Normal Toric Varieties

affine_normal_toric_variety(C::Cone)

Construct the affine normal toric variety $U_{C}$ corresponding to a polyhedral cone C.

Examples

Set C to be the positive orthant in two dimensions.

julia> C = positive_hull([1 0; 0 1])
Polyhedral cone in ambient dimension 2

julia> antv = affine_normal_toric_variety(C)
Normal toric variety

normal_toric_variety(C::Cone)

Construct the (affine) normal toric variety $X_{\Sigma}$ corresponding to a polyhedral fan $\Sigma = C$ consisting only of the cone C.

Examples

Set C to be the positive orthant in two dimensions.

julia> C = positive_hull([1 0; 0 1])
Polyhedral cone in ambient dimension 2

julia> ntv = normal_toric_variety(C)
Normal toric variety

affine_normal_toric_variety(v::NormalToricVariety)

For internal design, we make a strict distinction between normal toric varieties and affine toric varieties. Given an affine, normal toric variety v, this method turns it into an affine toric variety.

Examples

julia> v = normal_toric_variety(positive_hull([1 0; 0 1]))
Normal toric variety

julia> affineVariety = affine_normal_toric_variety(v)
Normal toric variety

normal_toric_variety(C::Cone)

Construct the (affine) normal toric variety $X_{\Sigma}$ corresponding to a polyhedral fan $\Sigma = C$ consisting only of the cone C.

Examples

Set C to be the positive orthant in two dimensions.

julia> C = positive_hull([1 0; 0 1])
Polyhedral cone in ambient dimension 2

julia> ntv = normal_toric_variety(C)
Normal toric variety

normal_toric_variety(max_cones::IncidenceMatrix, rays::AbstractCollection[RayVector]; non_redundant::Bool = false)

Construct a normal toric variety $X$ by providing the rays and maximal cones as vector of vectors. By default, this method allows redundancies in the input, e.g. duplicate rays and non-maximal cones. If the user is certain that no redundancy exists in the entered information, one can pass non_redundant = true as third argument. This will bypass these consistency checks. In addition, this will ensure that the order of the rays is not altered by the constructor.

Examples

julia> ray_generators = [[1,0], [0, 1], [-1, 5], [0, -1]]
4-element Vector{Vector{Int64}}:
 [1, 0]
 [0, 1]
 [-1, 5]
 [0, -1]

julia> max_cones = incidence_matrix([[1, 2], [2, 3], [3, 4], [4, 1]])
4×4 IncidenceMatrix
 [1, 2]
 [2, 3]
 [3, 4]
 [1, 4]

julia> normal_toric_variety(max_cones, ray_generators)
Normal toric variety

julia> normal_toric_variety(max_cones, ray_generators; non_redundant = true)
Normal toric variety

normal_toric_variety(PF::PolyhedralFan)

Construct the normal toric variety $X_{PF}$ corresponding to a polyhedral fan PF.

Examples

Take PF to be the normal fan of the square.

julia> square = cube(2)
Polytope in ambient dimension 2

julia> nf = normal_fan(square)
Polyhedral fan in ambient dimension 2

julia> ntv = normal_toric_variety(nf)
Normal toric variety

normal_toric_variety(P::Polyhedron)

Construct the normal toric variety $X_{\Sigma_P}$ corresponding to the normal fan $\Sigma_P$ of the given polyhedron P.

Note that this only coincides with the projective variety associated to P from the affine relations of the lattice points in P, if P is very ample.

Examples

Set P to be a square.

julia> square = cube(2)
Polytope in ambient dimension 2

julia> ntv = normal_toric_variety(square)
Normal toric variety

affine_space(::Type{NormalToricVariety}, d::Int)

Construct the (toric) affine space of dimension d.

Examples

julia> affine_space(NormalToricVariety, 2)
Normal toric variety

del_pezzo_surface(::Type{NormalToricVariety}, b::Int)

Construct the del Pezzo surface with b blowups for b at most 3.

