Normal Toric Varieties

affine_normal_toric_variety(C::Cone; set_attributes::Bool = true)

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, affine toric variety

normal_toric_variety(C::Cone; set_attributes::Bool = true)

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, affine toric variety

affine_normal_toric_variety(v::NormalToricVariety; set_attributes::Bool = true)

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, affine toric variety

julia> affineVariety = affine_normal_toric_variety(v)
Normal, affine toric variety

normal_toric_variety(rays::Vector{Vector{Int64}}, max_cones::Vector{Vector{Int64}}; non_redundant::Bool = false, set_attributes::Bool = true)

Construct a normal toric variety $X$ by providing the rays and maximal cones as vector of vectors. By default, this method assumes that the input is not non-redundant (e.g. that a ray was entered twice by accident). 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 = [[1, 2], [2, 3], [3, 4], [4, 1]]
4-element Vector{Vector{Int64}}:
 [1, 2]
 [2, 3]
 [3, 4]
 [4, 1]

julia> normal_toric_variety(ray_generators, max_cones)
Normal toric variety

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

normal_toric_variety(PF::PolyhedralFan; set_attributes::Bool = true)

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)
Polyhedron in ambient dimension 2

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

julia> ntv = normal_toric_variety(nf)
Normal toric variety

NormalToricVariety(P::Polyhedron; set_attributes::Bool = true)

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)
Polyhedron in ambient dimension 2

julia> ntv = normal_toric_variety(square)
Normal toric variety

affine_space(::Type{NormalToricVariety}, d::Int; set_attributes::Bool = true)

Constructs the (toric) affine space of dimension d.

Examples

julia> affine_space(NormalToricVariety, 2)
Normal, affine, 2-dimensional toric variety

del_pezzo_surface(::Type{NormalToricVariety}, b::Int; set_attributes::Bool = true)

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

Examples

julia> del_pezzo_surface(NormalToricVariety, 3)
Normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

hirzebruch_surface(::Type{NormalToricVariety}, r::Int; set_attributes::Bool = true)

Constructs the r-th Hirzebruch surface.

Examples

julia> hirzebruch_surface(NormalToricVariety, 5)
Normal, non-affine, smooth, projective, gorenstein, non-fano, 2-dimensional toric variety without torusfactor

projective_space(::Type{NormalToricVariety}, d::Int; set_attributes::Bool = true)

Construct the projective space of dimension d.

Examples

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

weighted_projective_space(::Type{NormalToricVariety}, w::Vector{T}; set_attributes::Bool = true) where {T <: IntegerUnion}

Construct the weighted projective space corresponding to the weights w.

Examples

julia> weighted_projective_space(NormalToricVariety, [2,3,1])
Normal, non-affine, simplicial, projective, 2-dimensional toric variety without torusfactor

blowup(v::AbstractNormalToricVariety, I::MPolyIdeal; coordinatename::String = "e", set_attributes::Bool = true)

Blowup the toric variety by subdividing the cone in the list of all cones of the fan of v which corresponds to the provided ideal I. Note that this cone need not be maximal.

By default, we pick "e" as the name of the homogeneous coordinate for the exceptional divisor. As third optional argument on can supply a custom variable name.

Examples

julia> P3 = projective_space(NormalToricVariety, 3)
Normal, non-affine, smooth, projective, gorenstein, fano, 3-dimensional toric variety without torusfactor

julia> (x1,x2,x3,x4) = gens(cox_ring(P3))
4-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
 x1
 x2
 x3
 x4

julia> I = ideal([x2,x3])
ideal(x2, x3)

julia> bP3 = blow_up(P3, I)
Normal toric variety

julia> cox_ring(bP3)
Multivariate polynomial ring in 5 variables over QQ graded by
  x1 -> [1 0]
  x2 -> [0 1]
  x3 -> [0 1]
  x4 -> [1 0]
  e -> [1 -1]

blowup(v::AbstractNormalToricVariety, n::Int; coordinatename::String = "e", set_attributes::Bool = true)

Blowup the toric variety by subdividing the n-th cone in the list of all cones of the fan of v. This cone need not be maximal.

By default, we pick "e" as the name of the homogeneous coordinate for the exceptional divisor. As third optional argument on can supply a custom variable name.

