diff --git a/experimental/Schemes/src/BlowupMorphism.jl b/experimental/Schemes/src/BlowupMorphism.jl
index 9498e0586c5..9a0aa5a5dff 100644
--- a/experimental/Schemes/src/BlowupMorphism.jl
+++ b/experimental/Schemes/src/BlowupMorphism.jl
@@ -644,18 +644,18 @@ function produce_object_on_affine_chart(I::StrictTransformIdealSheaf, U::AbsAffi
   f_loc = covering_morphism(f)[U]
   V = codomain(f_loc)
   IE_loc = IE(U)
-  @assert isone(ngens(IE_loc)) "ideal sheaf of exceptional locus is not principal"
-  tot = pullback(f_loc)(J(V))
-  #return saturation_with_index(tot, IE_loc)
   # It is usually better to pass to the simplified covering to do the computations
   simp_cov = simplified_covering(X)
   U_simp = first([V for V in patches(simp_cov) if original(V) === U])
   a, b = identification_maps(U_simp)
-  # This used to be the following line. But we don't use the index, so we 
-  # switch to the more performant version
-  # result, _ = saturation_with_index(pullback(a)(tot), pullback(a)(IE_loc))
-  result = _iterative_saturation(pullback(a)(tot), elem_type(OO(U_simp))[pullback(a)(u) for (u, _) in factor(lifted_numerator(first(gens(IE_loc))))])
-  return pullback(b)(result)
+  tot = pullback(f_loc)(J(V))
+  if isone(ngens(IE_loc))
+    result = _iterative_saturation(pullback(a)(tot), elem_type(OO(U_simp))[pullback(a)(u) for (u, _) in factor(lifted_numerator(first(gens(IE_loc))))])
+    return pullback(b)(result)
+  else
+    result, _ = saturation_with_index(pullback(a)(tot), pullback(a)(IE_loc))
+    return result
+  end
 end
 
 @attr Bool function is_prime(I::StrictTransformIdealSheaf)