Examples

julia> del_pezzo_surface(NormalToricVariety, 3)
Normal toric variety

hirzebruch_surface(::Type{NormalToricVariety}, r::Int)

Construct the r-th Hirzebruch surface.

Examples

julia> hirzebruch_surface(NormalToricVariety, 5)
Normal toric variety

projective_space(::Type{NormalToricVariety}, d::Int)

Construct the projective space of dimension d.

Examples

julia> projective_space(NormalToricVariety, 2)
Normal toric variety

weighted_projective_space(::Type{NormalToricVariety}, w::Vector{T}) where {T <: IntegerUnion}

Construct the weighted projective space corresponding to the weights w.

Examples

julia> weighted_projective_space(NormalToricVariety, [2, 3, 1])
Normal toric variety

normal_toric_variety_from_star_triangulation(P::Polyhedron)

Return a toric variety that was obtained from a fine regular star triangulation of the lattice points of the polyhedron P. This is particularly useful when the lattice points of the polyhedron in question admit many triangulations.

Examples

julia> P = convex_hull([0 0 0; 0 0 1; 1 0 1; 1 1 1; 0 1 1])
Polyhedron in ambient dimension 3

julia> v = normal_toric_variety_from_star_triangulation(P)
Normal toric variety

normal_toric_varieties_from_star_triangulations(P::Polyhedron)

Return the list of toric varieties obtained from fine regular star triangulations of the polyhedron P. With this we can compute the two phases of the famous conifold transition.

Examples

julia> P = convex_hull([0 0 0; 0 0 1; 1 0 1; 1 1 1; 0 1 1])
Polyhedron in ambient dimension 3

julia> (v1, v2) = normal_toric_varieties_from_star_triangulations(P)
2-element Vector{NormalToricVariety}:
 Normal toric variety
 Normal toric variety

julia> stanley_reisner_ideal(v1)
Ideal generated by
  x1*x4

julia> stanley_reisner_ideal(v2)
Ideal generated by
  x2*x3

normal_toric_variety_from_glsm(charges::ZZMatrix)

Return one toric variety with the desired GLSM charges. This can be particularly useful provided that there are many such toric varieties.

Examples

julia> charges = [[1, 1, 1]]
1-element Vector{Vector{Int64}}:
 [1, 1, 1]

julia> normal_toric_variety_from_glsm(charges)
Normal toric variety

For convenience, we also support:

• normal_toric_variety_from_glsm(charges::Vector{Vector{Int}})
• normal_toric_variety_from_glsm(charges::Vector{Vector{ZZRingElem}})

normal_toric_varieties_from_glsm(charges::ZZMatrix)

Return all toric variety with the desired GLSM charges. This computation may take a long time if there are many such toric varieties.

Examples

julia> charges = [[1, 1, 1]]
1-element Vector{Vector{Int64}}:
 [1, 1, 1]

julia> normal_toric_varieties_from_glsm(charges)
1-element Vector{NormalToricVariety}:
 Normal toric variety

julia> varieties = normal_toric_varieties_from_glsm(matrix(ZZ, [1 2 3 4 6 0; -1 -1 -2 -2 -3 1]))
1-element Vector{NormalToricVariety}:
 Normal toric variety

julia> cox_ring(varieties[1])
Multivariate polynomial ring in 6 variables over QQ graded by
  x1 -> [1 -1]
  x2 -> [2 -1]
  x3 -> [3 -2]
  x4 -> [4 -2]
  x5 -> [6 -3]
  x6 -> [0 1]

For convenience, we also support:

• normal_toric_varieties_from_glsm(charges::Vector{Vector{Int}})
• normal_toric_varieties_from_glsm(charges::Vector{Vector{ZZRingElem}})

Base.:*(v::NormalToricVarietyType, w::NormalToricVarietyType)

Return the Cartesian/direct product of two normal toric varieties v and w.

By default, we prepend an "x" to all homogeneous coordinate names of the first factor v and a "y" to all homogeneous coordinate names of the second factor w. This default can be overwritten by invoking set_coordinate_names after creating the variety (cf. set_coordinate_names(v::NormalToricVarietyType, coordinate_names::Vector{String})).

