Toric Divisor Classes
Introduction
Toric divisor classes are equivalence classes of Weil divisors modulo linear equivalence.
Constructors
General constructors
toric_divisor_class — Methodtoric_divisor_class(v::NormalToricVarietyType, class::FinGenAbGroupElem)Construct the toric divisor class associated to a group element of the class group of the normal toric variety v.
Examples
julia> P2 = projective_space(NormalToricVariety, 2)
Normal toric variety
julia> tdc = toric_divisor_class(P2, class_group(P2)([1]))
Divisor class on a normal toric varietytoric_divisor_class — Methodtoric_divisor_class(v::NormalToricVarietyType, coeffs::Vector{T}) where {T <: IntegerUnion}Construct the toric divisor class associated to a list of integers which specify an element of the class group of the normal toric variety v.
Examples
julia> P2 = projective_space(NormalToricVariety, 2)
Normal toric variety
julia> tdc = toric_divisor_class(P2, class_group(P2)([ZZRingElem(1)]))
Divisor class on a normal toric varietytoric_divisor_class — Methodtoric_divisor_class(td::ToricDivisor)Construct the toric divisor class associated to the element ... of the class group of the normal toric variety v.
Examples
julia> P2 = projective_space(NormalToricVariety, 2)
Normal toric variety
julia> td = toric_divisor(P2, [1, 2, 3])
Torus-invariant, non-prime divisor on a normal toric variety
julia> tdc = toric_divisor_class(td)
Divisor class on a normal toric varietyAddition, subtraction and scalar multiplication
Toric divisor classes can be added and subtracted via the usual + and - operators. Moreover, multiplication by scalars from the left is supported for scalars which are integers or of type ZZRingElem.
Special divisor classes
trivial_divisor_class — Methodtrivial_divisor_class(v::NormalToricVarietyType)Construct the trivial divisor class of a normal toric variety.
Examples
julia> v = projective_space(NormalToricVariety, 2)
Normal toric variety
julia> trivial_divisor_class(v)
Divisor class on a normal toric varietyanticanonical_divisor_class — Methodanticanonical_divisor_class(v::NormalToricVarietyType)Construct the anticanonical divisor class of a normal toric variety.
Examples
julia> v = projective_space(NormalToricVariety, 2)
Normal toric variety
julia> anticanonical_divisor_class(v)
Divisor class on a normal toric varietycanonical_divisor_class — Methodcanonical_divisor_class(v::NormalToricVarietyType)Construct the canonical divisor class of a normal toric variety.
Examples
julia> v = projective_space(NormalToricVariety, 2)
Normal toric variety
julia> canonical_divisor_class(v)
Divisor class on a normal toric varietyProperties
Equality of toric divisor classes can be tested via ==.
To check if a toric divisor class is trivial, one can invoke is_trivial.
is_effective — Methodis_effective(tdc::ToricDivisorClass)Determines whether the toric divisor class tdc is effective, that is if a toric divisor in this divisor class is linearly equivalent to an effective toric divisor.
Examples
julia> P2 = projective_space(NormalToricVariety,2)
Normal toric variety
julia> tdc = toric_divisor_class(P2, [1])
Divisor class on a normal toric variety
julia> is_effective(tdc)
true
julia> tdc2 = toric_divisor_class(P2, [-1])
Divisor class on a normal toric variety
julia> is_effective(tdc2)
falseAttributes
divisor_class — Methoddivisor_class(tdc::ToricDivisorClass)Return the element of the class group corresponding to the toric divisor class tdc.
Examples
julia> P2 = projective_space(NormalToricVariety, 2)
Normal toric variety
julia> tdc = toric_divisor_class(P2, class_group(P2)([1]))
Divisor class on a normal toric variety
julia> divisor_class(tdc)
Abelian group element [1]toric_variety — Methodtoric_variety(tdc::ToricDivisorClass)Return the toric variety on which the toric divisor class tdc is defined.
Examples
julia> P2 = projective_space(NormalToricVariety, 2)
Normal toric variety
julia> tdc = toric_divisor_class(P2, class_group(P2)([1]))
Divisor class on a normal toric variety
julia> toric_variety(tdc)
Normal toric varietytoric_divisor — Methodtoric_divisor(tdc::ToricDivisorClass)Constructs a toric divisor corresponding to the toric divisor class tdc.
Examples
julia> P2 = projective_space(NormalToricVariety, 2)
Normal toric variety
julia> tdc = toric_divisor_class(P2, class_group(P2)([1]))
Divisor class on a normal toric variety
julia> toric_divisor(tdc)
Torus-invariant, prime divisor on a normal toric variety