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Cremona :: Cremona

Cremona -- package for some computations on rational maps between projective varieties

Description

Cremona is a package to perform some basic computations on rational and birational maps between absolutely irreducible projective varieties over a field K. For instance, it provides general methods to compute degrees and projective degrees of rational maps (see degreeOfRationalMap and projectiveDegrees) and a general method to compute the push-forward to projective space of Segre classes (see SegreClass). Moreover, all the main methods are available both in version probabilistic and in version deterministic, and one can switch from one to the other with the boolean option MathMode.

Let Φ:X ---> Y be a rational map from a subvariety X=V(I)⊆ℙn=Proj(K[x0,...,xn]) to a subvariety Y=V(J)⊆ℙm=Proj(K[y0,...,ym]). We assume that the map Φ can be represented, although not uniquely, by a homogeneous ring map φ:K[y0,...,ym]/J →K[x0,...,xn]/I of quotients of polynomial rings by homogeneous ideals. These kinds of ring maps, together with the objects of the RationalMap class, are the typical inputs for the methods in this package. The method toMap (resp. rationalMap) constructs such a ring map (resp. rational map) from a list of m+1 homogeneous elements of the same degree in K[x0,...,xn]/I.

Below is an example using the methods provided by this package, dealing with a birational transformation Φ:ℙ6 ---> G(2,4)⊂ℙ9 of bidegree (3,3).

i1 : ZZ/300007[t_0..t_6];
i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
     -- used 0.00209206 seconds

           ZZ                                 ZZ                                               3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
o2 = map(------[t , t , t , t , t , t , t ],------[x , x , x , x , x , x , x , x , x , x ],{- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
         300007  0   1   2   3   4   5   6  300007  0   1   2   3   4   5   6   7   8   9      2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6

               ZZ                                      ZZ
o2 : RingMap ------[t , t , t , t , t , t , t ] <--- ------[x , x , x , x , x , x , x , x , x , x ]
             300007  0   1   2   3   4   5   6       300007  0   1   2   3   4   5   6   7   8   9
i3 : time J = kernel(phi,2)
     -- used 0.0902663 seconds

o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x 
             6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
     ------------------------------------------------------------------------
     - x x  + x x , x x  - x x  + x x )
        1 6    0 8   2 4    1 5    0 7

                ZZ
o3 : Ideal of ------[x , x , x , x , x , x , x , x , x , x ]
              300007  0   1   2   3   4   5   6   7   8   9
i4 : time degreeOfRationalMap phi
     -- used 0.0153314 seconds

o4 = 1
i5 : time projectiveDegrees phi
     -- used 0.284182 seconds

o5 = {1, 3, 9, 17, 21, 15, 5}

o5 : List
i6 : time projectiveDegrees(phi,NumDegrees=>0)
     -- used 0.0776131 seconds

o6 = {5}

o6 : List
i7 : time phi = toMap(phi,Dominant=>J)
     -- used 0.00185381 seconds

                                                                         ZZ
                                                                       ------[x , x , x , x , x , x , x , x , x , x ]
           ZZ                                                          300007  0   1   2   3   4   5   6   7   8   9                                 3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
o7 = map(------[t , t , t , t , t , t , t ],----------------------------------------------------------------------------------------------------,{- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
         300007  0   1   2   3   4   5   6  (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 )     2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
                                              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7

                                                                                  ZZ
                                                                                ------[x , x , x , x , x , x , x , x , x , x ]
               ZZ                                                               300007  0   1   2   3   4   5   6   7   8   9
o7 : RingMap ------[t , t , t , t , t , t , t ] <--- ----------------------------------------------------------------------------------------------------
             300007  0   1   2   3   4   5   6       (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 )
                                                       6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
i8 : time psi = inverseMap phi
     -- used 0.62872 seconds

                                      ZZ
                                    ------[x , x , x , x , x , x , x , x , x , x ]
                                    300007  0   1   2   3   4   5   6   7   8   9                               ZZ                                 3                2               2    2                        2                          2     2        2                               2                                   2               2             2                       3                                                 2                 2    2                                  2    2                 2                                                 3                         2      2    2      2                                              2
o8 = map(----------------------------------------------------------------------------------------------------,------[t , t , t , t , t , t , t ],{x  - 2x 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  - 2x 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 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 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  - 2x 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  - 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 ) 300007  0   1   2   3   4   5   6    2     1 2 3    0 3    1 2 5    0 5    1 6    0 2 6    0 4 6    1 7     0 2 7    0 4 7     0 9   2 3    1 3    1 2 6    0 3 6    0 5 6    1 8     0 2 8    0 4 8    0 1 9   2 3    1 3 6    0 6    0 3 8    1 9    0 2 9    0 4 9   3    1 3 8    0 6 8    1 2 9     0 3 9    0 5 9   3 6    2 3 8    0 8    2 9    1 3 9    0 6 9    0 7 9   3 6    3 8    2 6 8    1 8    2 3 9    2 5 9     1 6 9    1 7 9    0 8 9   6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
           6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7

