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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 degreeMap 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 Certify.

Let $\Phi:X \dashrightarrow Y$ be a rational map from a subvariety $X=V(I)\subseteq\mathbb{P}^n=Proj(K[x_0,\ldots,x_n])$ to a subvariety $Y=V(J)\subseteq\mathbb{P}^m=Proj(K[y_0,\ldots,y_m])$. Then the map $\Phi $ can be represented, although not uniquely, by a homogeneous ring map $\phi:K[y_0,\ldots,y_m]/J \to K[x_0,\ldots,x_n]/I$ 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 $K[x_0,...,x_n]/I$.

Below is an example using the methods provided by this package, dealing with a birational transformation $\Phi:\mathbb{P}^6 \dashrightarrow \mathbb{G}(2,4)\subset\mathbb{P}^9$ 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 0.00358274 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 ], ------[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   6   300007  0   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 ] <--- ------[x ..x ]
             300007  0   6       300007  0   9
i3 : time J = kernel(phi,2)
     -- used 0.0661065 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 ]
              300007  0   9
i4 : time degreeMap phi
     -- used 0.0257491 seconds

o4 = 1
i5 : time projectiveDegrees phi
     -- used 0.518089 seconds

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

o5 : List
i6 : time projectiveDegrees(phi,NumDegrees=>0)
     -- used 0.0673901 seconds

o6 = {5}

o6 : List
i7 : time phi = toMap(phi,Dominant=>J)
     -- used 0.00226517 seconds

                                                                       ZZ
                                                                     ------[x ..x ]
            ZZ                                                       300007  0   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  + 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   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 ]
               ZZ                                                           300007  0   9
o7 : RingMap ------[t ..t ] <--- ----------------------------------------------------------------------------------------------------
             300007  0   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 0.624896 seconds

                                                       ZZ
                                                     ------[x ..x ]
                                                     300007  0   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 ], {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   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 ]
                                                        300007  0   9                                                    ZZ
o8 : RingMap ---------------------------------------------------------------------------------------------------- <--- ------[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   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 0.00700234 seconds

o9 = true
i10 : time degreeMap psi
     -- used 0.202827 seconds

o10 = 1
i11 : time projectiveDegrees psi
     -- used 7.53333 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 0.00198092 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 0.0961269 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 0.535528 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 0.264045 seconds

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

o15 : List
i16 : time degrees phi
     -- used 0.0188628 seconds

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

o16 : List
i17 : time describe phi
     -- used 0.00277073 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 already calculated)
      projective degrees: {1, 3, 9, 17, 21, 15, 5}
      coefficient ring: ZZ/300007
i18 : time describe phi^(-1)
     -- used 0.0105889 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 already calculated)
      projective degrees: {5, 15, 21, 17, 9, 3, 1}
      number of minimal representatives: 1
      dimension base locus: 4
      degree base locus: 24
      coefficient ring: ZZ/300007
i19 : time (f,g) = graph phi^-1; f;
     -- used 0.0388233 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 1.10193 seconds

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

o21 : List
i22 : time degree f
     -- used 0.000011617 seconds

o22 = 1
i23 : time describe f
     -- used 0.000636723 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 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.

Version

This documentation describes version 5.2.1 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

For the programmer

The object Cremona is a package.