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 — Method

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

Construct the toric morphism with given domain and associated to the lattice morphism given by the mapping_matrix.

If the codomain is left out, it will be determined whether the image of the domain fan is itself a polyhedral fan. In that case the codomain is assumed to be the associated toric variety.

All checks can be disabled with check=false.

Examples

julia> domain = projective_space(NormalToricVariety, 1)
Normal toric variety

julia> codomain = hirzebruch_surface(NormalToricVariety, 2)
Normal toric variety

julia> mapping_matrix = matrix(ZZ, [0 1])
[0   1]

julia> toric_morphism(domain, mapping_matrix, codomain)
Toric morphism

source

toric_morphism — Method

toric_morphism(domain::NormalToricVarietyType, grid_morphism::FinGenAbGroupHom, codomain::NormalToricVarietyType; check=true)

Construct the toric morphism from the domain to the codomain with map given by the grid_morphism.

If the codomain is left out, it will be determined whether the image of the domain fan is itself a polyhedral fan. In that case the codomain is assumed to be the associated toric variety.

All checks can be disabled with check=false.

Examples

julia> domain = projective_space(NormalToricVariety, 1)
Normal toric variety

julia> codomain = hirzebruch_surface(NormalToricVariety, 2)
Normal toric variety

julia> mapping_matrix = matrix(ZZ, [[0, 1]])
[0   1]

julia> grid_morphism = hom(character_lattice(domain), character_lattice(codomain), mapping_matrix)
Map
  from Z
  to Z^2

julia> toric_morphism(domain, grid_morphism, codomain)
Toric morphism

source

Special constructors

toric_identity_morphism — Method

toric_identity_morphism(variety::NormalToricVarietyType)

Construct the toric identity morphism from variety to variety.

Examples

julia> toric_identity_morphism(hirzebruch_surface(NormalToricVariety, 2))
Toric morphism

source

Attributes of Toric Morphisms

General attributes

domain — Method

domain(tm::ToricMorphism)

Return the domain of the toric morphism tm.

Examples

julia> F4 = hirzebruch_surface(NormalToricVariety, 4)
Normal toric variety

julia> domain(toric_identity_morphism(F4))
Normal toric variety

source

image — Method

image(F::FreeMod{R}, A::MatElem{R}) where R

Return the image of A as an object of type SubquoModule with ambient free module F.

Examples

julia> R, (x,y,z) = polynomial_ring(QQ, [:x, :y, :z]);

julia> F = free_module(R, 2)
Free module of rank 2 over R

julia> A = R[x y; 2*x^2 3*y^2]
[    x       y]
[2*x^2   3*y^2]
 
julia> M = image(F, A)
Submodule with 2 generators
  1: x*e[1] + y*e[2]
  2: 2*x^2*e[1] + 3*y^2*e[2]
represented as subquotient with no relations

julia> ambient_free_module(M) === F
true

julia> Rg, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

julia> F = graded_free_module(Rg, [8,8])
Graded free module Rg^2([-8]) of rank 2 over Rg

julia> A = Rg[x y; 2*x^2 3*y^2]
[    x       y]
[2*x^2   3*y^2]
 
julia> M = image(F, A)
Graded submodule of F with 2 generators
  1: x*e[1] + y*e[2]
  2: 2*x^2*e[1] + 3*y^2*e[2]
represented as subquotient with no relations

julia> ambient_free_module(M) === F
true

julia> degrees_of_generators(M)
2-element Vector{FinGenAbGroupElem}:
 [9]
 [10]

source

codomain — Method

codomain(tm::ToricMorphism)

Return the codomain of the toric morphism tm.

Examples

julia> F4 = hirzebruch_surface(NormalToricVariety, 4)
Normal toric variety

julia> codomain(toric_identity_morphism(F4))
Normal toric variety

source

grid_morphism — Method

grid_morphism(tm::ToricMorphism)

Return the underlying grid morphism of the toric morphism tm.

Examples

julia> F4 = hirzebruch_surface(NormalToricVariety, 4)
Normal toric variety

julia> grid_morphism(toric_identity_morphism(F4))
Map
  from Z^2
  to Z^2

source

morphism_on_torusinvariant_weil_divisor_group — Method

morphism_on_torusinvariant_weil_divisor_group(tm::ToricMorphism)

For a given toric morphism tm, this method computes the corresponding map of the torusinvariant Weil divisors.

Examples

julia> F4 = hirzebruch_surface(NormalToricVariety, 4)
Normal toric variety

julia> morphism_on_torusinvariant_weil_divisor_group(toric_identity_morphism(F4))
Map
  from Z^4
  to Z^4

source

morphism_on_torusinvariant_cartier_divisor_group — Method

morphism_on_torusinvariant_cartier_divisor_group(tm::ToricMorphism)

For a given toric morphism tm, this method computes the corresponding map of the Cartier divisors.

Examples

julia> F4 = hirzebruch_surface(NormalToricVariety, 4)
Normal toric variety

julia> morphism_on_torusinvariant_cartier_divisor_group(toric_identity_morphism(F4))
Map
  from Z^4
  to Z^4

source

morphism_on_class_group — Method

morphism_on_class_group(tm::ToricMorphism)

For a given toric morphism tm, this method computes the corresponding map of the Class groups.

Examples

julia> F4 = hirzebruch_surface(NormalToricVariety, 4)
Normal toric variety

julia> morphism_on_class_group(toric_identity_morphism(F4))
Map
  from Z^2
  to Z^2

source

morphism_on_picard_group — Method

morphism_on_picard_group(tm::ToricMorphism)

For a given toric morphism tm, this method computes the corresponding map of the Picard groups.

Examples

julia> F4 = hirzebruch_surface(NormalToricVariety, 4)
Normal toric variety

julia> morphism_on_picard_group(toric_identity_morphism(F4))
Map
  from Z^2
  to Z^2

source

covering_morphism — Method

covering_morphism(f::ToricMorphism)

For a given toric morphism tm, we can compute the corresponding morphism of covered schemes. The following demonstrates this for the blow-up morphism of a blow-up of the projective space.

Examples

julia> IP2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> bl = blow_up(IP2, [1, 1]);

julia> cov_bl = covering_morphism(bl);

julia> domain(cov_bl)
Covering
  described by patches
    1: normal toric variety
    2: normal toric variety
    3: normal toric variety
    4: normal toric variety
  in the coordinate(s)
    1: [x_1_1, x_2_1]
    2: [x_1_2, x_2_2]
    3: [x_1_3, x_2_3]
    4: [x_1_4, x_2_4]

julia> codomain(cov_bl)
Covering
  described by patches
    1: normal toric variety
    2: normal toric variety
    3: normal toric variety
  in the coordinate(s)
    1: [x_1_1, x_2_1]
    2: [x_1_2, x_2_2]
    3: [x_1_3, x_2_3]

source

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 — Method

morphism_from_cox_variety(variety::NormalToricVarietyType)

Return the quotient morphism from the Cox variety to the toric variety in question.

Examples

julia> F4 = hirzebruch_surface(NormalToricVariety, 4)
Normal toric variety

julia> morphism_from_cox_variety(F4)
Toric morphism

source

cox_variety — Method

cox_variety(variety::NormalToricVarietyType)

Return the Cox variety of the toric variety in question.

Examples

julia> F4 = hirzebruch_surface(NormalToricVariety, 4)
Normal toric variety

julia> cox_variety(F4)
Normal toric variety

source