Implement equality for toric varieties

docs/src/AlgebraicGeometry/ToricVarieties/NormalToricVarieties.md

@@ -18,31 +18,10 @@ the affine and non-affine case:
 
 ## Equality of Normal Toric Varieties
 
-!!! warning
-    Equality `==` of normal toric varieties currently checks equality of
-    memory locations. We recommend using `===`, which always checks
-    equality of memory locations in OSCAR.
-
-To check that the fans considered as sets of cones are equal, you may use the method below:
-
-```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
-```
-
-Polyhedral fans can be stored in Polymake using either rays or
-hyperplanes. In the former case, cones are stored as sets of indices of
-rays, corresponding to polyhedral hulls, while in the latter case, cones
-are stored as sets of indices of hyperplanes, corresponding to
-intersections of hyperplanes. Converting between these two formats can
-be expensive. In the case where the polyhedral fans of two normal toric
-varieties are both stored using rays, the above method `slow_equal` has
-computational complexity $O(n \log n)$, where $n$ is the number of
-cones.
+Equality `==` of normal toric varieties checks equality of the
+polyhedral fans (sets of cones). This computes the rays of both of the
+toric varieties, which can be expensive if they are not already computed.
+Triple-equality `===` always checks equality of memory locations in OSCAR.
 
 
 ## Constructors

src/AlgebraicGeometry/ToricVarieties/NormalToricVarieties/constructors.jl

@@ -181,9 +181,16 @@ 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.")
+function Base.:(==)(X::NormalToricVariety, Y::NormalToricVariety)
+  X === Y && return true
+  ambient_dim(X) == ambient_dim(Y) || return false
+  f_vector(X) == f_vector(Y) || return false
+
+  # p is a permutation such that the i-th ray of X is the p(i)-th ray of Y
+  p = inv(perm(sortperm(rays(X)))) * perm(sortperm(rays(Y)))
+
+  @inline rows(Z) = [row(maximal_cones(IncidenceMatrix, Z), i) for i in 1:n_maximal_cones(Z)]
+  return Set(map(r -> Set(p.(r)), rows(X))) == Set(rows(Y))
 end
 
 function Base.hash(tv::NormalToricVariety, h::UInt)

test/AlgebraicGeometry/ToricVarieties/normal_toric_varieties.jl

@@ -41,6 +41,14 @@
   @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
+    @test p2 != f2
+
+    X = projective_space(NormalToricVariety, 2)
+    X = domain(blow_up(X, [3, 4]))
+    X = domain(blow_up(X, [-2, -3]))
+    Y = weighted_projective_space(NormalToricVariety, [1, 2, 3])
+    Y = domain(blow_up(Y, [-1, -1]))
+    Y = domain(blow_up(Y, [3, 4]))
+    @test X == Y
   end
 end