[ToricVarieties] Overhaul equality of normal toric varieties

docs/src/AlgebraicGeometry/ToricVarieties/NormalToricVarieties.md

@@ -16,6 +16,46 @@ the affine and non-affine case:
     Transformations for non-standard lattices will have to be done by the user.
 
 
+## Equality of Normal Toric Varieties
+
+!!! warning
+    Equality of normal toric varieties is computationally very demanding.
+    We have therefore made special design decisions for the `==` method.
+
+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 specially designed the `==` method.
+It checks if the memory locations of the two objects in question are identical. If 
+so, our `==` method returns `true`. Otherwise, it raise an error.
+
+Note that triple equality `===` (i.e. the check of equal the memory locations) is always
+supported for normal toric varieties. We recommend using it.
+
+However, if you truly need to check for two normal toric varieties to be mathematically
+identical, then you will need to add a custom method. This method could look as follows:
+
+```julia
+function slow_equal(tv1::NormalToricVariety, tv2::NormalToricVariety)
+  tv1 === tv2 && return true
+  ambient_dim(tv1) == ambient_dim(tv2) || return false
+  f_vector(tv1) == f_vector(tv2) || return false
+  return Set(maximal_cones(tv1)) == Set(maximal_cones(tv2))
+end
+```
+
+Please note that this method `slow_equal` is not performant, that we currently (summer 2024)
+have no intentions in adding this function to OSCAR nor to make improvements to its performance.
+Rather, expect this method to be slow, potentially painfully so.
+
+
+
 ## Constructors
 
 ### Affine Toric Varieties

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

src/AlgebraicGeometry/ToricVarieties/NormalToricVarieties/constructors.jl

@@ -178,6 +178,17 @@ end
 
 
 
+######################
+# Equality
+######################
+
+function Base.:(==)(tv1::NormalToricVariety, tv2::NormalToricVariety)
+  tv1 === tv2 && return true
+  error("Equality of normal toric varieties is 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/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/normal_toric_varieties.jl

@@ -34,4 +34,13 @@
     ntv5 = normal_toric_variety(polarize(polyhedron(Polymake.polytope.rand_sphere(5, 60; seed=42))))
     @test ngens(stanley_reisner_ideal(ntv5)) == 1648
   end
+
+  p2 = projective_space(NormalToricVariety, 2)
+  f2 = hirzebruch_surface(NormalToricVariety, 2)
+
+  @testset "Equality of normal toric varieties" begin
+    @test (p2 === f2) == false
+    @test p2 === p2
+    @test_throws ErrorException("Equality of normal toric varieties is computationally very demanding. More details in the documentation.") p2 == f2
+  end
 end

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)

test/Serialization/ToricGeometry.jl

@@ -16,7 +16,7 @@
       test_save_load_roundtrip(path, vtd) do loaded
         @test coefficients(td0) == coefficients(loaded[1])
         @test coefficients(td1) == coefficients(loaded[2])
-        @test toric_variety(loaded[1]) == toric_variety(loaded[2])
+        @test toric_variety(loaded[1]) === toric_variety(loaded[2])
       end
     end
 
@@ -28,7 +28,7 @@
       test_save_load_roundtrip(path, vtd) do loaded
         @test divisor_class(tdc0) == divisor_class(loaded[1])
         @test divisor_class(tdc1) == divisor_class(loaded[2])
-        @test toric_variety(loaded[1]) == toric_variety(loaded[2])
+        @test toric_variety(loaded[1]) === toric_variety(loaded[2])
       end
     end