Examples

julia> P3 = projective_space(NormalToricVariety, 3)
Normal, non-affine, smooth, projective, gorenstein, fano, 3-dimensional toric variety without torusfactor

julia> cones(P3)
14×4 IncidenceMatrix
[1, 2, 3]
[2, 3, 4]
[1, 3, 4]
[1, 2, 4]
[2, 3]
[1, 3]
[1, 2]
[3, 4]
[2, 4]
[1, 4]
[2]
[3]
[1]
[4]

julia> bP3 = blow_up(P3, 5)
Normal toric variety

julia> cox_ring(bP3)
Multivariate polynomial ring in 5 variables over QQ graded by
  x1 -> [1 0]
  x2 -> [0 1]
  x3 -> [0 1]
  x4 -> [1 0]
  e -> [1 -1]

Base.:*(v::AbstractNormalToricVariety, w::AbstractNormalToricVariety; set_attributes::Bool = true)

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::AbstractNormalToricVariety, 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::AbstractNormalToricVariety)).

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, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

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]

normal_toric_varieties_from_star_triangulations(P::Polyhedron; set_attributes::Bool = true)

Returns 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(x2*x4)

julia> stanley_reisner_ideal(v2)
ideal(x1*x3)

normal_toric_varieties_from_glsm(charges::ZZMatrix; set_attributes::Bool = true)

Witten's Generalized-Sigma models (GLSM) Edward Witten (1988) originally sparked interest in the physics community in toric varieties. On a mathematical level, this establishes a construction of toric varieties for which a Z^n grading of the Cox ring is provided. See for example Huijun Fan, Tyler Jarvis, Yongbin Ruan (2017), which describes this as GIT construction David A. Cox, John B. Little, Henry K. Schenck (2011).

Explicitly, given the grading of the Cox ring, the map from the group of torus invariant Weil divisors to the class group is known. Under the assumption that the variety in question has no torus factor, we can then identify the map from the lattice to the group of torus invariant Weil divisors as the kernel of the map from the torus invariant Weil divisor to the class group. The latter is a map between free Abelian groups, i.e. is provided by an integer valued matrix. The rows of this matrix are nothing but the ray generators of the fan of the toric variety. It then remains to triangulate these rays, hence in general for a GLSM the toric variety is only unique up to fine regular star triangulations.

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

For convenience, we also support:

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

has_torusfactor(v::AbstractNormalToricVariety)

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

Examples

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

is_affine(v::AbstractNormalToricVariety)

Checks if the normal toric variety v is affine.

Examples

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

is_complete(v::AbstractNormalToricVariety)

Checks if the normal toric variety v is complete.

Examples

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

is_fano(v::AbstractNormalToricVariety)

Checks if the normal toric variety v is fano.

Examples

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

is_gorenstein(v::AbstractNormalToricVariety)

Checks if the normal toric variety v is Gorenstein.

Examples

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

is_simplicial(v::AbstractNormalToricVariety)

Checks 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::AbstractNormalToricVariety)

Checks if the normal toric variety v is smooth.

Examples

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

is_normal(v::AbstractNormalToricVariety)

Checks 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::AbstractNormalToricVariety)

Checks if the normal toric variety v is an orbifold.

Examples

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

is_projective(v::AbstractNormalToricVariety)

Checks 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::AbstractNormalToricVariety)

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

Examples

julia> F5 = hirzebruch_surface(NormalToricVariety, 5)
Normal, non-affine, smooth, projective, gorenstein, non-fano, 2-dimensional toric variety without torusfactor

julia> is_projective_space(F5)
false

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

is_q_gorenstein(v::AbstractNormalToricVariety)

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

Examples

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

affine_open_covering(v::AbstractNormalToricVariety)

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, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

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

torusinvariant_cartier_divisor_group(v::AbstractNormalToricVariety)

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

Examples

julia> p2 = projective_space(NormalToricVariety, 2)
Normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

julia> torusinvariant_cartier_divisor_group(p2)
GrpAb: Z^3

character_lattice(v::AbstractNormalToricVariety)

Return the character lattice of a normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> character_lattice(p2)
GrpAb: Z^2

class_group(v::AbstractNormalToricVariety)

