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 0.00664726 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 0.170468 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 0.0570263 seconds

o4 = 1

i5 : time projectiveDegrees phi
     -- used 1.18491 seconds

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

o5 : List

i6 : time projectiveDegrees(phi,NumDegrees=>0)
     -- used 0.135894 seconds

o6 = {5}

o6 : List

i7 : time phi = toMap(phi,Dominant=>J)
     -- used 0.00533788 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 1.01233 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 0.0154738 seconds

o9 = true

i10 : time degreeMap psi
     -- used 0.361406 seconds

o10 = 1

i11 : time projectiveDegrees psi
     -- used 13.4415 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 0.00519467 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 0.160102 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 0.833222 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 0.466942 seconds

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

o15 : List

i16 : time degrees phi
     -- used 0.0244792 seconds

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

o16 : List

i17 : time describe phi
     -- used 0.00430572 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 0.0494793 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 0.0167763 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 2.25191 seconds

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

o21 : List

i22 : time degree f
     -- used 0.00001861 seconds

o22 = 1

i23 : time describe f
     -- used 0.00114102 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

Giovanni Staglianò <[email protected]>

Certification

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

Types
- RationalMap -- the class of all rational maps between absolutely irreducible projective varieties over a field
Functions and commands
- abstractRationalMap -- make an abstract rational map
- approximateInverseMap -- random map related to the inverse of a birational map
- ChernSchwartzMacPherson -- Chern-Schwartz-MacPherson class of a projective scheme
- degreeMap -- degree of a rational map between projective varieties
- EulerCharacteristic -- topological Euler characteristic of a (smooth) projective variety
- exceptionalLocus -- exceptional locus of a birational map
- forceImage -- declare which is the image of a rational map
- forceInverseMap -- declare that two rational maps are one the inverse of the other
- graph -- closure of the graph of a rational map
- inverseMap -- inverse of a birational map
- isBirational -- whether a rational map is birational
- isDominant -- whether a rational map is dominant
- isInverseMap -- checks whether a rational map is the inverse of another
- isMorphism -- whether a rational map is a morphism
- parametrize -- parametrization of a rational projective variety
- point -- pick a random rational point on a projective variety
- projectiveDegrees -- projective degrees of a rational map between projective varieties
- quadroQuadricCremonaTransformation -- quadro-quadric Cremona transformations
- rationalMap -- makes a rational map
- segre -- Segre embedding
- SegreClass -- Segre class of a closed subscheme of a projective variety
- specialCremonaTransformation -- special Cremona transformations whose base locus has dimension at most three
- specialCubicTransformation -- special cubic transformations whose base locus has dimension at most three
- specialQuadraticTransformation -- special quadratic transformations whose base locus has dimension three
- toMap -- rational map defined by a linear system
Methods
- "abstractRationalMap(PolynomialRing,PolynomialRing,FunctionClosure)" -- see abstractRationalMap -- make an abstract rational map
- "abstractRationalMap(PolynomialRing,PolynomialRing,FunctionClosure,ZZ)" -- see abstractRationalMap -- make an abstract rational map
- "abstractRationalMap(RationalMap)" -- see abstractRationalMap -- make an abstract rational map
- "approximateInverseMap(RationalMap)" -- see approximateInverseMap -- random map related to the inverse of a birational map
- "approximateInverseMap(RationalMap,ZZ)" -- see approximateInverseMap -- random map related to the inverse of a birational map
- "approximateInverseMap(RingMap)" -- see approximateInverseMap -- random map related to the inverse of a birational map
- "approximateInverseMap(RingMap,ZZ)" -- see approximateInverseMap -- random map related to the inverse of a birational map
- "ChernSchwartzMacPherson(Ideal)" -- see ChernSchwartzMacPherson -- Chern-Schwartz-MacPherson class of a projective scheme
- coefficientRing(RationalMap) -- coefficient ring of a rational map
- coefficients(RationalMap) -- coefficient matrix of a rational map
- degree(RationalMap) -- degree of a rational map
