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Toric Line Bundles

Constructors

Generic constructors

toric_line_bundle — Method
toric_line_bundle(v::NormalToricVarietyType, picard_class::FinGenAbGroupElem)

Construct the line bundle on the abstract normal toric variety with given class in the Picard group of the toric variety in question.

Examples

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

julia> pc = picard_group_with_map(P2)[1];

julia> l = toric_line_bundle(P2, pc([1]))
Toric line bundle on a normal toric variety
source
toric_line_bundle — Method
toric_line_bundle(v::NormalToricVarietyType, picard_class::Vector{T}) where {T <: IntegerUnion}

Construct the line bundle on the abstract normal toric variety v with class c in the Picard group of v.

Examples

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

julia> l = toric_line_bundle(v, [ZZRingElem(2)])
Toric line bundle on a normal toric variety
source
toric_line_bundle — Method
toric_line_bundle(v::NormalToricVarietyType, d::ToricDivisor)

Construct the toric variety associated to a (Cartier) torus-invariant divisor d on the normal toric variety v.

Examples

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

julia> l = toric_line_bundle(v, toric_divisor(v, [1, 2, 3]))
Toric line bundle on a normal toric variety
source
toric_line_bundle — Method
toric_line_bundle(d::ToricDivisor)

Construct the toric variety associated to a (Cartier) torus-invariant divisor d.

Examples

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

julia> d = toric_divisor(v, [1, 2, 3]);

julia> l = toric_line_bundle(d)
Toric line bundle on a normal toric variety
source
toric_line_bundle — Method
toric_line_bundle(v::NormalToricVarietyType, dc::ToricDivisorClass)

Construct the toric variety associated to a divisor class in the class group of a toric variety.

Examples

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

julia> d = toric_divisor(v, [1, 2, 3])
Torus-invariant, non-prime divisor on a normal toric variety

julia> dc = toric_divisor_class(d)
Divisor class on a normal toric variety

julia> l = toric_line_bundle(v, dc)
Toric line bundle on a normal toric variety
source
toric_line_bundle — Method
toric_line_bundle(dc::ToricDivisorClass)

Construct the toric variety associated to a divisor class in the class group of a toric variety.

Examples

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

julia> d = toric_divisor(v, [1, 2, 3])
Torus-invariant, non-prime divisor on a normal toric variety

julia> dc = toric_divisor_class(d)
Divisor class on a normal toric variety

julia> l = toric_line_bundle(dc)
Toric line bundle on a normal toric variety
source

Tensor products

Toric line bundles can be tensored via *. The n-th tensor power can be computed via ^n. In particular, ^(-1) computes the inverse of a line bundle. Alternatively, one can compute the inverse by invoking inv.

Special line bundles

anticanonical_bundle — Method
anticanonical_bundle(v::NormalToricVarietyType)

Construct the anticanonical bundle of a normal toric variety.

Examples

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

julia> anticanonical_bundle(v)
Toric line bundle on a normal toric variety
source
canonical_bundle — Method
canonical_bundle(v::NormalToricVarietyType)

Construct the canonical bundle of a normal toric variety.

Examples

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

julia> canonical_bundle(v)
Toric line bundle on a normal toric variety
source
structure_sheaf — Method
structure_sheaf(v::NormalToricVarietyType)

Construct the structure sheaf of a normal toric variety.

Examples

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

julia> structure_sheaf(v)
Toric line bundle on a normal toric variety
source
trivial_line_bundle — Method
trivial_line_bundle(v::NormalToricVarietyType)

Construct the trivial line bundle on a normal toric variety.

Examples

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

julia> l = trivial_line_bundle(v)
Toric line bundle on a normal toric variety

julia> is_trivial(l)
true
source

Properties

Equality of toric line bundles can be tested via ==.

To check if a toric line bundle is trivial, one can invoke is_trivial. Beyond this, we support the following properties of toric line bundles:

is_ample — Method
is_ample(l::ToricLineBundle)

Return true if the toric line bundle l is ample and false otherwise.

Examples

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

julia> is_ample(toric_line_bundle(F4, [1,0]))
false
source
is_basepoint_free — Method
is_basepoint_free(l::ToricLineBundle)

Return true if the toric line bundle l is basepoint free and false otherwise.

Examples

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

julia> is_basepoint_free(toric_line_bundle(F4, [1, 0]))
true
source
is_immaculate — Method
is_immaculate(l::ToricLineBundle)

Return true if all sheaf cohomologies of l are trivial and false otherwise.

