i1 : (ZZ/190181)[x_0..x_4]; phi = rationalMap(minors(2,matrix{{x_0..x_3},{x_1..x_4}}),Dominant=>true)
o2 = -- rational map --
ZZ
source: Proj(------[x , x , x , x , x ])
190181 0 1 2 3 4
ZZ
target: subvariety of Proj(------[y , y , y , y , y , y ]) defined by
190181 0 1 2 3 4 5
{
y y - y y + y y
2 3 1 4 0 5
}
defining forms: {
2
- x + x x ,
1 0 2
- x x + x x ,
1 2 0 3
2
- x + x x ,
2 1 3
- x x + x x ,
1 3 0 4
- x x + x x ,
2 3 1 4
2
- x + x x
3 2 4
}
o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
|
i3 : time (p1,p2) = graph phi;
-- used 0.0480559 seconds
|
i4 : p1
o4 = -- rational map --
ZZ ZZ
source: subvariety of Proj(------[x , x , x , x , x ]) x Proj(------[y , y , y , y , y , y ]) defined by
190181 0 1 2 3 4 190181 0 1 2 3 4 5
{
y y - y y + y y ,
2 3 1 4 0 5
x y - x y + x y ,
4 2 3 4 2 5
x y - x y + x y ,
3 2 2 4 1 5
x y - x y + x y ,
4 1 3 3 1 5
x y - x y + x y ,
3 1 2 3 0 5
x y - x y - x y + x y ,
2 1 1 2 1 3 0 4
x y - x y + x y ,
4 0 2 3 1 4
x y - x y + x y ,
3 0 1 3 0 4
x y - x y + x y
2 0 1 1 0 2
}
ZZ
target: Proj(------[x , x , x , x , x ])
190181 0 1 2 3 4
defining forms: {
x ,
0
x ,
1
x ,
2
x ,
3
x
4
}
o4 : MultihomogeneousRationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4)
|
i5 : p2
o5 = -- rational map --
ZZ ZZ
source: subvariety of Proj(------[x , x , x , x , x ]) x Proj(------[y , y , y , y , y , y ]) defined by
190181 0 1 2 3 4 190181 0 1 2 3 4 5
{
y y - y y + y y ,
2 3 1 4 0 5
x y - x y + x y ,
4 2 3 4 2 5
x y - x y + x y ,
3 2 2 4 1 5
x y - x y + x y ,
4 1 3 3 1 5
x y - x y + x y ,
3 1 2 3 0 5
x y - x y - x y + x y ,
2 1 1 2 1 3 0 4
x y - x y + x y ,
4 0 2 3 1 4
x y - x y + x y ,
3 0 1 3 0 4
x y - x y + x y
2 0 1 1 0 2
}
ZZ
target: subvariety of Proj(------[y , y , y , y , y , y ]) defined by
190181 0 1 2 3 4 5
{
y y - y y + y y
2 3 1 4 0 5
}
defining forms: {
y ,
0
y ,
1
y ,
2
y ,
3
y ,
4
y
5
}
o5 : MultihomogeneousRationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)
|
i6 : assert(p1 * phi == p2 and p2 * phi^-1 == p1) |
i7 : describe p2
o7 = rational map defined by multiforms of degree {0, 1}
source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of degrees ({0, 2},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1})
target variety: smooth quadric hypersurface in PP^5
dominance: true
coefficient ring: ZZ/190181
|
i8 : projectiveDegrees p2
o8 = {51, 28, 14, 6, 2}
o8 : List
|
When the source of the rational map is a multi-projective variety, the method returns all the projections.
i9 : time g = graph p2;
-- used 0.135747 seconds
|
i10 : g_0; o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to PP^4) |
i11 : g_1; o11 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to PP^5) |
i12 : g_2; o12 : MultihomogeneousRationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to hypersurface in PP^5) |
i13 : describe g_0
o13 = rational map defined by multiforms of degree {1, 0, 0}
source variety: 4-dimensional subvariety of PP^4 x PP^5 x PP^5 cut out by 34 hypersurfaces of degrees ({0, 1, 1},{0, 0, 2},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{1, 0, 1},{1, 0, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{1, 0, 1},{1, 0, 1},{1, 0, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{1, 0, 1},{1, 0, 1},{1, 0, 1},{0, 2, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0})
target variety: PP^4
coefficient ring: ZZ/190181
|
The object graph is a method function with options.