# 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 David A. Cox, John B. Little, Henry K. Schenck (2011), 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_morphismMethod
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, non-affine, smooth, projective, gorenstein, fano, 1-dimensional toric variety without torusfactor

julia> codomain = hirzebruch_surface(NormalToricVariety, 2)
Normal, non-affine, smooth, projective, gorenstein, non-fano, 2-dimensional toric variety without torusfactor

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

julia> toric_morphism(domain, mapping_matrix, codomain)
A toric morphism
source
toric_morphismMethod
toric_morphism(domain::NormalToricVarietyType, grid_morphism::GrpAbFinGenMap, 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, non-affine, smooth, projective, gorenstein, fano, 1-dimensional toric variety without torusfactor

julia> codomain = hirzebruch_surface(NormalToricVariety, 2)
Normal, non-affine, smooth, projective, gorenstein, non-fano, 2-dimensional toric variety without torusfactor

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

julia> grid_morphism = hom(character_lattice(domain), character_lattice(codomain), mapping_matrix)
Map: GrpAb: Z -> GrpAb: Z^2

julia> toric_morphism(domain, grid_morphism, codomain)
A toric morphism
source

### Special constructors

toric_identity_morphismMethod
toric_identity_morphism(variety::NormalToricVarietyType)

Construct the toric identity morphism from variety to variety.

Examples

julia> toric_identity_morphism(hirzebruch_surface(NormalToricVariety, 2))
A toric morphism
source

## Attributes of Toric Morhpisms

### General attributes

domainMethod
domain(tm::ToricMorphism)

Return the domain of the toric morphism tm.

Examples

julia> F4 = hirzebruch_surface(NormalToricVariety, 4)
Normal, non-affine, smooth, projective, gorenstein, non-fano, 2-dimensional toric variety without torusfactor

julia> domain(toric_identity_morphism(F4))
Normal, non-affine, smooth, projective, gorenstein, non-fano, 2-dimensional toric variety without torusfactor
source
imageMethod
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 Multivariate polynomial ring in 3 variables over QQ

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 + y*e
2 -> 2*x^2*e + 3*y^2*e
represented as subquotient with no relations.

julia> ambient_free_module(M) === F
true
julia> R, _ = polynomial_ring(QQ, ["x", "y", "z"]);

julia> Z = abelian_group(0);

julia> Rg, (x, y, z) = grade(R, [Z, Z, Z]);

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)
1 -> x*e + y*e
2 -> 2*x^2*e + 3*y^2*e
represented as subquotient with no relations

julia> ambient_free_module(M) === F
true

julia> degrees_of_generators(M)
2-element Vector{GrpAbFinGenElem}:
Element of Z with components 
Element of Z with components 
source
codomainMethod
codomain(tm::ToricMorphism)

Return the codomain of the toric morphism tm.

Examples

julia> F4 = hirzebruch_surface(NormalToricVariety, 4)
Normal, non-affine, smooth, projective, gorenstein, non-fano, 2-dimensional toric variety without torusfactor

julia> codomain(toric_identity_morphism(F4))
Normal, non-affine, smooth, projective, gorenstein, non-fano, 2-dimensional toric variety without torusfactor
source
grid_morphismMethod
grid_morphism(tm::ToricMorphism)

Return the underlying grid morphism of the toric morphism tm.

Examples

julia> F4 = hirzebruch_surface(NormalToricVariety, 4)
Normal, non-affine, smooth, projective, gorenstein, non-fano, 2-dimensional toric variety without torusfactor

julia> grid_morphism(toric_identity_morphism(F4))
Map: GrpAb: Z^2 -> GrpAb: Z^2
source
morphism_on_torusinvariant_weil_divisor_groupMethod
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, non-affine, smooth, projective, gorenstein, non-fano, 2-dimensional toric variety without torusfactor

julia> morphism_on_torusinvariant_weil_divisor_group(toric_identity_morphism(F4))
Map: GrpAb: Z^4 -> GrpAb: Z^4
source
morphism_on_torusinvariant_cartier_divisor_groupMethod
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, non-affine, smooth, projective, gorenstein, non-fano, 2-dimensional toric variety without torusfactor

julia> morphism_on_torusinvariant_cartier_divisor_group(toric_identity_morphism(F4))
Map: GrpAb: Z^4 -> GrpAb: Z^4
source
morphism_on_class_groupMethod
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, non-affine, smooth, projective, gorenstein, non-fano, 2-dimensional toric variety without torusfactor

julia> morphism_on_class_group(toric_identity_morphism(F4))
Map: GrpAb: Z^2 -> GrpAb: Z^2
source
morphism_on_picard_groupMethod
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, non-affine, smooth, projective, gorenstein, non-fano, 2-dimensional toric variety without torusfactor

julia> morphism_on_picard_group(toric_identity_morphism(F4))
Map: GrpAb: Z^2 -> GrpAb: Z^2
source
covering_morphismMethod
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-down morphism of a blow-up of the projective space.

Examples

julia> IP2 = projective_space(NormalToricVariety, 2)
Normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

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

julia> cov_bl = covering_morphism(bl);

julia> domain(cov_bl)
Covering
described by patches
1: normal, affine toric variety
2: normal, affine toric variety
3: normal, affine toric variety
4: normal, affine 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, affine toric variety
2: normal, affine toric variety
3: normal, affine 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_varietyMethod
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, non-affine, smooth, projective, gorenstein, non-fano, 2-dimensional toric variety without torusfactor

julia> morphism_from_cox_variety(F4)
A toric morphism
source
cox_varietyMethod
cox_variety(variety::NormalToricVarietyType)

Return the Cox variety of the toric variety in question.

Examples

julia> F4 = hirzebruch_surface(NormalToricVariety, 4)
Normal, non-affine, smooth, projective, gorenstein, non-fano, 2-dimensional toric variety without torusfactor

julia> cox_variety(F4)
Normal toric variety
source