From 7beaaedbcf547d99d22d1de7e06d8e66b6cf85e8 Mon Sep 17 00:00:00 2001
From: Matthias Zach <85350711+HechtiDerLachs@users.noreply.github.com>
Date: Thu, 28 Sep 2023 17:15:25 +0200
Subject: [PATCH] Computeralgebra Rundbrief (#2854)

* Implement unions and intersections of projective schemes,  closed embeddings for projective schemes.
conversion of projective closed embeddings to covered closed embeddings, composition of closed embeddings.

* Partially implement fiber products of closed embeddings.
---
 .../Schemes/CoveredProjectiveSchemes.jl       | 42 +++++++++++++
 .../CoveredSchemes/Morphisms/Methods.jl       |  1 +
 .../ProjectiveSchemes/Morphisms/Attributes.jl | 19 +++++-
 .../Morphisms/Constructors.jl                 | 43 ++++++++++++-
 .../ProjectiveSchemes/Morphisms/Methods.jl    | 41 ++++++++++---
 .../ProjectiveSchemes/Morphisms/Types.jl      | 61 ++++++++++++++++++-
 .../ProjectiveSchemes/Objects/Constructors.jl |  3 +
 .../ProjectiveSchemes/Objects/Methods.jl      | 32 ++++++++++
 .../Schemes/ProjectiveSchemes.jl              | 33 ++++++++++
 9 files changed, 264 insertions(+), 11 deletions(-)

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 a905af3dedb6..6639e88f4096 100644
--- a/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/Methods.jl
@@ -158,3 +158,4 @@ function Base.show(io::IO, ::MIME"text/plain", f::AbsCoveredSchemeMorphism)
     Oscar._show_semi_compact(io, covering_morphism(f))
   end
 end
+
diff --git a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Attributes.jl b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Attributes.jl
index 8a5c71d75c69..5bddd55bee8e 100644
--- a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Attributes.jl
+++ b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Attributes.jl
@@ -1,5 +1,14 @@
+### generic getters
+domain(f::AbsProjectiveSchemeMorphism) = domain(underlying_morphism(f))
+codomain(f::AbsProjectiveSchemeMorphism) = codomain(underlying_morphism(f))
+pullback(f::AbsProjectiveSchemeMorphism) = pullback(underlying_morphism(f))
+base_ring_morphism(f::AbsProjectiveSchemeMorphism) = base_ring_morphism(underlying_morphism(f))
+base_map(f::AbsProjectiveSchemeMorphism) = base_map(underlying_morphism(f))
+map_on_affine_cones(f::AbsProjectiveSchemeMorphism) = map_on_affine_cones(underlying_morphism(f))
 
-### getters
+underlying_morphism(f::AbsProjectiveSchemeMorphism) = error("no underlying morphism for morphisms of type $(typeof(f))")
+
+### getters for the minimal concrete type
 domain(phi::ProjectiveSchemeMor) = phi.domain
 codomain(phi::ProjectiveSchemeMor) = phi.codomain
 @doc raw"""
@@ -158,3 +167,11 @@ function map_on_affine_cones(phi::ProjectiveSchemeMor{<:AbsProjectiveScheme{<:Sp
   return phi.map_on_affine_cones::SpecOpenMor
 end
 
+
+########################################################################
+# Special implementations for closed embeddings
+########################################################################
+
+underlying_morphism(f::ProjectiveClosedEmbedding) = f.underlying_morphism
+image_ideal(f::ProjectiveClosedEmbedding) = f.ideal_of_image 
+
diff --git a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Constructors.jl b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Constructors.jl
index 81c731a65af6..8fbb1d946d02 100644
--- a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Constructors.jl
+++ b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Constructors.jl
@@ -44,7 +44,7 @@ function ProjectiveSchemeMor(
   return ProjectiveSchemeMor(X, Y, hom(P, Q, a, check=false))
 end
 
-function compose(f::ProjectiveSchemeMor, g::ProjectiveSchemeMor)
+function compose(f::AbsProjectiveSchemeMorphism, g::AbsProjectiveSchemeMorphism)
   return ProjectiveSchemeMor(domain(f), codomain(g), compose(pullback(g), pullback(f)))
 end
 
@@ -149,3 +149,44 @@ identity_map(P::AbsProjectiveScheme) = ProjectiveSchemeMor(P, P,
                                                         check=false
                                                        )
 
