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

- 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.

## See also

- working with toric maps -- information about toric maps and the induced operations
- map(NormalToricVariety,NormalToricVariety,Matrix) -- make a torus-equivariant map between normal toric varieties
- map(NormalToricVariety,NormalToricVariety,ZZ) -- make a torus-equivariant map between normal toric varieties
- isComplete(NormalToricVariety) -- whether a toric variety is complete

## Ways to use this method:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NormalToricVarieties/ToricMapsDocumentation.m2:1080:0.