Important: Recall that the coordinate names can only be changed as long as the toric variety in question is not finalized (cf. is_finalized(v::NormalToricVarietyType)).

Crucially, the order of the homogeneous coordinates is not shuffled. To be more specific, assume that v has $n_1$ and w has $n_2$ homogeneous coordinates. Then v * w has $n_1 + n_2$ homogeneous coordinates. The first $n_1$ of these coordinates are those of v and appear in the very same order as they do for v. The remaining $n_2$ homogeneous coordinates are those of w and appear in the very same order as they do for w.

Examples

julia> P2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> v1 = P2 * P2
Normal toric variety

julia> cox_ring(v1)
Multivariate polynomial ring in 6 variables over QQ graded by
  xx1 -> [1 0]
  xx2 -> [1 0]
  xx3 -> [1 0]
  yx1 -> [0 1]
  yx2 -> [0 1]
  yx3 -> [0 1]

julia> v2 = P2 * P2
Normal toric variety

julia> set_coordinate_names(v2, ["x1", "x2", "x3", "y1", "y2", "y3"])


julia> cox_ring(v2)
Multivariate polynomial ring in 6 variables over QQ graded by
  x1 -> [1 0]
  x2 -> [1 0]
  x3 -> [1 0]
  y1 -> [0 1]
  y2 -> [0 1]
  y3 -> [0 1]

projectivization(E::ToricLineBundle...)

This function computes the projectivization of a direct sum of line bundles or divisors. Please see [OM78] for more background information.

Examples

Let us construct the projective bundles $X=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(1))$ and $Y=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(2))$.

julia> P1 = projective_space(NormalToricVariety, 1);

julia> D0 = toric_divisor(P1, [0,0]);

julia> D1 = toric_divisor(P1, [1,0]);

julia> X = projectivization(D0, D1)
Normal toric variety

julia> L0 = toric_line_bundle(P1, [0]);

julia> L1 = toric_line_bundle(P1, [2]);

julia> Y = projectivization(L0, L1)
Normal toric variety

total_space(E::ToricLineBundle...)

This function computes the total space of a direct sum of line bundles or divisors. Please see [OM78] for more background information.

Examples

Let us construct the toric Calabi-Yau varieties given by the total space of $\mathcal{O}_{\mathbb{P}^1}(2)\oplus\mathcal{O}_{\mathbb{P}^1}(-4)$ and $\omega_{\mathbb{P}^2}$.

julia> P1 = projective_space(NormalToricVariety, 1);

julia> L1 = toric_line_bundle(P1, [2]);

julia> L2 = toric_line_bundle(P1, [-4]);

julia> X = total_space(L1, L2)
Normal toric variety

julia> degree(canonical_bundle(X))
0

julia> P2 = projective_space(NormalToricVariety, 2);

julia> D = canonical_divisor(P2);

julia> Y = total_space(D)
Normal toric variety

julia> degree(canonical_bundle(Y))
0

has_torusfactor(v::NormalToricVarietyType)

Check if the normal toric variety v has a torus factor.

Examples

julia> has_torusfactor(projective_space(NormalToricVariety, 2))
false

is_affine(v::NormalToricVarietyType)

Check if the normal toric variety v is affine.

Examples

julia> is_affine(projective_space(NormalToricVariety, 2))
false

is_complete(v::NormalToricVarietyType)

Check if the normal toric variety v is complete.

Examples

julia> is_complete(projective_space(NormalToricVariety, 2))
true

is_fano(v::NormalToricVarietyType)

Check if the normal toric variety v is fano.

Examples

julia> is_fano(projective_space(NormalToricVariety, 2))
true

is_gorenstein(v::NormalToricVarietyType)

Check if the normal toric variety v is Gorenstein.

Examples

julia> is_gorenstein(projective_space(NormalToricVariety, 2))
true

is_simplicial(v::NormalToricVarietyType)

Check if the normal toric variety v is simplicial. Hence, this function works just as is_orbifold. It is implemented for user convenience.

