# Toric Divisor Classes

## Introduction

Toric divisor classes are equivalence classes of Weil divisors modulo linear equivalence.

## Constructors

### General constructors

toric_divisor_classMethod
toric_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 variety
source
toric_divisor_classMethod
toric_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 variety
source
toric_divisor_classMethod
toric_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 variety
source

### Addition, 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_classMethod
trivial_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 variety
source
anticanonical_divisor_classMethod
anticanonical_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 variety
source
canonical_divisor_classMethod
canonical_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 variety
source

## Properties

Equality of toric divisor classes can be tested via ==.

To check if a toric divisor class is trivial, one can invoke is_trivial.

is_effectiveMethod
is_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)
false
source

## Attributes

divisor_classMethod
divisor_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]
source
toric_varietyMethod
toric_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 variety
source
toric_divisorMethod
toric_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
source