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 varietytoric_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 varietytoric_line_bundle — Method
toric_line_bundle(v::NormalToricVarietyType, d::ToricDivisor)Construct the toric line bundle 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 varietytoric_line_bundle — Method
toric_line_bundle(d::ToricDivisor)Construct the toric line bundle 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 varietytoric_line_bundle — Method
toric_line_bundle(v::NormalToricVarietyType, dc::ToricDivisorClass)Construct the toric line bundle associated to a Cartier divisor class on the normal toric variety v.
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 varietytoric_line_bundle — Method
toric_line_bundle(dc::ToricDivisorClass)Construct the toric line bundle associated to a Cartier divisor class.
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 varietyTensor 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 varietycanonical_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 varietystructure_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 varietytrivial_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)
trueProperties
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_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]))
trueis_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> sheaf_cohomology(l)
3-element Vector{ZZRingElem}:
2
0
0is_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]))
falseAttributes
coefficients — Method
coefficients(l::ToricLineBundle)Return the coefficients of the chosen toric divisor representative of l.
degree — Method
degree(l::ToricLineBundle)Return the degree of the toric line bundle l when the Picard group of the underlying toric variety is free of rank one. The degree is the coefficient of the Picard class with respect to the chosen generator of the Picard group. An error is raised otherwise.
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)
2divisor_class — Method
divisor_class(l::ToricLineBundle)Return the element of the class group represented by the toric line bundle l.
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]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))
truetoric_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))
truetoric_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 torusfactorsMethods
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_
1basis_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
x1^2generic_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