The initial method of expressing some maps between toric varieties is to derive them from maps between their associated lattices. These can also then be presented in terms of the variables of the Cox ring; this is the usual method for describing toric maps between projective spaces, for instance. This often results in radical expressions such as
(u, v) |-> (Sqrt(u), v, vSqrt(u)).
This example could be describing a map from P1 to P(1, 2, 3), for example: it is a `monomial' map which observes the gradings. In that case, we could represent the same map by (u, v) |-> (u, uv, u2v) or (u, v) |-> (1, v/u, v/u), or a host of other expressions. These two expressions have the benefit that they are polynomial or rational functions in u and v, and so they automatically define rational maps, but they have disadvantages too: for example, evaluating the map at the point (0, 1) is not defined for these expressions, whereas it gives image (1, 0, 1) in the original radical expression. (Notice that the choice of root does not matter, as long as it is assumed that the same choice a = Sqrt(u) is made at each coordinate.)
More generally, one can define all maps between toric varieties (not just those arising from maps of lattices) using an appropriate notion of `rational radical function', defined in terms of the polynomial Cox coordinates. This is very common when describing maps between standard projective spaces: one writes down a sequence of homogeneous polynomials of the same degree, without demanding that they are monomials.
The key point is that a rational map between varieties pulls rational functions (that are defined on the image) back to rational functions. It is enough to test this on a basis of rational functions. In the example above, if x, y, z are the coordinates on P(1, 2, 3) then y/x2 and z/x3 form a basis, and these both pull back to v/u, which is a rational function on P1.
We allow maps to be constructed from maps of the underlying toric lattices of fans. When displayed, they are described in these radical polynomial terms.
The rational map between toric varieties X and Y determined by the map f between their respective toric lattices (that is, the lattices underlying their respective fans). If the map f is not specified, it is assumed to be the identity map (and X and Y are assumed to have the same toric lattice).
The blowup of the toric variety X at the toric lattice point v of the toric lattice containing the fan of X; the natural map from the blowup to X is also returned.
The identity map on the toric variety X.
Return true if and only if the map f between toric varieties is regular.
A sequence of subschemes of the toric variety that is the domain of the map f between toric varieties at which f is not defined. (Note that these subschemes may in fact be empty.)
> F2<u,v,x,y> := HirzebruchSurface(Rationals(),2);
> D := Divisor(F2,x);
> Y,f := Proj(D);
> Y;
Toric variety of dimension 2
Variables: $.1, $.2, $.3
The irrelevant ideal is:
($.3, $.2, $.1)
The grading is:
1, 1, 2
> f;
Mapping from: 2-dimensional toric lattice N to 2-dimensional toric lattice N
given by a rule
The image variety Y is clearly the weighted projective space P(1, 1, 2).
The map f returned is a map of underlying lattices. We can convert
it into a map of the toric varieties, after which it will be
presented in Cox coordinates.
> F := ToricVarietyMap(F2,Y,f);
> F;
A map between toric varieties described by:
1,
(v)*(u)^(-1),
(x)*(y)^(-1)*(u)^(-2)
Now we can ask whether this map F is a morphism.
> IsRegular(F); true