From 4abd1b756076eb8986a378fc13b9d6187b6edb3a Mon Sep 17 00:00:00 2001
From: Matthias Zach <zach@math.uni-hannover.de>
Date: Tue, 26 Sep 2023 16:57:19 +0200
Subject: [PATCH] Partially implement fiber products of closed embeddings.

---
 .../CoveredSchemes/Morphisms/Methods.jl       | 47 +++++++++++++++++++
 .../ProjectiveSchemes/Morphisms/Types.jl      |  1 +
 .../Schemes/ProjectiveSchemes.jl              |  8 ++++
 3 files changed, 56 insertions(+)

diff --git a/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl b/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl
index 24030bd10235..20c3ca8307e6 100644
--- a/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl
@@ -158,3 +158,50 @@ function Base.show(io::IO, ::MIME"text/plain", f::AbsCoveredSchemeMorphism)
     Oscar._show_semi_compact(io, covering_morphism(f))
   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/Types.jl b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Types.jl
index 46645bc9e46c..9dec25c8f7d4 100644
--- a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Types.jl
+++ b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Types.jl
@@ -154,6 +154,7 @@ end
     )
     Y = codomain(f)
     SY = homogeneous_coordinate_ring(Y) 
+    ambient_coordinate_ring(Y) === ambient_coordinate_ring(domain(f)) || error("ambient coordinate rings are not compatible")
     base_ring(I) === SY || error("ideal does not belong to the correct ring")
     @check begin
       pbf = pullback(f)
diff --git a/test/AlgebraicGeometry/Schemes/ProjectiveSchemes.jl b/test/AlgebraicGeometry/Schemes/ProjectiveSchemes.jl
index 3bc767457995..9e2d05374567 100644
--- a/test/AlgebraicGeometry/Schemes/ProjectiveSchemes.jl
+++ b/test/AlgebraicGeometry/Schemes/ProjectiveSchemes.jl
@@ -320,6 +320,14 @@ end
   I = ideal(S, [x^2 + y^2 + z^2])
   X, inc = sub(IP2, I)
   X, inc = sub(IP2, gens(I))
+
+  J = ideal(S, [x+y+z])
+  X2, inc2 = sub(IP2, J)
+  inc_cov = covered_scheme_morphism(inc)
+  inc2_cov = covered_scheme_morphism(inc2)
+  j1, j2 = fiber_product(inc_cov, inc2_cov)
+  @test pushforward(inc_cov)(image_ideal(j2)) == pushforward(inc2_cov)(image_ideal(j1))
+  
   @test X === domain(inc)
   @test IP2 === codomain(inc)
   T = homogeneous_coordinate_ring(X)