A morphism $f : X \to Y$ is surjective if $f(X) = Y$ as sets. To be surjective toric map, the dimension of $X$ must be greater than or equal to $Y$ and the image of the algebraic torus in $X$ must be equal to the algebraic torus in $Y$. Since $f$ is torus-equivariant, it follows that $f$ is surjective if and only if its image contains a point in each torus orbit in $Y$. This method checks whether all of the cones in the target fan contain a point from the relative interior of a cone in the source fan.
The canonical projections from a product to the factors are surjective.
i1 : X = toricProjectiveSpace 2; |
i2 : Y = hirzebruchSurface 2; |
i3 : XY = X ** Y; |
i4 : pi0 = XY^[0] o4 = | 1 0 0 0 | | 0 1 0 0 | o4 : ToricMap X <--- XY |
i5 : isSurjective pi0 o5 = true |
i6 : assert (isWellDefined pi0 and isSurjective pi0) |
i7 : pi1 = XY^[1] o7 = | 0 0 1 0 | | 0 0 0 1 | o7 : ToricMap Y <--- XY |
i8 : isSurjective pi1 o8 = true |
i9 : assert (isWellDefined pi1 and isSurjective pi1) |
We demonstrate that the natural inclusion from the affine plane into the projective plane is a dominant, but not surjective
i10 : A = affineSpace 2; |
i11 : f = map(X, A, 1) o11 = | 1 0 | | 0 1 | o11 : ToricMap X <--- A |
i12 : isDominant f o12 = true |
i13 : isSurjective f o13 = false |
i14 : assert (isWellDefined f and isDominant f and not isSurjective f) |
For a toric map to be surjective, the underlying map of fans need not be surjective.
i15 : Y = (toricProjectiveSpace 1) ** (toricProjectiveSpace 1); |
i16 : X = normalToricVariety( {{1,0},{1,1},{0,1},{-1,1},{-1,0},{-1,-1},{0,-1},{1,-1}}, {{0},{1},{2},{3},{4},{5},{6},{7}}); |
i17 : g = map(Y,X,1) o17 = | 1 0 | | 0 1 | o17 : ToricMap Y <--- X |
i18 : isSurjective g o18 = true |
i19 : isComplete X o19 = false |
i20 : assert (isWellDefined g and isSurjective g and not isComplete X) |
To avoid repeating a computation, the package caches the result in the CacheTable of the toric map.