ToricMorphisms.md

CurrentModule = Oscar

ToricMorphisms

A class of morphisms among toric varieties are described by certain lattice morphisms. Let $N_1$ and $N_2$ be lattices and $\Sigma_1$, $\Sigma_2$ fans in $N_1$ and $N_2$ respectively. A $\mathbb{Z}$-linear map

$$\overline{\phi} \colon N_1 \to N_2$$

is said to be compatible with the fans $\Sigma_1$ and $\Sigma_2$ if for every cone $\sigma_1 \in \Sigma_1$, there exists a cone $\sigma_2 \in \Sigma_2$ such that $\overline{\phi}_{\mathbb{R}}(\sigma_1) \subseteq \sigma_2$.

By theorem 3.3.4 CLS11, such a map $\overline{\phi}$ induces a morphism $\phi \colon X_{\Sigma_1} \to X_{\Sigma_2}$ of the toric varieties, and those morphisms are exactly the toric morphisms.

Constructors

Generic constructors with specified codomain

toric_morphism(domain::NormalToricVarietyType, mapping_matrix::ZZMatrix, codomain::NormalToricVarietyType; check=true)
toric_morphism(domain::NormalToricVarietyType, lattice_homomorphism::FinGenAbGroupHom, codomain::NormalToricVarietyType; check=true)

Special constructors

toric_identity_morphism(variety::NormalToricVarietyType)

Attributes of Toric Morphisms

General attributes

domain(tm::ToricMorphism)
image(tm::ToricMorphism)
codomain(tm::ToricMorphism)
lattice_homomorphism(tm::ToricMorphism)
morphism_on_torusinvariant_weil_divisor_group(tm::ToricMorphism)
morphism_on_torusinvariant_cartier_divisor_group(tm::ToricMorphism)
morphism_on_class_group(tm::ToricMorphism)
morphism_on_picard_group(tm::ToricMorphism)
covering_morphism(f::ToricMorphism)

Special attributes of toric varieties

To every toric variety $v$ we can associate a special toric variety, the Cox variety. By definition, the Cox variety is such that the mapping matrix of the toric morphism from the Cox variety to the variety $v$ is simply given by the ray generators of the variety $v$. Put differently, if there are exactly $N$ ray generators for the fan of $v$, then the Cox variety of $v$ has a fan for which the ray generators are the standard basis of $\mathbb{R}^N$ and the maximal cones are one to one to the maximal cones of the fan of $v$.

morphism_from_cox_variety(variety::NormalToricVarietyType)
cox_variety(variety::NormalToricVarietyType)