working with toric maps -- information about toric maps and the induced operations

A toric map is a morphism $f : X \to Y$ between normal toric varieties 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.

Although the primary method for creating a toric map is map(NormalToricVariety,NormalToricVariety,Matrix), there are a few other constructors.

Making toric maps

• map(NormalToricVariety,NormalToricVariety,Matrix) -- make a torus-equivariant map between normal toric varieties
• id _ NormalToricVariety -- make the identity map from a NormalToricVariety to itself
• NormalToricVariety ^ Array -- make a canonical projection map
• NormalToricVariety _ Array -- make a canonical inclusion into a product
• diagonalToricMap -- make a diagonal map into a Cartesian product
• ToricMap -- the class of all torus-equivariant maps between normal toric varieties
• isWellDefined(ToricMap) -- whether a toric map is well defined

Having made a toric map, one can access its basic invariants or test for some elementary properties by using the following methods.

Determining attributes and properties of toric maps

• source(ToricMap) -- get the source of the map
• target(ToricMap) -- get the target of the map
• matrix(ToricMap) -- get the underlying map of lattices
• ToricMap * ToricMap -- make the composition of two toric maps
• isProper(ToricMap) -- whether a toric map is proper
• isFibration(ToricMap) -- whether a toric map is a fibration
• isDominant(ToricMap) -- whether a toric map is dominant
• isSurjective(ToricMap) -- whether a toric map is surjective

Several functorial aspects of normal toric varieties are also available.

Functorial aspects of toric maps

• weilDivisorGroup(ToricMap) -- make the induced map between groups of Weil divisors
• classGroup(ToricMap) -- make the induced map between class groups
• cartierDivisorGroup(ToricMap) -- make the induced map between groups of Cartier divisors.
• picardGroup(ToricMap) -- make the induced map between Picard groups
• pullback(ToricMap,ToricDivisor) -- make the pullback of a Cartier divisor under a toric map
• pullback(ToricMap,CoherentSheaf) -- make the pullback of a coherent sheaf under a toric map
• inducedMap(ToricMap) -- make the induced map between total coordinate rings (a.k.a. Cox rings)
• ideal(ToricMap) -- make the ideal defining the closure of the image