Examples

julia> is_simplicial(projective_space(NormalToricVariety, 2))
true

is_smooth(v::NormalToricVarietyType)

Check if the normal toric variety v is smooth.

Examples

julia> is_smooth(projective_space(NormalToricVariety, 2))
true

is_normal(v::NormalToricVarietyType)

Check if the normal toric variety v is normal. (This function is somewhat tautological at this point.)

Examples

julia> is_normal(projective_space(NormalToricVariety, 2))
true

is_orbifold(v::NormalToricVarietyType)

Check if the normal toric variety v is an orbifold.

Examples

julia> is_orbifold(projective_space(NormalToricVariety, 2))
true

is_projective(v::NormalToricVarietyType)

Check if the normal toric variety v is projective, i.e. if the fan of v is the the normal fan of a polytope.

Examples

julia> is_projective(projective_space(NormalToricVariety, 2))
true

is_projective_space(v::NormalToricVarietyType)

Decides if the normal toric varieties v is a projective space.

Examples

julia> F5 = hirzebruch_surface(NormalToricVariety, 5)
Normal toric variety

julia> is_projective_space(F5)
false

julia> is_projective_space(projective_space(NormalToricVariety, 2))
true

is_q_gorenstein(v::NormalToricVarietyType)

Check if the normal toric variety v is Q-Gorenstein.

Examples

julia> is_q_gorenstein(projective_space(NormalToricVariety, 2))
true

affine_open_covering(v::NormalToricVarietyType)

Compute an affine open cover of the normal toric variety v, i.e. returns a list of affine toric varieties.

Examples

julia> p2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> affine_open_covering(p2)
3-element Vector{AffineNormalToricVariety}:
 Normal toric variety
 Normal toric variety
 Normal toric variety

torusinvariant_cartier_divisor_group(v::NormalToricVarietyType)

Return the Cartier divisor group of an abstract normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> torusinvariant_cartier_divisor_group(p2)
Z^3

character_lattice(v::NormalToricVarietyType)

Return the character lattice of a normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> character_lattice(p2)
Z^2

class_group(v::NormalToricVarietyType)

Return the class group of the normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> class_group(p2)
Z

map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(v::NormalToricVarietyType)

Return the embedding of the group of Cartier divisors into the group of torus-invariant Weil divisors of an abstract normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(p2)
Map
  from Z^3
  to Z^3

map_from_torusinvariant_cartier_divisor_group_to_picard_group(v::NormalToricVarietyType)

Return the map from the Cartier divisors to the Picard group of an abstract normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> map_from_torusinvariant_cartier_divisor_group_to_picard_group(p2)
Map
  from Z^3
  to Z

map_from_character_lattice_to_torusinvariant_weil_divisor_group(v::NormalToricVarietyType)

Return the map from the character lattice to the group of principal divisors of a normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> map_from_character_lattice_to_torusinvariant_weil_divisor_group(p2)
Map
  from Z^2
  to Z^3

map_from_torusinvariant_weil_divisor_group_to_class_group(v::NormalToricVarietyType)

Return the map from the group of Weil divisors to the class of group of a normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> map_from_torusinvariant_weil_divisor_group_to_class_group(p2)
Map
  from Z^3
  to Z

picard_group(v::NormalToricVarietyType)

Return the Picard group of an abstract normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> picard_group(p2)
Z

torusinvariant_weil_divisor_group(v::NormalToricVarietyType)

Return the torusinvariant divisor group of a normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> torusinvariant_weil_divisor_group(p2)
Z^3

torusinvariant_prime_divisors(v::NormalToricVarietyType)

Return the list of all torus invariant prime divisors in a normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> torusinvariant_prime_divisors(p2)
3-element Vector{ToricDivisor}:
 Torus-invariant, prime divisor on a normal toric variety
 Torus-invariant, prime divisor on a normal toric variety
 Torus-invariant, prime divisor on a normal toric variety

gorenstein_index(v::NormalToricVarietyType)

Return the Gorenstein index of a $\mathbb{Q}$-Gorenstein normal toric variety v. This is the smallest positive integer $l$ such that $-l K$ is Cartier, where $K$ is a canonical divisor on v. See exercise 8.3.10 and 8.3.11 in [CLS11] for more details.