Return the class group of the normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> class_group(p2)
GrpAb: Z

map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(v::AbstractNormalToricVariety)

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, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

julia> map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(p2)
Map with following data
Domain:
=======
Abelian group with structure: Z^3
Codomain:
=========
Abelian group with structure: Z^3

map_from_torusinvariant_cartier_divisor_group_to_picard_group(v::AbstractNormalToricVariety)

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, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

julia> map_from_torusinvariant_cartier_divisor_group_to_picard_group(p2)
Map with following data
Domain:
=======
Abelian group with structure: Z^3
Codomain:
=========
Abelian group with structure: Z

map_from_character_lattice_to_torusinvariant_weil_divisor_group(v::AbstractNormalToricVariety)

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 with following data
Domain:
=======
Abelian group with structure: Z^2
Codomain:
=========
Abelian group with structure: Z^3

map_from_torusinvariant_weil_divisor_group_to_class_group(v::AbstractNormalToricVariety)

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 with following data
Domain:
=======
Abelian group with structure: Z^3
Codomain:
=========
Abelian group with structure: Z

picard_group(v::AbstractNormalToricVariety)

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

Examples

julia> p2 = projective_space(NormalToricVariety, 2)
Normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

julia> picard_group(p2)
GrpAb: Z

torusinvariant_weil_divisor_group(v::AbstractNormalToricVariety)

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

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> torusinvariant_weil_divisor_group(p2)
GrpAb: Z^3

torusinvariant_prime_divisors(v::AbstractNormalToricVariety)

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

fan(v::AbstractNormalToricVariety)

Return the fan of an abstract normal toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2)
Normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

julia> 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

dual_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, affine toric variety

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

julia> polarize(cone(antv)) == dual_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, affine 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, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

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, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

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

julia> dim(nef)
1

dim(v::AbstractNormalToricVariety)

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::AbstractNormalToricVariety)

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::AbstractNormalToricVariety)

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::AbstractNormalToricVariety, 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 David A. Cox, John B. Little, Henry K. Schenck (2011). Note that this theorem requires that the normal toric variety v is both complete and simplicial.

Examples

julia> P3 = projective_space(NormalToricVariety, 3)
Normal, non-affine, smooth, projective, gorenstein, fano, 3-dimensional toric variety without torusfactor

julia> betti_number(P3,0)
1

julia> betti_number(P3, 1)
0

set_coordinate_names(v::AbstractNormalToricVariety, coordinate_names::Vector{String})

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::AbstractNormalToricVariety)).

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"

set_coordinate_names_of_torus(v::AbstractNormalToricVariety, coordinate_names::Vector{String})

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::AbstractNormalToricVariety)

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::AbstractNormalToricVariety)

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

cox_ring(v::AbstractNormalToricVariety)

Computes the Cox ring of the normal toric variety v. Note that David A. Cox, John B. Little, Henry K. Schenck (2011) refers to this ring as the "total coordinate 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]

irrelevant_ideal(v::AbstractNormalToricVariety)

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::AbstractNormalToricVariety)

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::AbstractNormalToricVariety)

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, affine toric variety

julia> toric_ideal(antv)
ideal(-x1*x2 + x3*x4)

coordinate_ring_of_torus(v::AbstractNormalToricVariety)

Computes 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 over QQ
  by ideal(y1*y1_ - 1, y2*y2_ - 1)

is_finalized(v::AbstractNormalToricVariety)

Checks 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::AbstractNormalToricVariety)

Computes the Cox ring of the normal toric variety v, in this case by adding the Cox grading to the given ring R. Note that David A. Cox, John B. Little, Henry K. Schenck (2011) 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::AbstractNormalToricVariety)

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::AbstractNormalToricVariety)

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::AbstractNormalToricVariety)

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, affine toric variety

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

julia> toric_ideal(R, antv)
ideal(-x1*x2 + x3*x4)

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

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

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

Computes 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::AbstractNormalToricVariety, character::Vector{ZZRingElem})

Computes 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(-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(x2*x3 - x4^2, -x1*x3 + x2^2*x4, -x1*x4 + x2^3, -x1*x3^2 + x2*x4^3, -x1*x3^3 + x4^5)