- "degreeMap(RingMap)" -- see degreeMap -- degree of a rational map between projective varieties
- degreeMap(RationalMap) -- degree of a rational map
- degrees(RationalMap) -- projective degrees of a rational map
- "multidegree(RationalMap)" -- see degrees(RationalMap) -- projective degrees of a rational map
- "? Ideal" -- see describe(RationalMap) -- describe a rational map
- describe(RationalMap) -- describe a rational map
- entries(RationalMap) -- the entries of the matrix associated to a rational map
- "EulerCharacteristic(Ideal)" -- see EulerCharacteristic -- topological Euler characteristic of a (smooth) projective variety
- "exceptionalLocus(RationalMap)" -- see exceptionalLocus -- exceptional locus of a birational map
- flatten(RationalMap) -- write source and target as nondegenerate varieties
- "forceImage(RationalMap,Ideal)" -- see forceImage -- declare which is the image of a rational map
- "forceInverseMap(RationalMap,RationalMap)" -- see forceInverseMap -- declare that two rational maps are one the inverse of the other
- "graph(RationalMap)" -- see graph -- closure of the graph of a rational map
- graph(RingMap) -- closure of the graph of a rational map
- ideal(RationalMap) -- base locus of a rational map
- image(RationalMap,String) -- closure of the image of a rational map using the F4 algorithm (experimental)
- "image(RationalMap)" -- see image(RationalMap,ZZ) -- closure of the image of a rational map
- image(RationalMap,ZZ) -- closure of the image of a rational map
- inverse(RationalMap) -- inverse of a birational map
- "inverse(RationalMap,Option)" -- see inverse(RationalMap) -- inverse of a birational map
- "inverseMap(RationalMap)" -- see inverseMap -- inverse of a birational map
- "inverseMap(RingMap)" -- see inverseMap -- inverse of a birational map
- "isBirational(RationalMap)" -- see isBirational -- whether a rational map is birational
- "isBirational(RingMap)" -- see isBirational -- whether a rational map is birational
- "isDominant(RationalMap)" -- see isDominant -- whether a rational map is dominant
- "isDominant(RingMap)" -- see isDominant -- whether a rational map is dominant
- "isInverseMap(RingMap,RingMap)" -- see isInverseMap -- checks whether a rational map is the inverse of another
- isInverseMap(RationalMap,RationalMap) -- checks whether two rational maps are one the inverse of the other
- isIsomorphism(RationalMap) -- whether a birational map is an isomorphism
- "isMorphism(RationalMap)" -- see isMorphism -- whether a rational map is a morphism
- kernel(RingMap,ZZ) -- homogeneous components of the kernel of a homogeneous ring map
- map(RationalMap) -- get the ring map defining a rational map
- "map(ZZ,RationalMap)" -- see map(RationalMap) -- get the ring map defining a rational map
- matrix(RationalMap) -- the matrix associated to a rational map
- "matrix(ZZ,RationalMap)" -- see matrix(RationalMap) -- the matrix associated to a rational map
- parametrize(Ideal) -- parametrization of linear varieties and hyperquadrics
- "parametrize(PolynomialRing)" -- see parametrize(Ideal) -- parametrization of linear varieties and hyperquadrics
- "parametrize(QuotientRing)" -- see parametrize(Ideal) -- parametrization of linear varieties and hyperquadrics
- "point(PolynomialRing)" -- see point(QuotientRing) -- pick a random rational point on a projective variety
- point(QuotientRing) -- pick a random rational point on a projective variety
- "projectiveDegrees(RingMap)" -- see projectiveDegrees -- projective degrees of a rational map between projective varieties
- projectiveDegrees(RationalMap) -- projective degrees of a rational map
- "quadroQuadricCremonaTransformation(Ring,ZZ,ZZ)" -- see quadroQuadricCremonaTransformation -- quadro-quadric Cremona transformations
- "quadroQuadricCremonaTransformation(ZZ,ZZ)" -- see quadroQuadricCremonaTransformation -- quadro-quadric Cremona transformations
- "quadroQuadricCremonaTransformation(ZZ,ZZ,Ring)" -- see quadroQuadricCremonaTransformation -- quadro-quadric Cremona transformations
- "rationalMap(List)" -- see rationalMap -- makes a rational map
- "rationalMap(Matrix)" -- see rationalMap -- makes a rational map
- "rationalMap(Ring)" -- see rationalMap -- makes a rational map
- "rationalMap(Ring,Ring)" -- see rationalMap -- makes a rational map
- "rationalMap(Ring,Ring,List)" -- see rationalMap -- makes a rational map
- "rationalMap(Ring,Ring,Matrix)" -- see rationalMap -- makes a rational map
- "rationalMap(RingMap)" -- see rationalMap -- makes a rational map
- RationalMap ! -- calculates every possible thing
- "compose(RationalMap,RationalMap)" -- see RationalMap * RationalMap -- composition of rational maps
- "compose(RingMap,RingMap)" -- see RationalMap * RationalMap -- composition of rational maps
- RationalMap * RationalMap -- composition of rational maps
