rationalMap(Ring,Tally) -- rational map defined by an effective divisor

Synopsis

Function: rationalMap
Usage:

rationalMap D

rationalMap(R,D)
Inputs:
- R, a ring, the coordinate ring of a locally factorial projective variety $X\subseteq\mathbb{P}^n$
- D, a tally, a multiset of homogeneous ideals in $R$ (or in ambient $R$) defining pure codimension 1 subschemes of $X$ with no embedded components; so that D is interpreted as an effective divisor on $X$
Optional inputs:
- Dominant => ..., default value null,
Outputs:
- a rational map, the rational map defined by the complete linear system $|D|$

Description

In the example below, we take a smooth complete intersection $X\subset\mathbb{P}^5$ of three quadrics containing a conic $C\subset\mathbb{P}^5$. Then we calculate the map defined by the linear system $|2H+C|$, where $H$ is the hyperplane section class of $X$.

i1 : P5 = ZZ/65521[x_0..x_5];

i2 : C = ideal(x_1^2-x_0*x_2,x_3,x_4,x_5)

             2
o2 = ideal (x  - x x , x , x , x )
             1    0 2   3   4   5

o2 : Ideal of P5

i3 : X = quotient ideal(-x_1^2+x_0*x_2-x_1*x_3+x_3^2-x_0*x_5+x_1*x_5+x_3*x_5,-x_0*x_3-x_1*x_3+x_2*x_4-x_3*x_4-x_4^2-x_1*x_5-x_2*x_5+x_5^2,-x_1^2+x_0*x_2+x_2*x_3+x_1*x_4-x_3*x_4-x_4*x_5);

i4 : H = ideal random(1,X)

o4 = ideal(- 32646x  - 28377x  + 26433x  - 29566x  + 3783x  + 26696x )
                   0         1         2         3        4         5

o4 : Ideal of X

i5 : D = new Tally from {H => 2,C => 1};

i6 : time phi = rationalMap D
     -- used 0.0326602 seconds

o6 = -- rational map --
                                  ZZ
     source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
                                65521  0   1   2   3   4   5
             {
                 2                  2
              - x  + x x  - x x  + x  - x x  + x x  + x x ,
                 1    0 2    1 3    3    0 5    1 5    3 5
              
                                             2                  2
              - x x  - x x  + x x  - x x  - x  - x x  - x x  + x ,
                 0 3    1 3    2 4    3 4    4    1 5    2 5    5
              
                 2
              - x  + x x  + x x  + x x  - x x  - x x
                 1    0 2    2 3    1 4    3 4    4 5
             }
                    ZZ
     target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  , y  , y  , y  , y  , y  , y  , y  , y  ])
                  65521  0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
     defining forms: {
                      x x x ,
                       0 4 5
                      
                         2
                      x x ,
                       0 5
                      
                      x x x ,
                       1 2 5
                      
                      x x x ,
                       1 4 5
                      
                         2
                      x x ,
                       1 5
                      
                       2
                      x x ,
                       2 5
                      
                      x x x ,
                       2 3 5
                      
                      x x x ,
                       2 4 5
                      
                         2
                      x x ,
                       2 5
                      
                       2
                      x x ,
                       3 5
                      
                      x x x ,
                       3 4 5
                      
                         2
                      x x ,
                       3 5
                      
                       2
                      x x ,
                       4 5
                      
                         2
                      x x ,
                       4 5
                      
                       3
                      x ,
                       5
                      
                       2      2                 2                       2        2      2       2    3    2                          2                2                          2        2       2       2       2    3
                      x x  + x x  + x x x  + x x  - 3x x x  - x x x  - x x  + x x  - x x  + 2x x  - x  - x x  + 2x x x  + 2x x x  + x x  + 2x x x  - x x  + 4x x x  - 3x x x  + x x  + x x  - 3x x  - 2x x  - 2x x  + x ,
                       0 1    0 2    0 1 2    0 2     0 2 4    1 2 4    2 4    0 4    1 4     2 4    4    0 5     0 2 5     1 2 5    2 5     2 3 5    3 5     1 4 5     3 4 5    4 5    0 5     1 5     2 5     4 5    5
                      
                       2                 2      2     2           3                                2                         2             2      2    2                                                     2                 2                                                           2             2       2      2           2       2         3
                      x x  + x x x  + x x  + x x  + 2x x  + 32760x  - 32760x x x  - x x x  - 32760x x  + 32757x x x  - 32760x x  + 32760x x  + x x  - x x  - 32760x x x  - 32758x x x  + 32760x x x  + 32760x x  + 2x x x  - 3x x  + 32758x x x  + 7x x x  + 32760x x x  - 4x x x  - 32760x x  - 32758x x  - 5x x  - x x  + 32759x x  - 4x x  - 32759x ,
                       0 2    0 1 2    0 2    1 2     2 3         3         0 2 4    1 2 4         2 4         2 3 4         3 4         2 4    3 4    0 5         0 1 5         0 2 5         1 2 5         2 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
                      
                                  2      2    3              2        2
                      x x x  + x x  + x x  + x  + x x x  - 2x x  + x x ,
                       0 1 2    0 2    1 2    2    1 2 4     2 4    2 4
                      
                                                                        2        2      2
                      - 2x x x  + x x x  - 2x x x  + x x x  + x x x  - x x  + x x  + x x ,
                          0 2 5    0 4 5     1 4 5    2 4 5    3 4 5    4 5    1 5    4 5
                      
                         2                          2                2                          2        2       2       2       2    3
                      - x x  + 2x x x  + 2x x x  + x x  + 2x x x  - x x  + 4x x x  - 3x x x  + x x  + x x  - 3x x  - 2x x  - 2x x  + x ,
                         0 5     0 2 5     1 2 5    2 5     2 3 5    3 5     1 4 5     3 4 5    4 5    0 5     1 5     2 5     4 5    5
                      
                                                  2                         2
                      x x x  + x x x  + x x x  + x x  + x x x  - 2x x x  + x x
                       0 1 5    0 2 5    1 2 5    2 5    1 4 5     2 4 5    4 5
                     }

o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)

i7 : time ? image(phi,"F4")
     -- used 0.512096 seconds

o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
     hypersurfaces of degree 2

See also the package Divisor, which provides general tools for working with divisors.

Ways to use this method: