rational map f.

i6 : assert( f == (Hom(source f,target f)) entries f )

Here it is shown how to make a dominant rational map.

i7 : H' = Hom(PP_(ZZ/3)^{1,2},Dominant);

o7 : Hom(PP^1 x PP^2,*,Dominant)

i8 : H' F

o8 = multi-rational map consisting of 3 rational maps
     source variety: PP^1 x PP^2
     target variety: threefold in PP^1 x PP^3 x PP^2 cut out by 3 hypersurfaces of multi-degrees (0,0,1)^1 (0,1,0)^1 (1,1,1)^1 
     dominance: true

o8 : MultirationalMap (dominant rational map from PP^1 x PP^2 to threefold in PP^1 x PP^3 x PP^2)

i9 : assert(image oo == target oo)

Ways to use this method: