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
) and a general method to compute the push-forward to projective space of Segre classes (see
). 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
.
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];
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i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
-- used 0.00497767 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
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i3 : time J = kernel(phi,2)
-- used 0.081242 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 0.0366919 seconds
o4 = 1
|
i5 : time projectiveDegrees phi
-- used 0.722517 seconds
o5 = {1, 3, 9, 17, 21, 15, 5}
o5 : List
|
i6 : time projectiveDegrees(phi,NumDegrees=>0)
-- used 0.0891258 seconds
o6 = {5}
o6 : List
|
i7 : time phi = toMap(phi,Dominant=>J)
-- used 0.00337015 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 0.595683 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
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i9 : time isInverseMap(phi,psi)
-- used 0.0109667 seconds
o9 = true
|
i10 : time degreeMap psi
-- used 0.238201 seconds
o10 = 1
|
i11 : time projectiveDegrees psi
-- used 9.10037 seconds
o11 = {5, 15, 21, 17, 9, 3, 1}
o11 : List
|
i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
-- used 0.0029061 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 0.0895697 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 0.626097 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 0.342227 seconds
o15 = {5, 15, 21, 17, 9, 3, 1}
o15 : List
|
i16 : time degrees phi
-- used 0.0258735 seconds
o16 = {1, 3, 9, 17, 21, 15, 5}
o16 : List
|
i17 : time describe phi
-- used 0.00340773 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 0.0153187 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 0.0362543 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 1.52866 seconds
o21 = {904, 508, 268, 130, 56, 20, 5}
o21 : List
|
i22 : time degree f
-- used 0.000011794 seconds
o22 = 1
|
i23 : time describe f
-- used 0.0007752 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
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