+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
+  
+function sub(P::AbsProjectiveScheme, f::RingElem)
+  I = ideal(homogeneous_coordinate_ring(P), f)
+  return sub(P, I)
+end
+
+function sub(P::AbsProjectiveScheme, f::Vector{<:RingElem})
+  I = ideal(homogeneous_coordinate_ring(P), f)
+  return sub(P, I)
+end
+
+function compose(f::ProjectiveClosedEmbedding, g::ProjectiveClosedEmbedding)
+  X = domain(f)
+  Y = codomain(f)
+  Y === domain(g) || error("domain and codomain not compatible")
+  Z = codomain(g)
+  pb = compose(pullback(g), pullback(f))
+  Ig = image_ideal(g)
+  SY = homogeneous_coordinate_ring(Y)
+  SZ = homogeneous_coordinate_ring(Z)
+  If = image_ideal(f)
+  push_If = ideal(SZ, [preimage(pullback(g), x) for x in gens(If)])
+  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
diff --git a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Methods.jl b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Methods.jl
index 34d34d2f33c5..e9f50bde7561 100644
--- a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Methods.jl
@@ -1,5 +1,5 @@
 
-function ==(f::ProjectiveSchemeMor, g::ProjectiveSchemeMor) 
+function ==(f::AbsProjectiveSchemeMorphism, g::AbsProjectiveSchemeMorphism) 
   domain(f) === domain(g) || return false
   codomain(f) === codomain(g) || return false
   for s in gens(homogeneous_coordinate_ring(codomain(f)))
@@ -8,15 +8,15 @@ function ==(f::ProjectiveSchemeMor, g::ProjectiveSchemeMor)
   return true
 end
 
-function ==(f::ProjectiveSchemeMor{<:AbsProjectiveScheme{<:Union{<:MPolyRing, <:MPolyQuoRing, <:MPolyLocRing, <:MPolyQuoLocRing}}},
-            g::ProjectiveSchemeMor{<:AbsProjectiveScheme{<:Union{<:MPolyRing, <:MPolyQuoRing, <:MPolyLocRing, <:MPolyQuoLocRing}}})
+function ==(f::AbsProjectiveSchemeMorphism{<:AbsProjectiveScheme{<:Union{<:MPolyRing, <:MPolyQuoRing, <:MPolyLocRing, <:MPolyQuoLocRing}}},
+            g::AbsProjectiveSchemeMorphism{<:AbsProjectiveScheme{<:Union{<:MPolyRing, <:MPolyQuoRing, <:MPolyLocRing, <:MPolyQuoLocRing}}})
   domain(f) === domain(g) || return false
   codomain(f) === codomain(g) || return false
   return map_on_affine_cones(f) == map_on_affine_cones(g)
 end
 
 @doc raw"""
-    covered_scheme_morphism(f::ProjectiveSchemeMor)
+    covered_scheme_morphism(f::AbsProjectiveSchemeMorphism)
 
