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NormalToricVarieties :: isDominant(ToricMap)

isDominant(ToricMap) -- whether a toric map is dominant

Synopsis

Description

A morphism of varieties is dominant if the image is dense. For a toric map, it suffices to check that the dimension of the image is the dimension of the target.

We demonstrate that the natural inclusion from the affine plane into the projective plane is a dominant, but not surjective

i1 : A = affineSpace 2;
i2 : P = toricProjectiveSpace 2;
i3 : f = map(P, A, 1)

o3 = | 1 0 |
     | 0 1 |

o3 : ToricMap P <--- A
i4 : isDominant f

o4 = true
i5 : isSurjective f

o5 = false
i6 : assert (isWellDefined f and isDominant f and not isSurjective f)

A toric map from the projective line to the projective plane is not dominant.

i7 : X = toricProjectiveSpace 1;
i8 : g = map(P, X, matrix{{2}, {1}})

o8 = | 2 |
     | 1 |

o8 : ToricMap P <--- X
i9 : isDominant g

o9 = false
i10 : I = ideal g

                    2
o10 = ideal(x x  - x )
             0 1    2

o10 : Ideal of QQ[x ..x ]
                   0   2
i11 : assert (isWellDefined g and not isDominant g and codim I === 1)

To avoid repeating a computation, the package caches the result in the CacheTable of the toric map.

See also