Implement equality for toric varieties (#4509)

[ToricVarieties] Implement equality for toric varieties

---------
Co-authored-by: Benjamin Lorenz <[email protected]>

docs/src/AlgebraicGeometry/ToricVarieties/NormalToricVarieties.md

@@ -18,31 +18,13 @@ 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
+corresponding polyhedral fans as sets of cones.
+This computes the rays of both of the toric varieties, which can be
+expensive if they are not already computed, meaning if
+`"RAYS" in Polymake.list_properties(Oscar.pm_object(polyhedral_fan(X)))`
+is false for one of the varieties.
+Triple-equality `===` always checks equality of memory locations in OSCAR.
 
 
 ## Constructors

src/AlgebraicGeometry/ToricVarieties/NormalToricVarieties/constructors.jl

@@ -181,16 +181,74 @@ 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
+@doc raw"""
+    (==)(X::NormalToricVariety, Y::NormalToricVariety) -> Bool
+
+Checks equality of the polyhedral fans as sets of cones.
+
+# Examples
+```jldoctest
+julia> H = hirzebruch_surface(NormalToricVariety, 0)
+Normal toric variety
 
-function Base.hash(tv::NormalToricVariety, h::UInt)
-  return hash(objectid(tv), h)
+julia> P1 = projective_space(NormalToricVariety, 1)
+Normal toric variety
+
+julia> H == P1 * P1
+true
+```
+"""
+function Base.:(==)(X::NormalToricVariety, Y::NormalToricVariety)
+  X === Y && return true
+  ambient_dim(X) == ambient_dim(Y) || return false
+  n_rays(X) == n_rays(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)))
+
+  for i in 1:n_rays(X)
+    rays(X)[i] == rays(Y)[p(i)] || return false
+  end
+  @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
 
+@doc raw"""
+    _id(X::NormalToricVariety)
+    -> Tuple{Vector{Vector{QQFieldElem}}, Vector{Vector{Int64}}}
+
+Given a toric variety `X`, returns a pair `Oscar._id(X)` with the
+following property: two toric varieties `X` and `Y` have equal
+polyhedral fans, taken as sets of cones, if and only if
+`Oscar._id(X) == Oscar._id(Y)`.
 
+# Examples
+```jldoctest
+julia> H = hirzebruch_surface(NormalToricVariety, 0)
+Normal toric variety
+
+julia> P1 = projective_space(NormalToricVariety, 1)
+Normal toric variety
+
+julia> Oscar._id(H) == Oscar._id(P1 * P1)
+true
+```
+"""
+function _id(X::NormalToricVariety)
+  p = inv(perm(sortperm(rays(X))))
+  sorted_rays = Vector.(permuted(collect(rays(X)), p))
+  @inline rows(Z) = [
+    row(maximal_cones(IncidenceMatrix, Z), i) for i in 1:n_maximal_cones(Z)
+  ]
+  sorted_maximal_cones = sort(map(r -> sort(Vector(p.(r))), rows(X)))
+  return (sorted_rays, sorted_maximal_cones)
+end
+
+function Base.hash(X::NormalToricVariety, h::UInt)
+  return hash(_id(X), h)
+end
 
 ######################
 # Display

test/AlgebraicGeometry/ToricVarieties/normal_toric_varieties.jl

@@ -41,6 +41,43 @@
   @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
+
+    Z = projective_space(NormalToricVariety, 2)
+    X = domain(blow_up(Z, [1, 1]))
+    Y = domain(blow_up(Z, [1, 2]))
+    @test X != Y
+
+    H = hirzebruch_surface(NormalToricVariety, 0)
+    P1 = projective_space(NormalToricVariety, 1)
+    ray_generators = [[1, 1], [1, 2]]
+    max_cones = incidence_matrix([[1, 2]])
+    X = normal_toric_variety(max_cones, ray_generators)
+    @test length(Set([H, P1 * P1, X])) == 2
+
+    @testset "Speed test hash (at most 0.5 seconds)" begin
+      success = false
+      ntv5 = normal_toric_variety(polarize(polyhedron(Polymake.polytope.rand_sphere(5, 60; seed=42))))
+      hash(ntv5)
+      for i in 1:5
+        stats = @timed hash(ntv5)
+        duration = stats.time - stats.gctime
+        if duration < 0.5
+          success = true
+          break
+        else
+          @warn "Hash took $duration > 0.5 seconds (i=$i)"
+        end
+      end
+      @test success == true
+    end
   end
 end