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];
|
i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
-- used 0.130083 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 1.8232 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 0.966113 seconds
o4 = 1
|
i5 : time projectiveDegrees phi
-- used 14.5595 seconds
o5 = {1, 3, 9, 17, 21, 15, 5}
o5 : List
|
i6 : time projectiveDegrees(phi,NumDegrees=>0)
-- used 1.80652 seconds
o6 = {5}
o6 : List
|
i7 : time phi = toMap(phi,Dominant=>J)
-- used 0.0805171 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 11.1199 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 0.229538 seconds
o9 = true
|
i10 : time degreeMap psi
-- used 5.58456 seconds
o10 = 1
|
i11 : time projectiveDegrees psi
-- used 168.764 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 0.10396 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 2.48996 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 11.7861 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 5.8261 seconds
o15 = {5, 15, 21, 17, 9, 3, 1}
o15 : List
|
i16 : time degrees phi
-- used 0.890818 seconds
o16 = {1, 3, 9, 17, 21, 15, 5}
o16 : List
|
i17 : time describe phi
-- used 0.101529 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 0.395946 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 0.66105 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 19.6303 seconds
o21 = {904, 508, 268, 130, 56, 20, 5}
o21 : List
|
i22 : time degree f
-- used 0.0006882 seconds
o22 = 1
|
i23 : time describe f
-- used 0.0262477 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
|