Affine Toric Schemes

Constructors

We provide the following constructors for affine toric schemes:

toric_spec(antv::AffineNormalToricVariety)

julia> C = positive_hull([1 0; 0 1])
Polyhedral cone in ambient dimension 2

julia> antv = affine_normal_toric_variety(C)
Normal, affine toric variety

julia> toric_spec(antv)
Spec of an affine toric variety with cone spanned by RayVector{QQFieldElem}[[1, 0], [0, 1]]

Attributes

An affine toric scheme has all attributes of a normal toric scheme. In addition, there are the following special attributes, that we overload from the corresponding affine toric variety:

• $cone(X::ToricSpec)$,
• $dual_cone(X::ToricSpec)$,
• $hilbert_basis(X::ToricSpec)$,
• $toric_ideal(R::MPolyRing, X::ToricSpec)$,
• $toric_ideal(X::ToricSpec)$.

Properties

For an affine toric scheme, all properties of normal toric schemes are supported.

A note on the torus inclusion and the torus action

Note that the torus_inclusions(X::ToricSpec) is not yet supported. For an affine toric scheme $X$, we envision that this function should return a list l containing the inclusions $Tʳ⁽ⁱ⁾ ↪ X$ of the different tori.

Similarly, torus_action(X::ToricSpec) is not yet supported. For an affine toric scheme $X$ with a dense open torus $T$ this method should returns a quintuple of morphisms (pT, pX, incX, mult) consisting of the following:

• the projection $T × X → T$ of the product with the torus $T$ to $T$,
• the projection $T × X → X$,
• the inclusion $X ↪ T × X$ taking $x$ to $(1, x)$,
• the group action $T × X → X$.