[ToricVarieites] Overhaul equality of normal toric varieties

docs/src/AlgebraicGeometry/ToricVarieties/NormalToricVarieties.md

@@ -15,6 +15,11 @@ the affine and non-affine case:
     The lattice is always assumed to be the standard lattice $\mathbb{Z}^n$.
     Transformations for non-standard lattices will have to be done by the user.
 
+!!! warning
+```@docs
+    Base.:(==)(tv1::NormalToricVariety, tv2::NormalToricVariety)
+```
+
 
 ## Constructors
 

experimental/FTheoryTools/src/AbstractFTheoryModels/methods.jl

@@ -224,7 +224,7 @@ Hypersurface model over a concrete base
 julia> hypersurface_equation(tuned_t) == new_tate_polynomial
 true
 
-julia> base_space(tuned_t) == base_space(t)
+julia> base_space(tuned_t) === base_space(t)
 true
 ```
 """

experimental/FTheoryTools/test/hypersurface_models.jl

@@ -14,10 +14,10 @@ h = hypersurface_model(B3, ambient_space_of_fiber, [D1, D2, D3], p; completeness
 @testset "Attributes and properties of hypersurface models over concrete base space and fiber ambient space P2" begin
   @test parent(hypersurface_equation(h)) == cox_ring(ambient_space(h))
   @test dim(base_space(h)) == dim(B3)
-  @test fiber_ambient_space(h) == ambient_space_of_fiber
+  @test fiber_ambient_space(h) === ambient_space_of_fiber
   @test is_smooth(fiber_ambient_space(h)) == false
   @test [string(g) for g in gens(cox_ring(fiber_ambient_space(h)))] == ["x", "y", "z"]
-  @test toric_variety(calabi_yau_hypersurface(h)) == ambient_space(h)
+  @test toric_variety(calabi_yau_hypersurface(h)) === ambient_space(h)
   @test is_base_space_fully_specified(h) == true
 end
 
@@ -32,9 +32,9 @@ end
   mktempdir() do path
     test_save_load_roundtrip(path, h) do loaded
       @test hypersurface_equation(h) == hypersurface_equation(loaded)
-      @test base_space(h) == base_space(loaded)
-      @test ambient_space(h) == ambient_space(loaded)
-      @test fiber_ambient_space(h) == fiber_ambient_space(loaded)
+      @test base_space(h) === base_space(loaded)
+      @test ambient_space(h) === ambient_space(loaded)
+      @test fiber_ambient_space(h) === fiber_ambient_space(loaded)
       @test is_base_space_fully_specified(h) == is_base_space_fully_specified(loaded)
       @test is_partially_resolved(h) == is_partially_resolved(loaded)
     end
@@ -49,7 +49,7 @@ h2 = tune(h, new_poly)
   @test h == tune(h, hypersurface_equation(h))
   @test hypersurface_equation(h2) == new_poly
   @test hypersurface_equation(h2) != hypersurface_equation(h)
-  @test fiber_ambient_space(h2) == fiber_ambient_space(h)
+  @test fiber_ambient_space(h2) === fiber_ambient_space(h)
 end
 
 B2 = projective_space(NormalToricVariety, 2)
@@ -104,7 +104,7 @@ h4 = hypersurface_model(auxiliary_base_vars, auxiliary_base_grading, d, ambient_
 @testset "Attributes and properties of hypersurface models over concrete base space and fiber ambient space P2" begin
   @test parent(hypersurface_equation(h4)) == coordinate_ring(ambient_space(h4))
   @test dim(base_space(h4)) == d
-  @test fiber_ambient_space(h4) == ambient_space_of_fiber_2
+  @test fiber_ambient_space(h4) === ambient_space_of_fiber_2
   @test is_smooth(fiber_ambient_space(h4)) == false
   @test is_simplicial(fiber_ambient_space(h4)) == true
   @test [string(g) for g in gens(cox_ring(fiber_ambient_space(h4)))] == ["x", "y", "z"]

experimental/Schemes/src/IdealSheaves.jl

@@ -466,7 +466,11 @@ end
 function ==(I::AbsIdealSheaf, J::AbsIdealSheaf)
   I === J && return true
   X = space(I)
-  X == space(J) || return false
+  if X isa NormalToricVariety
+    X === space(J) || return false
+  else
+    X == space(J) || return false
+  end
   for U in basic_patches(default_covering(X))
     is_subset(I(U), J(U)) && is_subset(J(U), I(U)) || return false
   end

experimental/Schemes/src/ToricBlowups/constructors.jl

@@ -234,7 +234,7 @@ Ideal generated by
 
 julia> b2P3 = blow_up(P3, I2);
 
-julia> codomain(b2P3) == P3
+julia> codomain(b2P3) === P3
 true
 ```
 """

src/AlgebraicGeometry/ToricVarieties/NormalToricVarieties/constructors.jl

@@ -178,6 +178,62 @@ end
 
 
 
+######################
+# Equality
+######################
+
+@doc raw"""
+    Base.:(==)(tv1::NormalToricVariety, tv2::NormalToricVariety)
+
+In OSCAR, the `==` operator is reserved to check if two normal toric varieties are identical,
+meaning their underlying polyhedral fans are the same. However, this check is computationally
+challenging due to several reasons:
+
+- The ray generators might be scaled.
+- The ray generators might be stored in different orders.
+- The maximal (polyhedral) cones of the polyhedral fan might be stored in different orders.
+- If we fall back on polyhedral fan equality, lineality of the cones must also be considered.
+
+To avoid this computational bottleneck, we have designed OSCAR to *not* implement the `==`
+method for toric varieties. Instead, using `==` will always raise an error, as shown below.
+
+However, triple equality `===` is supported. This returns `true` if the memory locations of
+the two objects in question are identical, and `false` otherwise. We recommend using `===`
+instead of `==`.
+
+# Examples
+```jldoctest
+julia> p2 = projective_space(NormalToricVariety, 2)
+Normal toric variety
+
+julia> f2 = hirzebruch_surface(NormalToricVariety, 2)
+Normal toric variety
+
+julia> p2 === f2
+false
+
+julia> p2 === p2
+true
+
+julia> function capture_error_message()
+         try
+           p2 == f2
+         catch e
+           println(e.msg)
+         end
+       end
+capture_error_message (generic function with 1 method)
+
+julia> capture_error_message()
+Equality of normal toric varieties is not implemented, as computationally very demanding. More details in the documentation.
+```
+"""
+function Base.:(==)(tv1::NormalToricVariety, tv2::NormalToricVariety)
+  error("Equality of normal toric varieties is not implemented, as computationally very demanding. More details in the documentation.")
+end
+
+
+
 ######################
 # Display
 ######################

src/AlgebraicGeometry/ToricVarieties/Proj/constructors.jl

@@ -42,7 +42,7 @@ function _projectivization_and_total_space(is_proj::Bool, E::Vector{T}) where T
 
   is_proj && length(E) == 1 && return v
 
-  @req all(i -> toric_variety(E[i]) == v, eachindex(E)) "The divisors are defined on different toric varieties."
+  @req all(i -> toric_variety(E[i]) === v, eachindex(E)) "The divisors are defined on different toric varieties."
 
   if is_proj
     PF_fiber = normal_fan(simplex(length(E) - 1))

src/AlgebraicGeometry/ToricVarieties/Subvarieties/attributes.jl

@@ -21,7 +21,7 @@ julia> (t1, x1, t2, x2) = gens(cox_ring(f2));
 julia> c = closed_subvariety_of_toric_variety(f2, [t1])
 Closed subvariety of a normal toric variety
 
-julia> toric_variety(c) == f2
+julia> toric_variety(c) === f2
 true
 ```
 """

src/AlgebraicGeometry/ToricVarieties/ToricMorphisms/constructors.jl

@@ -190,7 +190,7 @@ end
 ####################################################
 
 function Base.:(==)(tm1::ToricMorphism, tm2::ToricMorphism)
-    return domain(tm1) == domain(tm2) && codomain(tm1) == codomain(tm2) && grid_morphism(tm1) == grid_morphism(tm2)
+    return domain(tm1) === domain(tm2) && codomain(tm1) === codomain(tm2) && grid_morphism(tm1) == grid_morphism(tm2)
 end
 
 function Base.hash(tm::ToricMorphism, h::UInt)

test/AlgebraicGeometry/ToricVarieties/toric_blowdowns.jl

@@ -51,7 +51,7 @@
     @test grid_morphism(bl + bl) == grid_morphism(2 * bl)
     @test grid_morphism(bl - bl) == grid_morphism(0 * bl)
     @test grid_morphism(bl * toric_identity_morphism(codomain(bl))) == grid_morphism(bl)
-    @test bl != bl2
+    @test bl !== bl2
   end
 
   S = cox_ring(P2)

test/AlgebraicGeometry/ToricVarieties/toric_morphisms.jl

@@ -22,7 +22,7 @@
   end
 
   @testset "Basic attributes of toric morphisms" begin
-    @test codomain(tm1) == domain(tm2)
+    @test codomain(tm1) === domain(tm2)
     @test matrix(grid_morphism(tm2)) == matrix(ZZ, [[1, 0], [0, 1]])
     @test morphism_on_torusinvariant_weil_divisor_group(tm2).map == identity_matrix(ZZ, 4)
     @test morphism_on_torusinvariant_cartier_divisor_group(tm2).map == identity_matrix(ZZ, 4)