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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 MathMode.

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 0.00495234 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 0.145836 seconds

o3 = 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 )
             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
o3 : Ideal of ------[x ..x ]
              300007  0   9
i4 : time degreeMap phi
     -- used 0.04453 seconds

o4 = 1
i5 : time projectiveDegrees phi
     -- used 1.13334 seconds

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

o5 : List
i6 : time projectiveDegrees(phi,NumDegrees=>0)
     -- used 0.131033 seconds

o6 = {5}

o6 : List
i7 : time phi = toMap(phi,Dominant=>J)
     -- used 0.00433484 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 0.716094 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 0.0123132 seconds

o9 = true
i10 : time degreeMap psi
     -- used 0.282228 seconds

o10 = 1
i11 : time projectiveDegrees psi
     -- used 4.47283 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 0.00185586 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 0.0971709 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 0.452324 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 0.27759 seconds

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

o15 : List
i16 : time degrees phi
     -- used 0.000026579 seconds

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

o16 : List
i17 : time describe phi
     -- used 0.00218513 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 0.00996408 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 0.0129314 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 2.5559 seconds

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

o21 : List
i22 : time degree f
     -- used 0.000031229 seconds

o22 = 1
i23 : time describe f
     -- used 0.00252281 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 5.0 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.