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

Synopsis

Function: isDominant
Usage:

isDominant f
Inputs:
- f, a toric map,
Outputs:
- a Boolean value, that is true if the image is dense

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.

Ways to use this method: