Other minor improvements for toric varieties

docs/src/ToricVarieties/NormalToricVarieties.md

@@ -40,6 +40,7 @@ NormalToricVariety(P::Polyhedron)
 ### Famous Toric Vareties
 
 ```@docs
+toric_affine_space(d::Int)
 del_pezzo(b::Int)
 hirzebruch_surface(r::Int)
 toric_projective_space(d::Int)

src/ToricVarieties/NormalToricVarieties/attributes.jl

@@ -56,7 +56,8 @@ export euler_characteristic
 @doc Markdown.doc"""
     cox_ring(v::AbstractNormalToricVariety)
 
-Return the Cox ring of the normal toric variety `v`.
+Computes the Cox ring of the normal toric variety `v`.
+Note that [CLS11](@cite) refers to this ring as the "total coordinate ring".
 
 # Examples
 ```jdoctest
@@ -179,7 +180,12 @@ function toric_ideal(antv::AffineNormalToricVariety)
     end
 end
 export toric_ideal
-toric_ideal(ntv::NormalToricVariety) = toric_ideal(AffineNormalToricVariety(ntv))
+
+function toric_ideal(ntv::NormalToricVariety)
+    isaffine(ntv) || error("Cannot construct affine toric variety from non-affine input")    
+    return get_attribute!(() -> toric_ideal(AffineNormalToricVariety(ntv)), ntv, :toric_ideal)
+end
+export toric_ideal
 
 
 ############################
@@ -203,7 +209,7 @@ GrpAb: Z^2
 """
 function character_lattice(v::AbstractNormalToricVariety)
     return get_attribute!(v, :character_lattice) do
-        return abelian_group([0 for i in 1:pm_object(v).FAN_DIM])
+        return free_abelian_group(dim(fan(v)))
     end
 end
 export character_lattice
@@ -225,7 +231,7 @@ GrpAb: Z^3
 """
 function torusinvariant_divisor_group(v::AbstractNormalToricVariety)
     return get_attribute!(v, :torusinvariant_divisor_group) do
-        return abelian_group([0 for i in 1:pm_object(v).N_RAYS])
+        return free_abelian_group(nrays(fan(v)))
     end
 end
 export torusinvariant_divisor_group
@@ -409,8 +415,8 @@ function map_from_cartier_divisor_group_to_torus_invariant_divisor_group(v::Abst
 
         # compute the total map
         mapping_matrix = map_for_scalar_products * map_for_difference_of_elements
-        source = abelian_group(zeros(Int, nrows(mapping_matrix)))
-        target = abelian_group(zeros(Int, ncols(mapping_matrix)))
+        source = free_abelian_group(nrows(mapping_matrix))
+        target = free_abelian_group(ncols(mapping_matrix))
         total_map = hom(source, target, mapping_matrix)
 
         # identify the embedding of the cartier_data_group

src/ToricVarieties/NormalToricVarieties/constructors.jl

@@ -172,7 +172,7 @@ function AffineNormalToricVariety(v::NormalToricVariety)
     # set variety
     variety = AffineNormalToricVariety(pm_object(v), Dict())
 
-    # set properties
+    # set properties of variety
     set_attribute!(variety, :isaffine, true)
     set_attribute!(variety, :iscomplete, false)
     set_attribute!(variety, :isprojective, false)
@@ -187,6 +187,47 @@ end
 # 3: Special constructors
 ######################
 
+@doc Markdown.doc"""
+    toric_affine_space(d::Int)
+
+Constructs the (toric) affine space of dimension `d`.
+
+# Examples
+```jldoctest
+julia> toric_affine_space(2)
+A normal, affine, non-complete, 2-dimensional toric variety
+```
+"""
+function toric_affine_space(d::Int)
+    # construct the cone of the variety
+    m = zeros(Int, d, d)
+    for i in 1:d
+        m[i,i] = 1
+    end
+    C = positive_hull(m)
+
+    # construct the variety
+    fan = PolyhedralFan(C)
+    pmntv = Polymake.fulton.NormalToricVariety(Oscar.pm_object(fan))
+    variety = NormalToricVariety(pmntv, Dict())
+
+    # set known properties
+    set_attribute!(variety, :isaffine, true)
+    set_attribute!(variety, :iscomplete, false)
+    set_attribute!(variety, :isprojective, false)
+    set_attribute!(variety, :isprojective_space, false)
+
+    # set attributes
+    set_attribute!(variety, :fan, fan)
+    set_attribute!(variety, :dim, d)
+    set_attribute!(variety, :dim_of_torusfactor, 0)
+
+    # return the variety
+    return variety
+end
+export toric_affine_space
+
+
 @doc Markdown.doc"""
     toric_projective_space(d::Int)
 
@@ -221,13 +262,13 @@ function toric_projective_space(d::Int)
     set_attribute!(variety, :dim, d)
     set_attribute!(variety, :dim_of_torusfactor, 0)
     set_attribute!(variety, :euler_characteristic, d+1)
-    set_attribute!(variety, :character_lattice, abelian_group([0 for i in 1:d]))
-    set_attribute!(variety, :torusinvariant_divisor_group, abelian_group([0 for i in 1:d+1]))
+    set_attribute!(variety, :character_lattice, free_abelian_group(d))
+    set_attribute!(variety, :torusinvariant_divisor_group, free_abelian_group(d+1))
     set_attribute!(variety, :map_from_cartier_divisor_group_to_torus_invariant_divisor_group, Hecke.identity_map(torusinvariant_divisor_group(variety)))
     set_attribute!(variety, :map_from_cartier_divisor_group_to_picard_group, map_from_weil_divisors_to_class_group(variety))
     set_attribute!(variety, :stanley_reisner_ideal, ideal([prod(Hecke.gens(cox_ring(variety)))]))
     set_attribute!(variety, :irrelevant_ideal, ideal(Hecke.gens(cox_ring(variety))))
-    betti_numbers = [fmpz(1) for i in 0:d]
+    betti_numbers = fill(fmpz(1), d+1)
     set_attribute!(variety, :betti_number, betti_numbers)
 
     # return the variety