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 0.0033981 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 0.147974 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 0.0367172 seconds o4 = 1 |
i5 : time projectiveDegrees phi -- used 0.859528 seconds o5 = {1, 3, 9, 17, 21, 15, 5} o5 : List |
i6 : time projectiveDegrees(phi,NumDegrees=>0) -- used 0.107429 seconds o6 = {5} o6 : List |
i7 : time phi = toMap(phi,Dominant=>J) -- used 0.00333928 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 0.629261 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 0.00903579 seconds o9 = true |
i10 : time degreeMap psi -- used 0.236449 seconds o10 = 1 |
i11 : time projectiveDegrees psi -- used 6.82988 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 0.00218976 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 0.144145 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 0.536281 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 0.291182 seconds o15 = {5, 15, 21, 17, 9, 3, 1} o15 : List |
i16 : time degrees phi -- used 0.00002633 seconds o16 = {1, 3, 9, 17, 21, 15, 5} o16 : List |
i17 : time describe phi -- used 0.00230503 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 0.0118391 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 0.0157948 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 1.62858 seconds o21 = {904, 508, 268, 130, 56, 20, 5} o21 : List |
i22 : time degree f -- used 0.000016491 seconds o22 = 1 |
i23 : time describe f -- used 0.0012209 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).
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.
This documentation describes version 5.1 of Cremona.
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/.