 Given a morphism of `ProjectiveScheme`s ``f : X → Y``, construct and 
 return the same morphism as a `CoveredSchemeMorphism` of the `covered_scheme`s 
@@ -51,7 +51,7 @@ with default covering
     3: [(x//z), (y//z)]
 ```
 """
-@attr function covered_scheme_morphism(f::ProjectiveSchemeMor)
+@attr function covered_scheme_morphism(f::AbsProjectiveSchemeMorphism)
   PX = domain(f)
   PY = codomain(f)
   SX = ambient_coordinate_ring(PX)
@@ -90,13 +90,38 @@ with default covering
   return ff
 end
 
+@attr CoveredClosedEmbedding function covered_scheme_morphism(f::ProjectiveClosedEmbedding)
+  PX = domain(f)
+  PY = codomain(f)
+  SX = homogeneous_coordinate_ring(PX)
+  SY = homogeneous_coordinate_ring(PY)
+  pbf = pullback(f)
+
+  X = covered_scheme(PX)
+  Y = covered_scheme(PY)
+
+  mor_dict = IdDict{AbsSpec, ClosedEmbedding}()
+  U = affine_charts(X)
+  # TODO: The code below does not run when we have empty affine charts. Adjust accordingly when cleaning up!
+  II = IdealSheaf(PY, image_ideal(f))
+  for i in 1:ngens(SX)
+    U_i = U[i]
+    V_i = affine_charts(Y)[i]
+    mor_dict[U_i] = ClosedEmbedding(SpecMor(U_i, V_i, gens(OO(U_i)), check=false), II(V_i))
+  end
+  f_cov = CoveringMorphism(default_covering(X), default_covering(Y), mor_dict, check=false)
+
+  result = CoveredClosedEmbedding(X, Y, f_cov, check=false)
+  return result
+end
+
 ###############################################################################
 #
 #  Printing
 #
 ###############################################################################
 
-function Base.show(io::IO, f::ProjectiveSchemeMor)
+function Base.show(io::IO, f::AbsProjectiveSchemeMorphism)
   if get(io, :supercompact, false)
     print(io, "Morphism")
   else
@@ -105,7 +130,7 @@ function Base.show(io::IO, f::ProjectiveSchemeMor)
   end
 end
 
-function Base.show(io::IO, ::MIME"text/plain", f::ProjectiveSchemeMor)
+function Base.show(io::IO, ::MIME"text/plain", f::AbsProjectiveSchemeMorphism)
   io = pretty(io)
   X = domain(f)
   Y = codomain(f)
@@ -130,7 +155,7 @@ function Base.show(io::IO, ::MIME"text/plain", f::ProjectiveSchemeMor)
   end
 end
 
-function _show_semi_compact(io::IO, f::ProjectiveSchemeMor)
+function _show_semi_compact(io::IO, f::AbsProjectiveSchemeMorphism)
   io = pretty(io)
   print(io, "Morphism of projective schemes")
   if has_attribute(f, :covered_scheme_morphism)
diff --git a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Types.jl b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Types.jl
index 8400c0c4c1af..9dec25c8f7d4 100644
--- a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Types.jl
+++ b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Types.jl
@@ -1,3 +1,18 @@
+########################################################################
+# Abstract type for morphisms of projective schemes for which the 
+# generic interface is defined.
+########################################################################
+abstract type AbsProjectiveSchemeMorphism{
+    DomainType,
+    CodomainType,
+    SelfType, # The concrete type itself as required by the generic `Map` implementation
+    BaseMorType
+  } <: SchemeMor{DomainType, CodomainType,
+                 SelfType,
+                 BaseMorType
+                }
+end
+
 ########################################################################
 # Morphisms of projective schemes                                      #
 ########################################################################
@@ -28,7 +43,7 @@ space over the same ring with the identity on the base.
     CodomainType<:AbsProjectiveScheme,
     PullbackType<:Map,
     BaseMorType
-  } <: SchemeMor{DomainType, CodomainType,
+  } <: AbsProjectiveSchemeMorphism{DomainType, CodomainType,
                  ProjectiveSchemeMor,
                  BaseMorType
                 }
@@ -106,3 +121,47 @@ space over the same ring with the identity on the base.
   end
 end
 
+@attributes mutable struct ProjectiveClosedEmbedding{
+    DomainType<:AbsProjectiveScheme,
+    CodomainType<:AbsProjectiveScheme,
+    PullbackType<:Map,
+    BaseMorType, 
+    IdealType<:Ideal
+  } <: AbsProjectiveSchemeMorphism{DomainType, CodomainType,
+                 ProjectiveClosedEmbedding,
+                 BaseMorType
+                }
+  underlying_morphism::ProjectiveSchemeMor{DomainType, CodomainType, PullbackType, Nothing}
+  ideal_of_image::IdealType
+
+  function ProjectiveClosedEmbedding(
+      P::DomainType,
+      I::IdealType,
+      check::Bool=true