                                          ZZ
                                        ------[x , x , x , x , x , x , x , x , x , x ]
                                        300007  0   1   2   3   4   5   6   7   8   9                                    ZZ
o8 : RingMap ---------------------------------------------------------------------------------------------------- <--- ------[t , t , t , t , t , t , t ]
             (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 )      300007  0   1   2   3   4   5   6
               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
i9 : time isInverseMap(phi,psi)
     -- used 0.00591944 seconds

o9 = true
i10 : time degreeOfRationalMap psi
     -- used 0.0666944 seconds

o10 = 1
i11 : time projectiveDegrees psi
     -- used 0.991954 seconds

o11 = {5, 15, 21, 17, 9, 3, 1}

o11 : List

We repeat the example using the type RationalMap and using deterministic methods.

i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
     -- used 0.00284441 seconds

o12 = -- rational map --
                     ZZ
      source: Proj(------[t , t , t , t , t , t , t ])
                   300007  0   1   2   3   4   5   6
                     ZZ
      target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
                   300007  0   1   2   3   4   5   6   7   8   9
      defining forms: {
                          3                2    2
                       - t  + 2t t t  - t t  - t t  + t t t ,
                          2     1 2 3    0 3    1 4    0 2 4
                       
                          2        2                      2
                       - t t  + t t  + t t t  - t t t  - t t  + t t t ,
                          2 3    1 3    1 2 4    0 3 4    1 5    0 2 5
                       
                            2    2                 2
                       - t t  + t t  + t t t  - t t  - t t t  + t t t ,
                          2 3    2 4    1 3 4    0 4    1 2 5    0 3 5
                       
                          3                2    2
                       - t  + 2t t t  - t t  - t t  + t t t ,
                          3     2 3 4    1 4    2 5    1 3 5
                       
                          2                                 2
                       - t t  + t t t  + t t t  - t t t  - t t  + t t t ,
                          2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6
                       
                                     2    2
                       - t t t  + t t  + t t  - t t t  - t t t  + t t t ,
                          2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6
                       
                          2        2                      2
                       - t t  + t t  + t t t  - t t t  - t t  + t t t ,
                          3 4    2 4    2 3 5    1 4 5    2 6    1 3 6
                       
                            2                        2
                       - t t  + t t t  + t t t  - t t  - t t t  + t t t ,
                          2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6
                       
                            2    2                 2
                       - t t  + t t  + t t t  - t t  - t t t  + t t t ,
                          3 4    3 5    2 4 5    1 5    2 3 6    1 4 6
                       
                          3                2    2
                       - t  + 2t t t  - t t  - t t  + t t t
                          4     3 4 5    2 5    3 6    2 4 6
                      }

o12 : RationalMap (cubic rational map from PP^6 to PP^9)
i13 : time phi = rationalMap(phi,Dominant=>2)
     -- used 0.126082 seconds

o13 = -- rational map --
                     ZZ
      source: Proj(------[t , t , t , t , t , t , t ])
                   300007  0   1   2   3   4   5   6
                                   ZZ
      target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
                                 300007  0   1   2   3   4   5   6   7   8   9
              {
               x x  - x x  + x x ,
                6 7    5 8    4 9
               
               x x  - x x  + x x ,
                3 7    2 8    1 9
               
               x x  - x x  + x x ,
                3 5    2 6    0 9
               
               x x  - x x  + x x ,
                3 4    1 6    0 8
               
               x x  - x x  + x x
                2 4    1 5    0 7
              }
      defining forms: {
                          3                2    2
                       - t  + 2t t t  - t t  - t t  + t t t ,
                          2     1 2 3    0 3    1 4    0 2 4
                       
                          2        2                      2
                       - t t  + t t  + t t t  - t t t  - t t  + t t t ,
                          2 3    1 3    1 2 4    0 3 4    1 5    0 2 5
                       
                            2    2                 2
                       - t t  + t t  + t t t  - t t  - t t t  + t t t ,
                          2 3    2 4    1 3 4    0 4    1 2 5    0 3 5
                       
                          3                2    2
                       - t  + 2t t t  - t t  - t t  + t t t ,
                          3     2 3 4    1 4    2 5    1 3 5
                       
                          2                                 2
                       - t t  + t t t  + t t t  - t t t  - t t  + t t t ,
                          2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6
                       
                                     2    2
                       - t t t  + t t  + t t  - t t t  - t t t  + t t t ,
                          2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6
                       
                          2        2                      2
                       - t t  + t t  + t t t  - t t t  - t t  + t t t ,
                          3 4    2 4    2 3 5    1 4 5    2 6    1 3 6
                       
                            2                        2
                       - t t  + t t t  + t t t  - t t  - t t t  + t t t ,
                          2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6
                       