Examples

julia> gorenstein_index(weighted_projective_space(NormalToricVariety, [2,3,5]))
3

picard_index(v::NormalToricVarietyType)

Return the index of the Picard group in the class group of a simplicial normal toric variety v. Here, the Picard group embeds as the group of Cartier divisor classes into the class group via map_from_picard_group_to_class_group. See [HHS11] for more details.

Examples

julia> picard_index(weighted_projective_space(NormalToricVariety, [2,3,5]))
30

polyhedral_fan(v::NormalToricVarietyType)

Return the fan of an abstract normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> polyhedral_fan(p2)
Polyhedral fan in ambient dimension 2

cone(v::AffineNormalToricVariety)

Return the cone of the affine normal toric variety v.

Examples

julia> cone(affine_normal_toric_variety(Oscar.positive_hull([1 1; -1 1])))
Polyhedral cone in ambient dimension 2

weight_cone(v::AffineNormalToricVariety)

Return the dual cone of the affine normal toric variety v.

Examples

julia> C = positive_hull([1 0; 0 1])
Polyhedral cone in ambient dimension 2

julia> antv = affine_normal_toric_variety(C)
Normal toric variety

julia> weight_cone(antv)
Polyhedral cone in ambient dimension 2

julia> polarize(cone(antv)) == weight_cone(antv)
true

hilbert_basis(v::AffineNormalToricVariety)

For an affine toric variety $v$, this returns the Hilbert basis of the cone dual to the cone of $v$.

Examples

julia> C = positive_hull([-1 1; 1 1])
Polyhedral cone in ambient dimension 2

julia> antv = affine_normal_toric_variety(C)
Normal toric variety

julia> hilbert_basis(antv)
[-1   1]
[ 1   1]
[ 0   1]

mori_cone(v::NormalToricVariety)

Return the mori cone of the normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> mori = mori_cone(p2)
Polyhedral cone in ambient dimension 1

julia> dim(mori)
1

nef_cone(v::NormalToricVariety)

Return the nef cone of the normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> nef = nef_cone(p2)
Polyhedral cone in ambient dimension 1

julia> dim(nef)
1

dim(v::NormalToricVarietyType)

Return the dimension of the normal toric variety v.

Examples

julia> C = Oscar.positive_hull([1 0]);

julia> antv = affine_normal_toric_variety(C);

julia> dim(antv)
1

dim_of_torusfactor(v::NormalToricVarietyType)

Return the dimension of the torus factor of the normal toric variety v.

Examples

julia> C = Oscar.positive_hull([1 0]);

julia> antv = affine_normal_toric_variety(C);

julia> dim_of_torusfactor(antv)
1

euler_characteristic(v::NormalToricVarietyType)

Return the Euler characteristic of the normal toric variety v.

Examples

julia> C = Oscar.positive_hull([1 0]);

julia> antv = affine_normal_toric_variety(C);

julia> euler_characteristic(antv)
1

betti_number(v::NormalToricVarietyType, i::Int)

Compute the i-th Betti number of the normal toric variety v. Specifically, this method returns the dimension of the i-th simplicial homology group (with rational coefficients) of v. The employed algorithm is derived from theorem 12.3.12 in [CLS11]. Note that this theorem requires that the normal toric variety v is both complete and simplicial.

Examples

julia> P3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> betti_number(P3,0)
1

julia> betti_number(P3, 1)
0

set_coordinate_names(v::NormalToricVarietyType, coordinate_names::AbstractVector{<:VarName})

Allows to set the names of the homogeneous coordinates as long as the toric variety in question is not yet finalized (cf. is_finalized(v::NormalToricVarietyType)).

