Toric varieties

gkm_graph_of_toric(v::Union{AffineNormalToricVariety, NormalToricVariety}; big_torus::Bool=true)

Construct the GKM graph of the smooth toric variety v.

If the variety is projective, all flags will be connected to edges. If the variety is non-projective, codimension-1 faces that are not shared by two maximal cones will correspond to standalone flags.

Dimension of the torus

If the optional argument big_torus is true (default value) then the torus rank of the result is the number of rays of v. If big_torus is false then the torus rank of the result is the dimension of v.

Examples

julia> P2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> gkm_graph_of_toric(P2)
GKM graph with 3 nodes, valency 2 and axial function:
2 -> 1 => (-1, 0, 1)
3 -> 1 => (0, -1, 1)
3 -> 2 => (1, -1, 0)
Algorithmic connection for GKM graph with 3 nodes and valency 2

julia> gkm_graph_of_toric(P2; big_torus=false)
GKM graph with 3 nodes, valency 2 and axial function:
2 -> 1 => (-1, 0)
3 -> 1 => (0, -1)
3 -> 2 => (1, -1)
Algorithmic connection for GKM graph with 3 nodes and valency 2

julia> F = hirzebruch_surface(NormalToricVariety, 3)
Normal toric variety

julia> gkm_graph_of_toric(F)
GKM graph with 4 nodes, valency 2 and axial function:
2 -> 1 => (-1, 0, 1, 0)
3 -> 2 => (-3, -1, 0, 1)
4 -> 1 => (0, -1, -3, 1)
4 -> 3 => (1, 0, -1, 0)
Algorithmic connection for GKM graph with 4 nodes and valency 2

julia> gkm_graph_of_toric(F; big_torus=false)
GKM graph with 4 nodes, valency 2 and axial function:
2 -> 1 => (-1, 0)
3 -> 2 => (-3, -1)
4 -> 1 => (0, -1)
4 -> 3 => (1, 0)
Algorithmic connection for GKM graph with 4 nodes and valency 2

julia> gkm_graph_of_toric(affine_space(NormalToricVariety, 3))
GKM graph with 1 nodes, valency 3 and axial function:
Standalone flags:
Algorithmic connection for GKM graph with 1 nodes and valency 3

gkm_graph_of_orbifold_toric(v::T; big_torus::Bool=true) where {T <: Union{AbstractStackyFan, AbstractStackyCone}}

Construct the GKM graph of the orbifold stacky fan or variety v.

stacky_weighted_projective_space_fan(x::Vector{Int})

Takes a vector of positive integer weights x and returns:

1. ray_mat: An n x (n+1) matrix where the columns represent the rays.
2. inc_mat: A boolean Incidence matrix (rows = cones, columns = rays), ordered with bigger cones first.