                            2    2                 2
                       - t t  + t t  + t t t  - t t  - t t t  + t t t ,
                          3 4    3 5    2 4 5    1 5    2 3 6    1 4 6
                       
                          3                2    2
                       - t  + 2t t t  - t t  - t t  + t t t
                          4     3 4 5    2 5    3 6    2 4 6
                      }

o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
i14 : time phi^(-1)
     -- used 0.616267 seconds

o14 = -- rational map --
                                   ZZ
      source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
                                 300007  0   1   2   3   4   5   6   7   8   9
              {
               x x  - x x  + x x ,
                6 7    5 8    4 9
               
               x x  - x x  + x x ,
                3 7    2 8    1 9
               
               x x  - x x  + x x ,
                3 5    2 6    0 9
               
               x x  - x x  + x x ,
                3 4    1 6    0 8
               
               x x  - x x  + x x
                2 4    1 5    0 7
              }
                     ZZ
      target: Proj(------[t , t , t , t , t , t , t ])
                   300007  0   1   2   3   4   5   6
      defining forms: {
                        3                2               2    2                        2                          2
                       x  - 2x 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  - 2x x ,
                        2     1 2 3    0 3    1 2 5    0 5    1 6    0 2 6    0 4 6    1 7     0 2 7    0 4 7     0 9
                       
                        2        2                               2
                       x x  - x x  - x x x  + x x x  + x x x  + x x  - 2x x x  - x x x  + x x x ,
                        2 3    1 3    1 2 6    0 3 6    0 5 6    1 8     0 2 8    0 4 8    0 1 9
                       
                          2               2             2
                       x x  - x x x  + x x  - x x x  + x x  - x x x  - x x x ,
                        2 3    1 3 6    0 6    0 3 8    1 9    0 2 9    0 4 9
                       
                        3
                       x  - x x x  + x x x  + x x x  - 2x x x  - x x x ,
                        3    1 3 8    0 6 8    1 2 9     0 3 9    0 5 9
                       
                        2                 2    2
                       x x  - x x x  + x x  + x x  - x x x  - x x x  - x x x ,
                        3 6    2 3 8    0 8    2 9    1 3 9    0 6 9    0 7 9
                       
                          2    2                 2
                       x x  - x x  - x x x  + x x  + x x x  + x x x  - 2x x x  - x x x  + x x x ,
                        3 6    3 8    2 6 8    1 8    2 3 9    2 5 9     1 6 9    1 7 9    0 8 9
                       
                        3                         2      2    2      2                                              2
                       x  - 2x 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  - 2x x
                        6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
                      }

o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
i15 : time degrees phi^(-1)
     -- used 0.285662 seconds

o15 = {5, 15, 21, 17, 9, 3, 1}

o15 : List
i16 : time degrees phi
     -- used 0.000027058 seconds

o16 = {1, 3, 9, 17, 21, 15, 5}

o16 : List
i17 : time describe phi
     -- used 0.00162444 seconds

o17 = rational map defined by forms of degree 3
      source variety: PP^6
      target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
      dominance: true
      birationality: true (the inverse map is known)
      projective degrees: {1, 3, 9, 17, 21, 15, 5}
      coefficient ring: ZZ/300007
i18 : time describe phi^(-1)
     -- used 0.00122355 seconds

o18 = rational map defined by forms of degree 3
      source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
      target variety: PP^6
      dominance: true
      birationality: true (the inverse map is known)
      projective degrees: {5, 15, 21, 17, 9, 3, 1}
      coefficient ring: ZZ/300007
i19 : time (f,g) = graph phi^-1; f;
     -- used 0.00954049 seconds

o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
i21 : time degrees f
     -- used 1.22938 seconds

o21 = {904, 508, 268, 130, 56, 20, 5}

o21 : List
i22 : time degree f
     -- used 0.000010529 seconds

o22 = 1
i23 : time describe f
     -- used 0.000823854 seconds

o23 = rational map defined by multiforms of degree {1, 0}
      source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20 hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2, 0},{2, 0})
      target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
      dominance: true
      birationality: true
      projective degrees: {904, 508, 268, 130, 56, 20, 5}
      coefficient ring: ZZ/300007

A rudimentary version of Cremona has been already used in an essential way in the paper doi:10.1016/j.jsc.2015.11.004 (it was originally named bir.m2).

Author

Certification a gold star

Version 4.2.2 of this package was accepted for publication in volume 8 of the journal The Journal of Software for Algebra and Geometry on 11 June 2018, in the article A Macaulay2 package for computations with rational maps. That version can be obtained from the journal or from the Macaulay2 source code repository, http://github.com/Macaulay2/M2/blob/master/M2/Macaulay2/packages/Cremona.m2, commit number 2e87a29e4b5b68af1bd8917a9c76d4008ff9fc5b.

Version

This documentation describes version 4.2.2 of Cremona.

Source code

The source code from which this documentation is derived is in the file Cremona.m2. The auxiliary files accompanying it are in the directory Cremona/.

Exports