- RationalMap ** Ring -- change the coefficient ring of a rational map
- RationalMap == RationalMap -- equality of rational maps
- "RationalMap == ZZ" -- see RationalMap == RationalMap -- equality of rational maps
- "ZZ == RationalMap" -- see RationalMap == RationalMap -- equality of rational maps
- RationalMap ^ ZZ -- power
- "RationalMap ^*" -- see RationalMap ^** Ideal -- inverse image via a rational map
- RationalMap ^** Ideal -- inverse image via a rational map
- RationalMap _* -- direct image via a rational map
- "RationalMap Ideal" -- see RationalMap _* -- direct image via a rational map
- RationalMap | Ideal -- restriction of a rational map
- "RationalMap | Ring" -- see RationalMap | Ideal -- restriction of a rational map
- "RationalMap | RingElement" -- see RationalMap | Ideal -- restriction of a rational map
- RationalMap || Ideal -- restriction of a rational map
- "RationalMap || Ring" -- see RationalMap || Ideal -- restriction of a rational map
- "RationalMap || RingElement" -- see RationalMap || Ideal -- restriction of a rational map
- "rationalMap(Ideal)" -- see rationalMap(Ideal,ZZ,ZZ) -- makes a rational map from an ideal
- "rationalMap(Ideal,List)" -- see rationalMap(Ideal,ZZ,ZZ) -- makes a rational map from an ideal
- "rationalMap(Ideal,ZZ)" -- see rationalMap(Ideal,ZZ,ZZ) -- makes a rational map from an ideal
- rationalMap(Ideal,ZZ,ZZ) -- makes a rational map from an ideal
- rationalMap(PolynomialRing,List) -- rational map defined by the linear system of hypersurfaces passing through random points with multiplicity
- rationalMap(Ring,Tally) -- rational map defined by an effective divisor
- "rationalMap(Tally)" -- see rationalMap(Ring,Tally) -- rational map defined by an effective divisor
- "segre(PolynomialRing)" -- see segre -- Segre embedding
- "segre(QuotientRing)" -- see segre -- Segre embedding
- "segre(RationalMap)" -- see segre -- Segre embedding
- "SegreClass(Ideal)" -- see SegreClass -- Segre class of a closed subscheme of a projective variety
- "SegreClass(RationalMap)" -- see SegreClass -- Segre class of a closed subscheme of a projective variety
- "SegreClass(RingMap)" -- see SegreClass -- Segre class of a closed subscheme of a projective variety
- source(RationalMap) -- coordinate ring of the source for a rational map
- "specialCremonaTransformation(Ring,ZZ)" -- see specialCremonaTransformation -- special Cremona transformations whose base locus has dimension at most three
- "specialCremonaTransformation(ZZ)" -- see specialCremonaTransformation -- special Cremona transformations whose base locus has dimension at most three
- "specialCremonaTransformation(ZZ,Ring)" -- see specialCremonaTransformation -- special Cremona transformations whose base locus has dimension at most three
- "specialCubicTransformation(Ring,ZZ)" -- see specialCubicTransformation -- special cubic transformations whose base locus has dimension at most three
- "specialCubicTransformation(ZZ)" -- see specialCubicTransformation -- special cubic transformations whose base locus has dimension at most three
- "specialCubicTransformation(ZZ,Ring)" -- see specialCubicTransformation -- special cubic transformations whose base locus has dimension at most three
- "specialQuadraticTransformation(Ring,ZZ)" -- see specialQuadraticTransformation -- special quadratic transformations whose base locus has dimension three
- "specialQuadraticTransformation(ZZ)" -- see specialQuadraticTransformation -- special quadratic transformations whose base locus has dimension three
- "specialQuadraticTransformation(ZZ,Ring)" -- see specialQuadraticTransformation -- special quadratic transformations whose base locus has dimension three
- substitute(RationalMap,PolynomialRing,PolynomialRing) -- substitute the ambient projective spaces of source and target
- "rationalMap(RationalMap)" -- see super(RationalMap) -- get the rational map whose target is a projective space
- super(RationalMap) -- get the rational map whose target is a projective space
- target(RationalMap) -- coordinate ring of the target for a rational map
- toExternalString(RationalMap) -- convert to a readable string
- "toMap(Ideal)" -- see toMap -- rational map defined by a linear system
- "toMap(Ideal,List)" -- see toMap -- rational map defined by a linear system
- "toMap(Ideal,ZZ)" -- see toMap -- rational map defined by a linear system
- "toMap(Ideal,ZZ,ZZ)" -- see toMap -- rational map defined by a linear system
- "toMap(List)" -- see toMap -- rational map defined by a linear system
- "toMap(Matrix)" -- see toMap -- rational map defined by a linear system
- "toMap(RingMap)" -- see toMap -- rational map defined by a linear system
Symbols
- BlowUpStrategy
- Certify -- whether to ensure correctness of output
- CodimBsInv
- Dominant
- NumDegrees

For the programmer

The object Cremona is a package.