Examples

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

julia> l = toric_line_bundle(F4, [1,0])
Toric line bundle on a normal toric variety

julia> is_immaculate(toric_line_bundle(F4, [1,0]))
false

julia> all_cohomologies(l)
3-element Vector{ZZRingElem}:
 2
 0
 0
source
is_very_ample — Method
is_very_ample(l::ToricLineBundle)

Return true if the toric line bundle l is very ample and false otherwise.

Examples

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

julia> is_very_ample(toric_line_bundle(F4, [1,0]))
false
source

Attributes

degree — Method
degree(l::ToricLineBundle)

Return the degree of the toric line bundle l.

Examples

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

julia> l = toric_line_bundle(v, [ZZRingElem(2)])
Toric line bundle on a normal toric variety

julia> degree(l)
2
source
picard_class — Method
picard_class(l::ToricLineBundle)

Return the class in the Picard group which defines the toric line bundle l.

Examples

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

julia> l = toric_line_bundle(v, [ZZRingElem(2)])
Toric line bundle on a normal toric variety

julia> picard_class(l)
Abelian group element [2]
source
toric_divisor — Method
toric_divisor(l::ToricLineBundle)

Return a toric divisor corresponding to the toric line bundle l.

Examples

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

julia> l = toric_line_bundle(v, [ZZRingElem(2)])
Toric line bundle on a normal toric variety

julia> toric_divisor(l)
Torus-invariant, cartier, non-prime divisor on a normal toric variety

julia> is_cartier(toric_divisor(l))
true
source
toric_divisor_class — Method
toric_divisor_class(l::ToricLineBundle)

Return a divisor class in the Class group corresponding to the toric line bundle l.

Examples

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

julia> l = toric_line_bundle(v, [ZZRingElem(2)])
Toric line bundle on a normal toric variety

julia> toric_divisor(l)
Torus-invariant, cartier, non-prime divisor on a normal toric variety

julia> is_cartier(toric_divisor(l))
true
source
toric_variety — Method
toric_variety(l::ToricLineBundle)

Return the toric variety over which the toric line bundle l is defined.

Examples

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

julia> l = toric_line_bundle(v, [ZZRingElem(2)])
Toric line bundle on a normal toric variety

julia> toric_variety(l)
Normal toric variety without torusfactor
source

Methods

basis_of_global_sections_via_rational_functions — Method
basis_of_global_sections_via_rational_functions(l::ToricLineBundle)

Return a basis of the global sections of the toric line bundle l in terms of rational functions.

Examples

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

julia> l = toric_line_bundle(v, [ZZRingElem(2)])
Toric line bundle on a normal toric variety

julia> basis_of_global_sections_via_rational_functions(l)
6-element Vector{MPolyQuoRingElem{QQMPolyRingElem}}:
 x1_^2
 x2*x1_^2
 x2^2*x1_^2
 x1_
 x2*x1_
 1
source
basis_of_global_sections_via_homogeneous_component — Method
basis_of_global_sections_via_homogeneous_component(l::ToricLineBundle)

Return a basis of the global sections of the toric line bundle l in terms of a homogeneous component of the Cox ring of toric_variety(l). For convenience, this method can also be called via basis_of_global_sections(l::ToricLineBundle).

Examples

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

julia> l = toric_line_bundle(v, [ZZRingElem(2)])
Toric line bundle on a normal toric variety

julia> basis_of_global_sections_via_homogeneous_component(l)
6-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
 x3^2
 x2*x3
 x2^2
 x1*x3
 x1*x2
 x1^2

julia> basis_of_global_sections(l)
6-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
 x3^2
 x2*x3
 x2^2
 x1*x3
 x1*x2
 x1^2
source
generic_section — Method
generic_section(l::ToricLineBundle; range::UnitRange{Int64} = -10000:10000, rng::AbstractRNG = Random.default_rng())

Return a generic section of the toric line bundle l, that is return the sum of all elements basis_of_global_sections(l), each multiplied by a random integer.

The optional keyword argument range can be used to set the range of the random integers, e.g., generic_section(l, range = -100:100)

The random source used to create random coefficients can be set with the optional argument rng.

Examples

julia> using Random;

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

julia> l = toric_line_bundle(v, [ZZRingElem(2)])
Toric line bundle on a normal toric variety

julia> s = generic_section(l, rng = Random.Xoshiro(1234));

julia> parent(s) == cox_ring(toric_variety(l))
true
source