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Cremona :: toMap

toMap -- rational map defined by a linear system

Synopsis

Description

When the input represents a list of homogeneous elements F0,...,Fm∈R=K[t0,...,tn]/I of the same degree, then the method returns the ring map φ:K[x0,...,xm] →R that sends xi into Fi.

i1 : QQ[t_0,t_1];
i2 : linSys=gens (ideal(t_0,t_1))^5

o2 = | t_0^5 t_0^4t_1 t_0^3t_1^2 t_0^2t_1^3 t_0t_1^4 t_1^5 |

                        1                  6
o2 : Matrix (QQ[t , t ])  <--- (QQ[t , t ])
                 0   1              0   1
i3 : phi=toMap linSys

                                                 5   4     3 2   2 3     4   5
o3 = map(QQ[t , t ],QQ[x , x , x , x , x , x ],{t , t t , t t , t t , t t , t })
             0   1      0   1   2   3   4   5    0   0 1   0 1   0 1   0 1   1

o3 : RingMap QQ[t , t ] <--- QQ[x , x , x , x , x , x ]
                 0   1           0   1   2   3   4   5

If a positive integer d is passed to the option Dominant, then the method returns the induced map on K[x0,...,xm]/Jd, where Jd is the ideal generated by all homogeneous elements of degree d of the kernel of φ (in this case kernel(RingMap,ZZ) is called).

i4 : phi'=toMap(linSys,Dominant=>2)

                                                                    QQ[x , x , x , x , x , x ]
                                                                        0   1   2   3   4   5                                                    5   4     3 2   2 3     4   5
o4 = map(QQ[t , t ],--------------------------------------------------------------------------------------------------------------------------,{t , t t , t t , t t , t t , t })
             0   1    2                                                 2                                    2                       2           0   0 1   0 1   0 1   0 1   1
                    (x  - x x , x x  - x x , x x  - x x , x x  - x x , x  - x x , x x  - x x , x x  - x x , x  - x x , x x  - x x , x  - x x )
                      4    3 5   3 4    2 5   2 4    1 5   1 4    0 5   3    1 5   2 3    0 5   1 3    0 4   2    0 4   1 2    0 3   1    0 2

                                                                             QQ[x , x , x , x , x , x ]
                                                                                 0   1   2   3   4   5
o4 : RingMap QQ[t , t ] <--- --------------------------------------------------------------------------------------------------------------------------
                 0   1         2                                                 2                                    2                       2
                             (x  - x x , x x  - x x , x x  - x x , x x  - x x , x  - x x , x x  - x x , x x  - x x , x  - x x , x x  - x x , x  - x x )
                               4    3 5   3 4    2 5   2 4    1 5   1 4    0 5   3    1 5   2 3    0 5   1 3    0 4   2    0 4   1 2    0 3   1    0 2

If the input is a pair consisting of a homogeneous ideal I and an integer v, then the output will be the map defined by the linear system of hypersurfaces of degree v which contain the projective subscheme defined by I.

i5 : I=kernel phi

             2                                                 2                                    2                       2
o5 = ideal (x  - x x , x x  - x x , x x  - x x , x x  - x x , x  - x x , x x  - x x , x x  - x x , x  - x x , x x  - x x , x  -
             4    3 5   3 4    2 5   2 4    1 5   1 4    0 5   3    1 5   2 3    0 5   1 3    0 4   2    0 4   1 2    0 3   1  
     ----------------------------------------------------------------------------------------------------------------------------
     x x )
      0 2

o5 : Ideal of QQ[x , x , x , x , x , x ]
                  0   1   2   3   4   5
i6 : toMap(I,2)

                                                                                 2                                                 2                                    2                       2
o6 = map(QQ[x , x , x , x , x , x ],QQ[y , y , y , y , y , y , y , y , y , y ],{x  - x x , x x  - x x , x x  - x x , x x  - x x , x  - x x , x x  - x x , x x  - x x , x  - x x , x x  - x x , x  - x x })
             0   1   2   3   4   5      0   1   2   3   4   5   6   7   8   9    4    3 5   3 4    2 5   2 4    1 5   1 4    0 5   3    1 5   2 3    0 5   1 3    0 4   2    0 4   1 2    0 3   1    0 2

o6 : RingMap QQ[x , x , x , x , x , x ] <--- QQ[y , y , y , y , y , y , y , y , y , y ]
                 0   1   2   3   4   5           0   1   2   3   4   5   6   7   8   9

This is identical to toMap(I,v,1), while the output of toMap(I,v,e) will be the map defined by the linear system of hypersurfaces of degree v having points of multiplicity e along the projective subscheme defined by I.

i7 : toMap(I,2,1)

                                                                                 2                                                 2                                    2                       2
o7 = map(QQ[x , x , x , x , x , x ],QQ[y , y , y , y , y , y , y , y , y , y ],{x  - x x , x x  - x x , x x  - x x , x x  - x x , x  - x x , x x  - x x , x x  - x x , x  - x x , x x  - x x , x  - x x })
             0   1   2   3   4   5      0   1   2   3   4   5   6   7   8   9    4    3 5   3 4    2 5   2 4    1 5   1 4    0 5   3    1 5   2 3    0 5   1 3    0 4   2    0 4   1 2    0 3   1    0 2

o7 : RingMap QQ[x , x , x , x , x , x ] <--- QQ[y , y , y , y , y , y , y , y , y , y ]
                 0   1   2   3   4   5           0   1   2   3   4   5   6   7   8   9
i8 : toMap(I,2,2)

o8 = map(QQ[x , x , x , x , x , x ],QQ[],{})
             0   1   2   3   4   5

o8 : RingMap QQ[x , x , x , x , x , x ] <--- QQ[]
                 0   1   2   3   4   5
i9 : toMap(I,3,2)

                                                         3                2    2                2    2                 2                     2        2                      2              3                2    2
o9 = map(QQ[x , x , x , x , x , x ],QQ[y , y , y , y ],{x  - 2x x x  + x x  + x x  - x x x , x x  - x x  - x x x  + x x  + x x x  - x x x , x x  - x x  - x x x  + x x x  + x x  - x x x , x  - 2x x x  + x x  + x x  - x x x })
             0   1   2   3   4   5      0   1   2   3    3     2 3 4    1 4    2 5    1 3 5   2 3    2 4    1 3 4    0 4    1 2 5    0 3 5   2 3    1 3    1 2 4    0 3 4    1 5    0 2 5   2     1 2 3    0 3    1 4    0 2 4

o9 : RingMap QQ[x , x , x , x , x , x ] <--- QQ[y , y , y , y ]
                 0   1   2   3   4   5           0   1   2   3

Ways to use toMap :