invertBirationalMap -- Computes the inverse of a birational map.

Synopsis

Usage:

invertBirationalMap(I,phi)
Inputs:
- I, an ideal, defining the source variety
- phi, a matrix, a row matrix over the ring of I with linear system defining the map.
Outputs:
- a list, of a matrix with the inverse of the birational map and the ideal of its image.

Description

Computes the inverse of a birational map.

i1 : K=QQ;

i2 : R=K[v,u,z];

i3 : I=ideal(v^8-u^3*(z+u)^5);

o3 : Ideal of R

i4 : linsys=matrix {{u^6+4*u^5*z+6*u^4*z^2+4*u^3*z^3+u^2*z^4,v*u^5+3*v*u^4*z+3*v*u^3*z^2+v*u^2*z^3,v^2*u^4+3*v^2*u^3*z+3*v^2*u^2*z^2+v^2*u*z^3,v^3*u^3+2*v^3*u^2*z+v^3*u*z^2,v^4*u^2+v^4*u*z,v^5*u+v^5*z,v^6}};

             1       7
o4 : Matrix R  <--- R

i5 : invertBirationalMap(I,linsys)

                                                                             
o5 = {| -t_5t_6       |, ideal (t t  - t t , t t  - t t , t t  - t t , t t  -
      | -t_4t_6       |          4 5    3 6   3 5    2 6   1 5    0 6   3 4  
      | -t_5^2+t_4t_6 |
     ------------------------------------------------------------------------
                                2   2                              2        
     t t , t t  - t t , t t  - t , t  - t t , t t  - t t , t t  - t , t t  -
      1 6   2 4    0 6   0 4    6   3    0 6   2 3    0 5   1 3    6   0 3  
     ------------------------------------------------------------------------
                                2   2                       2
     t t , t t  - t t , t t  - t , t  - t t , t t  - t t , t  - t t )}
      5 6   1 2    5 6   0 2    5   1    4 6   0 1    3 6   0    2 6

o5 : List

Caveat

Speed this up. Perhaps give as argument just phi in R/I. Does not check whether phi is birational.

Ways to use invertBirationalMap:

invertBirationalMap(Ideal,Matrix)

For the programmer

The object invertBirationalMap is a method function.