Toric Divisor Classes

Toric Divisor Classes

Introduction

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

Constructors

General constructors

ToricDivisorClass — Method

ToricDivisorClass(v::AbstractNormalToricVariety, 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)
A normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

julia> tdc = ToricDivisorClass(P2, class_group(P2)([fmpz(1)]))
A divisor class on a normal toric variety

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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 fmpz.

Special divisor classes

trivial_divisor_class — Method

trivial_divisor_class(v::AbstractNormalToricVariety)

Construct the trivial divisor class of a normal toric variety.

Examples

julia> v = projective_space(NormalToricVariety, 2)
A normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

julia> trivial_divisor_class(v)
A divisor class on a normal toric variety

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anticanonical_divisor_class — Method

anticanonical_divisor_class(v::AbstractNormalToricVariety)

Construct the anticanonical divisor class of a normal toric variety.

Examples

julia> v = projective_space(NormalToricVariety, 2)
A normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

julia> anticanonical_divisor_class(v)
A divisor class on a normal toric variety

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canonical_divisor_class — Method

canonical_divisor_class(v::AbstractNormalToricVariety)

Construct the canonical divisor class of a normal toric variety.

Examples

julia> v = projective_space(NormalToricVariety, 2)
A normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

julia> canonical_divisor_class(v)
A divisor class on a normal toric variety

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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_effective — Method

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)
A normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

julia> tdc = ToricDivisorClass(P2, [1])
A divisor class on a normal toric variety

julia> is_effective(tdc)
true

julia> tdc2 = ToricDivisorClass(P2, [-1])
A divisor class on a normal toric variety

julia> is_effective(tdc2)
false

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Attributes

divisor_class — Method

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)
A normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

julia> tdc = ToricDivisorClass(P2, class_group(P2)([1]))
A divisor class on a normal toric variety

julia> divisor_class(tdc)
Element of
GrpAb: Z
with components [1]

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toric_variety — Method

toric_variety(tdc::ToricDivisorClass)

Return the toric variety on which the toric divisor class tdc is defined.

Examples

julia> P2 = projective_space(NormalToricVariety, 2)
A normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

julia> tdc = ToricDivisorClass(P2, class_group(P2)([1]))
A divisor class on a normal toric variety

julia> toric_variety(tdc)
A normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

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toric_divisor — Method

toric_divisor(tdc::ToricDivisorClass)

Constructs a toric divisor corresponding to the toric divisor class tdc.

Examples

julia> P2 = projective_space(NormalToricVariety, 2)
A normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor

julia> tdc = ToricDivisorClass(P2, class_group(P2)([1]))
A divisor class on a normal toric variety

julia> toric_divisor(tdc)
A torus-invariant, prime divisor on a normal toric variety

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