Examples

julia> C = Oscar.positive_hull([1 0]);

julia> antv = affine_normal_toric_variety(C);

julia> set_coordinate_names(antv, [:u])

julia> coordinate_names(antv)
1-element Vector{String}:
 "u"

julia> set_coordinate_names(antv, ["v"])

julia> coordinate_names(antv)
1-element Vector{String}:
 "v"

julia> set_coordinate_names(antv, ['w'])

julia> coordinate_names(antv)
1-element Vector{String}:
 "w"

set_coordinate_names_of_torus(v::NormalToricVarietyType, coordinate_names::AbstractVector{<:VarName})

Allows to set the names of the coordinates of the torus.

Examples

julia> F3 = hirzebruch_surface(NormalToricVariety, 3);

julia> set_coordinate_names_of_torus(F3, ["u", "v"])

julia> coordinate_names_of_torus(F3)
2-element Vector{String}:
 "u"
 "v"

coordinate_names(v::NormalToricVarietyType)

Return the names of the homogeneous coordinates of the normal toric variety v. The default is x1, ..., xn.

Examples

julia> C = Oscar.positive_hull([1 0]);

julia> antv = affine_normal_toric_variety(C);

julia> coordinate_names(antv)
1-element Vector{String}:
 "x1"

coordinate_names_of_torus(v::NormalToricVarietyType)

Return the names of the coordinates of the torus of the normal toric variety v. The default is x1, ..., xn.

cox_ring(v::NormalToricVarietyType)

Compute the Cox ring of the normal toric variety v. Note that [CLS11] refers to this ring as the "total coordinate ring". For uniformity with schemes, we also support the function coordinate_ring to refer to the Cox ring.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> set_coordinate_names(p2, ["y1", "y2", "y3"])

julia> cox_ring(p2)
Multivariate polynomial ring in 3 variables over QQ graded by
  y1 -> [1]
  y2 -> [1]
  y3 -> [1]

julia> cox_ring(p2) == coordinate_ring(p2)
true

irrelevant_ideal(v::NormalToricVarietyType)

Return the irrelevant ideal of a normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> length(gens(irrelevant_ideal(p2)))
3

ideal_of_linear_relations(v::NormalToricVarietyType)

Return the ideal of linear relations of the simplicial and complete toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> ngens(ideal_of_linear_relations(p2))
2

stanley_reisner_ideal(v::NormalToricVarietyType)

Return the Stanley-Reisner ideal of a normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> ngens(stanley_reisner_ideal(p2))
1

toric_ideal(antv::AffineNormalToricVariety)

Return the toric ideal defining the affine normal toric variety.

Examples

Take the cone over the square at height one. The resulting toric variety has one defining equation. In projective space this corresponds to $\mathbb{P}^1\times\mathbb{P}^1$. Note that this cone is self-dual, the toric ideal comes from the dual cone.

julia> C = positive_hull([1 0 0; 1 1 0; 1 0 1; 1 1 1])
Polyhedral cone in ambient dimension 3

julia> antv = affine_normal_toric_variety(C)
Normal toric variety

julia> toric_ideal(antv)
Ideal generated by
  -x1*x2 + x3*x4

coordinate_ring_of_torus(v::NormalToricVarietyType)

Compute the coordinate ring of the torus of the normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> set_coordinate_names_of_torus(p2, ["y1", "y2"])

julia> coordinate_ring_of_torus(p2)
Quotient
  of multivariate polynomial ring in 4 variables y1, y2, y1_, y2_
    over rational field
  by ideal (y1*y1_ - 1, y2*y2_ - 1)

is_finalized(v::NormalToricVarietyType)

Check if the Cox ring, the coordinate ring of the torus, the cohomology_ring, the Chow ring, the Stanley-Reisner ideal, the irrelevant ideal, the ideal of linear relations or the toric ideal has been cached. If any of these has been cached, then this function returns true and otherwise false.

Examples

julia> is_finalized(del_pezzo_surface(NormalToricVariety, 3))
false

cox_ring(R::MPolyRing, v::NormalToricVarietyType)

Compute the Cox ring of the normal toric variety v, in this case by adding the Cox grading to the given ring R. Note that [CLS11] refers to this ring as the "total coordinate ring".

