rationalMap(Ideal,ZZ,ZZ) -- makes a rational map from an ideal

Synopsis

Function: rationalMap
Usage:

rationalMap(I,d,e)

rationalMap(I,d)

rationalMap I
Inputs:
- I, an ideal, a homogeneous ideal in the coordinate ring of a projective variety $X\subseteq\mathbb{P}^n$
- d, an integer, a positive integer (or a list of integers in the case $X$ is embedded in a product of projective spaces)
- e, an integer, a positive integer (if omitted, it is taken to be 1)
Optional inputs:
- Dominant => ..., default value null,
Outputs:
- a rational map, the rational map $X\dashrightarrow \mathbb{P}^m$ defined by the linear system of hypersurfaces of degree $d$ having points of multiplicity $e$ along the projective subscheme of $X$ defined by $I$.

Description

In most cases, the command rationalMap(I,d,e) yields the same output as rationalMap(saturate(I^e),d), but the former is implemented using pure linear algebra.

The command rationalMap I is basically equivalent to rationalMap(I,max degrees I).

In the following example, we calculate the rational map defined by the linear system of cubic hypersurfaces in $\mathbb{P}^6$ having double points along a Veronese surface $V\subset\mathbb{P}^5\subset\mathbb{P}^6$.

i1 : ZZ/33331[x_0..x_6]; V = ideal(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,x_6);

                ZZ
o2 : Ideal of -----[x ..x ]
              33331  0   6

i3 : time phi = rationalMap(V,3,2)
     -- used 0.115324 seconds

o3 = -- rational map --
                    ZZ
     source: Proj(-----[x , x , x , x , x , x , x ])
                  33331  0   1   2   3   4   5   6
                    ZZ
     target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  , y  ])
                  33331  0   1   2   3   4   5   6   7   8   9   10   11   12   13
     defining forms: {
                       3
                      x ,
                       6
                      
                         2
                      x x ,
                       5 6
                      
                         2
                      x x ,
                       4 6
                      
                         2
                      x x ,
                       3 6
                      
                         2
                      x x ,
                       2 6
                      
                         2
                      x x ,
                       1 6
                      
                         2
                      x x ,
                       0 6
                      
                       2
                      x x  - x x x ,
                       4 6    3 5 6
                      
                      x x x  - x x x ,
                       2 4 6    1 5 6
                      
                      x x x  - x x x ,
                       2 3 6    1 4 6
                      
                       2
                      x x  - x x x ,
                       2 6    0 5 6
                      
                      x x x  - x x x ,
                       1 2 6    0 4 6
                      
                       2
                      x x  - x x x ,
                       1 6    0 3 6
                      
                       2                  2    2
                      x x  - 2x x x  + x x  + x x  - x x x
                       2 3     1 2 4    0 4    1 5    0 3 5
                     }

o3 : RationalMap (cubic rational map from PP^6 to PP^13)

i4 : describe phi!

o4 = rational map defined by forms of degree 3
     source variety: PP^6
     target variety: PP^13
     image: 6-dimensional variety of degree 16 in PP^13 cut out by 21 hypersurfaces of degree 2
     dominance: false
     birationality: false
     degree of map: 1
     projective degrees: {1, 3, 6, 12, 16, 16, 16}
     number of minimal representatives: 1
     dimension base locus: 4
     degree base locus: 3
     coefficient ring: ZZ/33331

Ways to use this method: