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# ToricMap -- the class of all torus-equivariant maps between normal toric varieties

## Description

Let $X$ and $Y$ be normal toric varieties whose underlying lattices are $N_X$ and $N_Y$ respectively. A toric map is a morphism $f : X \to Y$ that induces a morphism of algebraic groups $g : T_X \to T_Y$ such that $f$ is $T_X$-equivariant with respect to the $T_X$-action on $Y$ induced by $g$. Every toric map $f : X \to Y$ corresponds to a unique map $f_N : N_X \to N_Y$ between the underlying lattices such that, for every cone $\sigma$ in the fan of $X$, there is a cone in the fan of $Y$ that contains the image $f_N(\sigma)$. For details see Theorem 3.3.4 in Cox-Little-Schenck.

To specify a map of normal toric varieties, the target and source normal toric varieties need to be specified as well as a matrix which maps from $N_X$ to $N_Y$.

The primary constructor of a toric map is map(NormalToricVariety,NormalToricVariety,Matrix).

## See also

- working with toric maps -- information about toric maps and the induced operations
- NormalToricVariety -- the class of all normal toric varieties
- id _ NormalToricVariety -- make the identity map from a NormalToricVariety to itself
- isWellDefined(ToricMap) -- whether a toric map is well defined

## Functions and methods returning a toric map:

- `diagonalToricMap(NormalToricVariety)` -- see diagonalToricMap -- make a diagonal map into a Cartesian product
- `diagonalToricMap(NormalToricVariety,ZZ)` -- see diagonalToricMap -- make a diagonal map into a Cartesian product
- `diagonalToricMap(NormalToricVariety,ZZ,Array)` -- see diagonalToricMap -- make a diagonal map into a Cartesian product
- 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
- NormalToricVariety ^ Array -- make a canonical projection map
- NormalToricVariety _ Array -- make a canonical inclusion into a product
- ToricMap * ToricMap -- make the composition of two toric maps

## Methods that use a toric map:

- cartierDivisorGroup(ToricMap) -- make the induced map between groups of Cartier divisors.
- classGroup(ToricMap) -- make the induced map between class groups
- ideal(ToricMap) -- make the ideal defining the closure of the image
- inducedMap(ToricMap) -- make the induced map between total coordinate rings (a.k.a. Cox rings)
- isDominant(ToricMap) -- whether a toric map is dominant
- isFibration(ToricMap) -- whether a toric map is a fibration
- isProper(ToricMap) -- whether a toric map is proper
- isSurjective(ToricMap) -- whether a toric map is surjective
- isWellDefined(ToricMap) -- whether a toric map is well defined
- matrix(ToricMap) -- get the underlying map of lattices
- picardGroup(ToricMap) -- make the induced map between Picard groups
- pullback(ToricMap,CoherentSheaf) -- make the pullback of a coherent sheaf under a toric map
- `pullback(ToricMap,Module)` -- see pullback(ToricMap,CoherentSheaf) -- make the pullback of a coherent sheaf under a toric map
- pullback(ToricMap,ToricDivisor) -- make the pullback of a Cartier divisor under a toric map
- source(ToricMap) -- get the source of the map
- target(ToricMap) -- get the target of the map
- ToricMap == ToricMap -- whether to toric maps are equal
- weilDivisorGroup(ToricMap) -- make the induced map between groups of Weil divisors

## For the programmer

The object ToricMap is a type, with ancestor classes HashTable < Thing.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NormalToricVarieties/ToricMapsDocumentation.m2:103:0.