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> R, _ = polynomial_ring(QQ, 3);

julia> cox_ring(R, p2)
Multivariate polynomial ring in 3 variables over QQ graded by
  x1 -> [1]
  x2 -> [1]
  x3 -> [1]

irrelevant_ideal(R::MPolyRing, v::NormalToricVarietyType)

Return the irrelevant ideal of a normal toric variety v as an ideal in R.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> R, _ = polynomial_ring(QQ, 3);

julia> length(gens(irrelevant_ideal(R, p2)))
3

ideal_of_linear_relations(R::MPolyRing, v::NormalToricVarietyType)

Return the ideal of linear relations of the simplicial and complete toric variety v in the ring R.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> R, _ = polynomial_ring(QQ, 3);

julia> ngens(ideal_of_linear_relations(R, p2))
2

stanley_reisner_ideal(R::MPolyRing, v::NormalToricVarietyType)

Return the Stanley-Reisner ideal of a normal toric variety v as an ideal of R.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> R, _ = polynomial_ring(QQ, 3);

julia> ngens(stanley_reisner_ideal(R, p2))
1

toric_ideal(R::MPolyRing, antv::AffineNormalToricVariety)

Return the toric ideal defining the affine normal toric variety as an ideal in R.

Examples

Take the cone over the square at height one. The resulting toric variety has one defining equation. In projective space this corresponds to $\mathbb{P}^1\times\mathbb{P}^1$. Note that this cone is self-dual, the toric ideal comes from the dual cone.

julia> C = positive_hull([1 0 0; 1 1 0; 1 0 1; 1 1 1])
Polyhedral cone in ambient dimension 3

julia> antv = affine_normal_toric_variety(C)
Normal toric variety

julia> R, _ = polynomial_ring(QQ, 4);

julia> toric_ideal(R, antv)
Ideal generated by
  -x1*x2 + x3*x4

coordinate_ring_of_torus(R::MPolyRing, v::NormalToricVarietyType)

Compute the coordinate ring of the torus of the normal toric variety v in the given polynomial ring R.

character_to_rational_function(v::NormalToricVarietyType, character::Vector{ZZRingElem})

Compute the rational function corresponding to a character of the normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> character_to_rational_function(p2, [-1, 2])
x2^2*x1_

character_to_rational_function(R::MPolyRing, v::NormalToricVarietyType, character::Vector{ZZRingElem})

Compute the rational function corresponding to a character of the normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> R, _ = polynomial_ring(QQ, 4);

julia> character_to_rational_function(R, p2, [-1, 2])
x2^2*x3

binomial_exponents_to_ideal(binoms::Union{AbstractMatrix, ZZMatrix})

This function converts the rows of a matrix to binomials. Each row $r$ is written as $r=u-v$ with $u, v\ge 0$ by splitting into positive and negative entries. Then the row $r$ corresponds to $x^u-x^v$. The resulting ideal is returned.

Examples

julia> A = [-1 -1 0 2; 2 3 -2 -1]
2×4 Matrix{Int64}:
 -1  -1   0   2
  2   3  -2  -1

julia> binomial_exponents_to_ideal(A)
Ideal generated by
  -x1*x2 + x4^2
  x1^2*x2^3 - x3^2*x4

toric_ideal(pts::ZZMatrix)

Return the toric ideal generated from the linear relations between the points pts. This is the ideal generated by the set of binomials $\{x^u-x^v\ |\ u, v\in\mathbb{Z}^n_{\ge 0}\ (pts)^T\cdot(u-v) = 0\}$

Examples

julia> C = positive_hull([-2 5; 1 0]);

julia> H = hilbert_basis(C);

julia> toric_ideal(H)
Ideal generated by
  x2*x3 - x4^2
  -x1*x3 + x2^2*x4
  -x1*x4 + x2^3
  -x1*x3^2 + x2*x4^3
  -x1*x3^3 + x4^5