diff --git a/experimental/Schemes/CoveredProjectiveSchemes.jl b/experimental/Schemes/CoveredProjectiveSchemes.jl
index c804dc36f868..aecd5354731b 100644
--- a/experimental/Schemes/CoveredProjectiveSchemes.jl
+++ b/experimental/Schemes/CoveredProjectiveSchemes.jl
@@ -1756,3 +1756,45 @@ end
 #
 #
 
+function fiber_product(
+    i1::CoveredClosedEmbedding,
+    i2::CoveredClosedEmbedding
+  )
+  X1 = domain(i1)
+  X2 = domain(i2)
+  Y = codomain(i1)
+  Y === codomain(i2) || error("codomains do not coincide")
+  i1_cov = covering_morphism(i1)
+  i2_cov = covering_morphism(i2)
+  codomain(i1_cov) === codomain(i2_cov) || error("case of different coverings in codomain not implemented")
+  cod_cov = codomain(i1_cov)
+  #=
+  cod_ref, ref1, ref2 = common_refinement(codomain(i1_cov), codomain(i2_cov))
+  dom_ref1, i1_res = fiber_product(i1, ref1)
+  dom_ref2, i2_res = fiber_product(i1, ref2)
+  # etc. etc.... This is roughly the generic code to come.
+  =#
+  I1 = image_ideal(i1)
+  pb_I1 = pullback(i2, I1)
+  I2 = image_ideal(i2)
+  pb_I2 = pullback(i1, I2)
+
+  j1 = Oscar.CoveredClosedEmbedding(domain(i2), pb_I1)
+  Z = domain(j1)
+  morphism_dict = IdDict{AbsSpec, ClosedEmbedding}()
+  for U in affine_charts(Z)
+    V2 = codomain(j1[U])
+    W = codomain(i2[V2])
+    V1_candidates = maps_with_given_codomain(i1, W)
+    @assert length(V1_candidates) == 1 "not the correct number of patches found"
+    V1 = domain(first(V1_candidates))
+    x = gens(OO(V1))
+    lift_x = [preimage(pullback(i1[V1]), f) for f in x]
+    pb_x = pullback(i2[V2]).(lift_x)
+    pb_x = pullback(j1[U]).(pb_x)
+    morphism_dict[U] = ClosedEmbedding(SpecMor(U, V1, pb_x, check=false), pb_I2(V1), check=false)
+  end
+  j2_cov = CoveringMorphism(default_covering(Z), domain(i1_cov), morphism_dict, check=false)
+  j2 = Oscar.CoveredClosedEmbedding(Z, X1, j2_cov)
+  return j1, j2
+end
diff --git a/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl b/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl
index 20c3ca8307e6..a6bb844d521c 100644
--- a/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl
@@ -159,49 +159,3 @@ function Base.show(io::IO, ::MIME"text/plain", f::AbsCoveredSchemeMorphism)
   end
 end
 
-function fiber_product(
-    i1::CoveredClosedEmbedding,
-    i2::CoveredClosedEmbedding
-  )
-  X1 = domain(i1)
-  X2 = domain(i2)
-  Y = codomain(i1)
-  Y === codomain(i2) || error("codomains do not coincide")
-  i1_cov = covering_morphism(i1)
-  i2_cov = covering_morphism(i2)
-  codomain(i1_cov) === codomain(i2_cov) || error("case of different coverings in codomain not implemented")
-  cod_cov = codomain(i1_cov)
-  #=
-  cod_ref, ref1, ref2 = common_refinement(codomain(i1_cov), codomain(i2_cov))
-  dom_ref1, i1_res = fiber_product(i1, ref1)
-  dom_ref2, i2_res = fiber_product(i1, ref2)
-  # etc. etc.... This is roughly the generic code to come.
-  =#
-  I1 = image_ideal(i1)
-  pb_I1 = pullback(i2, I1)
-  I2 = image_ideal(i2)
-  pb_I2 = pullback(i1, I2)
-
-  j1 = Oscar.CoveredClosedEmbedding(domain(i2), pb_I1)
-  Z = domain(j1)
-  morphism_dict = IdDict{AbsSpec, ClosedEmbedding}()
-  for U in affine_charts(Z)
-    V2 = codomain(j1[U])
-    W = codomain(i2[V2])
-    V1_candidates = maps_with_given_codomain(i1, W)
-    @assert length(V1_candidates) == 1 "not the correct number of patches found"
-    V1 = domain(first(V1_candidates))
-    x = gens(OO(V1))
-    @show x
-    lift_x = [preimage(pullback(i1[V1]), f) for f in x]
-    @show lift_x
-    pb_x = pullback(i2[V2]).(lift_x)
-    @show pb_x
-    pb_x = pullback(j1[U]).(pb_x)
-    @show pb_x
-    morphism_dict[U] = ClosedEmbedding(SpecMor(U, V1, pb_x, check=false), pb_I2(V1), check=false)
-  end
-  j2_cov = CoveringMorphism(default_covering(Z), domain(i1_cov), morphism_dict, check=false)
-  j2 = Oscar.CoveredClosedEmbedding(Z, X1, j2_cov)
-  return j1, j2
-end
diff --git a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Constructors.jl b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Constructors.jl
index bf55c64354e0..8fbb1d946d02 100644
--- a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Constructors.jl
+++ b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Constructors.jl
@@ -152,6 +152,7 @@ identity_map(P::AbsProjectiveScheme) = ProjectiveSchemeMor(P, P,
 function sub(P::AbsProjectiveScheme, I::Ideal)
   @req base_ring(I) === homogeneous_coordinate_ring(P) "ideal must be defined in the homogeneous coordinate ring of the scheme"
   inc = ProjectiveClosedEmbedding(P, I)
+  set_attribute!(domain(inc), :ambient_space, ambient_space(P))
   return domain(inc), inc
 end
   
@@ -179,3 +180,13 @@ function compose(f::ProjectiveClosedEmbedding, g::ProjectiveClosedEmbedding)
   J = push_If + Ig
   return ProjectiveClosedEmbedding(compose(underlying_morphism(f), underlying_morphism(g)), J, check=false)
 end
+
+function ambient_embedding(X::AbsProjectiveScheme)
+  IP = ambient_space(X)
+  S = homogeneous_coordinate_ring(IP)
+  T = homogeneous_coordinate_ring(X)
+  I = defining_ideal(X)
+  pb = hom(S, T, gens(T))
+  inc_sub = ProjectiveSchemeMor(X, IP, pb, check=false)
+  return ProjectiveClosedEmbedding(inc_sub, I, check=false)
+end