Normal Toric Varieties

Introduction

We introduce two main types of normal toric varieties, distinguishing between the affine and non-affine case:

AffineNormalToricVariety is the type for toric varieties associated to a cone $\sigma$ , denoted by $U_{\sigma}$ in [CLS11]
NormalToricVariety is the type for toric varieties associated to a polyhedral fan $\Sigma$ , denoted by $X_{\Sigma}$ in [CLS11]

Warning

The lattice is always assumed to be the standard lattice $\mathbb{Z}^n$ . Transformations for non-standard lattices will have to be done by the user.

Equality of Normal Toric Varieties

Equality == of normal toric varieties checks equality of the corresponding polyhedral fans as sets of cones. This computes the rays of both of the toric varieties, which can be expensive if they are not already computed, meaning if "RAYS" in Polymake.list_properties(Oscar.pm_object(polyhedral_fan(X))) is false for one of the varieties. Triple-equality === always checks equality of memory locations in OSCAR.

Constructors

Affine Toric Varieties

affine_normal_toric_variety — Method

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 — Method

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 — Method

Normal Toric Varieties

normal_toric_variety — Function

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

Famous Toric Varieties

The constructors of del_pezzo_surface, hirzebruch_surface, projective_space and weighted_projective_space always make a default/standard choice for the grading of the Cox ring.

affine_space — Method

del_pezzo_surface — Method

hirzebruch_surface — Method

projective_space — Method

weighted_projective_space — Method

Constructions based on triangulations

It is possible to associate toric varieties to star triangulations of the lattice points of polyhedrons. Specifically, we can associate to any full star triangulation of the lattice points of the polyhedron in question a toric variety. For this task, we provide the following constructors.

normal_toric_variety_from_star_triangulation — Method

normal_toric_varieties_from_star_triangulations — Method

An application of this functionality exists in the physics. Witten's Generalized-Sigma models (GLSM) [Wit88] 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 [FJR17], which describes this as GIT construction [CLS11].

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. We provide the following two constructors:

normal_toric_variety_from_glsm — Method

normal_toric_varieties_from_glsm — Method

Further Constructions

* — Method

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 — Method

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 — Method

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

Properties of Toric Varieties

has_torusfactor — Method

is_affine — Method

is_complete — Method

is_fano — Method

is_gorenstein — Method

is_simplicial — Method

is_smooth — Method

is_normal — Method

is_orbifold — Method

is_projective — Method

is_projective_space — Method

is_q_gorenstein — Method

Operations for Toric Varieties

Affine Open Covering

affine_open_covering — Method

Characters, Weil Divisors, Cartier Divisors, Class Group and Picard Group

torusinvariant_cartier_divisor_group — Method

character_lattice — Method

class_group — Method

map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group — Method

map_from_torusinvariant_cartier_divisor_group_to_picard_group — Method

map_from_character_lattice_to_torusinvariant_weil_divisor_group — Method

map_from_torusinvariant_weil_divisor_group_to_class_group — Method

picard_group — Method

torusinvariant_weil_divisor_group — Method

torusinvariant_prime_divisors — Method

Gorenstein and Picard index

gorenstein_index — Method

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 — Method

Cones and Fans

polyhedral_fan — Method

cone — Method

weight_cone — Method

hilbert_basis — Method

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 — Method

nef_cone — Method

Dimensions

dim — Method

dim_of_torusfactor — Method

euler_characteristic — Method

betti_number — Method

Rings and ideals

We support the following rings and ideals for toric varieties:

Cox ring (also termed the "total coordinate ring" in [CLS11]),
coordinate ring of torus,
cohomology_ring,
Chow ring,
irrelevant ideal,
Stanley-Reisner ideal,
ideal of linear relations,
toric ideal.

Of course, for any of these coordinate names and the coefficient ring have to be chosen. The coefficient ring is fixed to Q. Therefore, the method coefficient_ring(v::NormalToricVarietyType) always return the field of rational numbers. For the coordinate names, we provide the following setter functions:

set_coordinate_names — Method

set_coordinate_names_of_torus — Method

The following methods allow to extract the chosen coordinates:

coordinate_names — Method

coordinate_names_of_torus — Method

In order to efficiently construct algebraic cycles (elements of the Cox ring), cohomology classes (elements of the cohomology ring), or in order to compare ideals, it is imperative to fix choices of the coordinate names. The default value for coordinate names is [x1, x2, ... ]. The choice of coordinate names is fixed, once one of the above-mentioned rings is computed via one the following methods:

cox_ring — Method

irrelevant_ideal — Method

ideal_of_linear_relations — Method

stanley_reisner_ideal — Method

toric_ideal — Method

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 — Method

One can check the status as follows:

is_finalized — Method

After the variety finalized, one can enforce to obtain the above ideals in different rings. Also, one can opt to compute the above rings with a different choice of coordinate names and different coefficient ring. To this end, one provides a custom ring (which reflects the desired choice of coordinate names and coefficient ring) as first argument. However, note that the cached ideals and rings are not altered.

cox_ring — Method

irrelevant_ideal — Method

ideal_of_linear_relations — Method

stanley_reisner_ideal — Method

toric_ideal — Method

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 — Method

Along the same lines, characters can be turned into rational functions:

character_to_rational_function — Method

character_to_rational_function — Method

Auxiliary Methods

binomial_exponents_to_ideal — Method

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 — Method

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