+    ) where {DomainType<:AbsProjectiveScheme, IdealType<:Ideal}
+    S = homogeneous_coordinate_ring(P)
+    @req base_ring(I) === S "ideal must be defined in the homogeneous coordinate ring of the scheme"
+    T, pr = quo(S, I)
+    Q = ProjectiveScheme(T)
+    f = ProjectiveSchemeMor(Q, P, pr, check=false)
+    return new{typeof(Q), DomainType, typeof(pr), Nothing, IdealType}(f, I)
+  end
+
+  function ProjectiveClosedEmbedding(
+      f::ProjectiveSchemeMor,
+      I::Ideal;
+      check::Bool=true
+    )
+    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)
+      kernel(pbf) == I || error("ideal does not coincide with the kernel of the pullback")
+    end
+    return new{typeof(domain(f)), typeof(Y), typeof(pullback(f)), Nothing, typeof(I)}(f, I)
+  end
+end
+
+
diff --git a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Constructors.jl b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Constructors.jl
index bf1f519912c3..9d86861292d3 100644
--- a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Constructors.jl
+++ b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Constructors.jl
@@ -22,6 +22,7 @@ function subscheme(P::AbsProjectiveScheme, f::RingElem)
   if isdefined(P, :Y) 
     set_base_scheme!(result, base_scheme(P))
   end
+  set_attribute!(result, :ambient_space, ambient_space(P))
   return result
 end
 
@@ -39,6 +40,7 @@ function subscheme(
   if isdefined(P, :Y) 
     set_base_scheme!(result, base_scheme(P))
   end
+  set_attribute!(result, :ambient_space, ambient_space(P))
   return result
 end
 
@@ -52,6 +54,7 @@ function subscheme(P::AbsProjectiveScheme,
   if isdefined(P, :Y) 
     set_base_scheme!(result, base_scheme(P))
   end
+  set_attribute!(result, :ambient_space, ambient_space(P))
   return result
 end
 
diff --git a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Methods.jl b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Methods.jl
index 40fd3ea3acdc..b282d61d8be3 100644
--- a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Methods.jl
+++ b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Methods.jl
@@ -399,3 +399,35 @@ function issubset(X::AbsProjectiveScheme, Y::AbsProjectiveScheme)
   IYsat = saturation(IX, irrelevant_ideal)
   return issubset(IYsat, IXsat)
 end
+
+function Base.intersect(X::AbsProjectiveScheme, Y::AbsProjectiveScheme)
+  return intersect([X, Y])
+end
+
+function Base.intersect(comp::Vector{<:AbsProjectiveScheme})
+  @assert length(comp) > 0 "list of schemes must not be empty"
+  IP = ambient_space(first(comp))
+  @assert all(x->ambient_space(x)===IP, comp[2:end]) "schemes must have the same ambient space"
+  S = homogeneous_coordinate_ring(IP)
+  I = sum([defining_ideal(x) for x in comp])
+  result = subscheme(IP, I)
+  set_attribute!(result, :ambient_space, IP)
+  return result
+end
+
+function Base.union(X::AbsProjectiveScheme, Y::AbsProjectiveScheme)
+  return union([X, Y])
+end
+
+function Base.union(comp::Vector{<:AbsProjectiveScheme})
+  @assert length(comp) > 0 "list of schemes must not be empty"
+  IP = ambient_space(first(comp))
+  @assert all(x->ambient_space(x)===IP, comp[2:end]) "schemes must have the same ambient space"
+  S = homogeneous_coordinate_ring(IP)
+  I = intersect([defining_ideal(x) for x in comp])
+  result = subscheme(IP, I)
+  set_attribute!(result, :ambient_space, IP)
+  return result
+end
+
+
diff --git a/test/AlgebraicGeometry/Schemes/ProjectiveSchemes.jl b/test/AlgebraicGeometry/Schemes/ProjectiveSchemes.jl
index 493a20baf496..9e2d05374567 100644
--- a/test/AlgebraicGeometry/Schemes/ProjectiveSchemes.jl
+++ b/test/AlgebraicGeometry/Schemes/ProjectiveSchemes.jl
@@ -312,3 +312,36 @@ end
   @test issubset(X1, P2)
   @test issubset(X1, X2)
 end
+
+@testset "closed embeddings" begin
+  IP2 = projective_space(QQ, 2)
+  S = homogeneous_coordinate_ring(IP2)
+  (x, y, z) = gens(S)
+  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)
+  Y, inc_Y = sub(X, T[1]*T[2] - T[3]^2)
+  @test domain(inc_Y) === Y
+  @test codomain(inc_Y) === X
+  @test image_ideal(inc_Y) == ideal(T, T[1]*T[2] - T[3]^2)
+  map_on_affine_cones(inc_Y)
+  inc_comp = compose(inc_Y, inc)
+  @test inc_comp isa Oscar.ProjectiveClosedEmbedding
+  phi = hom(homogeneous_coordinate_ring(codomain(inc_comp)), 
+            homogeneous_coordinate_ring(domain(inc_comp)),
+            pullback(inc_comp).(gens(homogeneous_coordinate_ring(codomain(inc_comp))))
+           )
+  K = kernel(phi)
+  @test K == I + ideal(S, S[1]*S[2] - S[3]^2)
+end