Disjoint gluings

experimental/Schemes/src/LazyGluing.jl

@@ -129,4 +129,3 @@ end
 # Method for compatibility with internal methods of the gluings
 
 Gluing(g::LazyGluing) = Gluing(underlying_gluing(g))
-

src/AlgebraicGeometry/Schemes/CoveredSchemes/Objects/Constructors.jl

@@ -148,6 +148,8 @@
   return X, injections
 end
 
+disjoint_union(X::AbsCoveredScheme, Y::AbsCoveredScheme) = disjoint_union([X,Y])
+
 ### Conversion of an affine scheme into a covered scheme
 CoveredScheme(X::AbsAffineScheme) = CoveredScheme(Covering(X))
 

src/AlgebraicGeometry/Schemes/Covering/Objects/Constructors.jl

@@ -96,6 +96,22 @@ function disjoint_union(C1::Covering, C2::Covering)
   for (X, Y) in keys(gluings(C2))
     add_gluing!(C, C2[X, Y])
   end
+  for X in patches(C1)
+    for Y in patches(C2)
+      add_gluing!(C, disjoint_gluing(X, Y))
+    end
+  end
   return C
 end
 
+# legacy for the test
+function _disjoint_union_raw(C1::Covering, C2::Covering)
+  C = Covering(vcat(patches(C1), patches(C2)))
+  for (X, Y) in keys(gluings(C1))
+    add_gluing!(C, C1[X, Y])
+  end
+  for (X, Y) in keys(gluings(C2))
+    add_gluing!(C, C2[X, Y])
+  end
+  return C
+end

src/AlgebraicGeometry/Schemes/Covering/Objects/Methods.jl

@@ -64,7 +64,9 @@ function update_gluing_graph(C::Covering; all_dense::Bool=false)
       add_edge!(gg, C[X], C[Y])
       add_edge!(gg, C[Y], C[X])
     else
-      (U, V) = gluing_domains(C[X,Y])
+      G = C[X,Y]
+      has_is_disjoint_gluing(G) && is_disjoint_gluing(G) && continue
+      (U, V) = gluing_domains(G)
       is_dense(U) && add_edge!(gg, C[X], C[Y])
       is_dense(V) && add_edge!(gg, C[Y], C[X])
     end
@@ -81,11 +83,13 @@ function pruned_gluing_graph(C::Covering)
   m = length(v)
   gt = Graph{Undirected}(m)
   for (X, Y) in keys(gluings(C))
+    G = C[X,Y]
+    has_is_disjoint_gluing(G) && is_disjoint_gluing(G) && continue
     i = findfirst(==(C[X]),v)
     !isnothing(i) || continue
     j = findfirst(==(C[Y]),v)
     !isnothing(j) || continue
-    (U, V) = gluing_domains(C[X,Y])
+    (U, V) = gluing_domains(G)
     is_dense(U) && add_edge!(gt, i, j)
     is_dense(V) && add_edge!(gt, j, i)
   end

src/AlgebraicGeometry/Schemes/Gluing/Attributes.jl

@@ -51,6 +51,7 @@
   return Ginv
 end
 
+
 ########################################################################
 # Getters for Gluing                                                  #
 ########################################################################
@@ -76,4 +77,15 @@
   return Ginv
 end
 
+########################################################################
+# Further methods                                                      #
+########################################################################
+
+@attr function is_disjoint_gluing(G::AbsGluing)
+  U = gluing_domains(G)[1]
+  return is_empty(U)
+end
 
+function has_is_disjoint_gluing(G::AbsGluing)
+  return has_attribute(G, :is_disjoint_gluing)
+end

src/AlgebraicGeometry/Schemes/Gluing/Constructors.jl

@@ -100,6 +100,21 @@ function Gluing(
   return SimpleGluing(X, Y, f, g, check=check)
 end
 
+########################################################################
+# Disjoint gluing                                                      #
+########################################################################
+
+function disjoint_gluing(X::AbsAffineScheme, Y::AbsAffineScheme)
+  U = EmptyPrincipalOpenSubset(X)
+  V = EmptyPrincipalOpenSubset(Y)
+  f = morphism(U,V, [zero(OO(U)) for i in 1:ngens(OO(V))])
+  g = morphism(V,U, [zero(OO(V)) for i in 1:ngens(OO(U))])
+  S = SimpleGluing(X,Y,f,g)
+  set_attribute!(S, :is_disjoint_gluing=>true)
+  return S
+end
+
+
 ########################################################################
 # Restrictions of Gluings to closed subschemes                        #
 ########################################################################

src/AlgebraicGeometry/Schemes/Gluing/Types.jl

@@ -118,4 +118,3 @@ Concrete instance of an `AbsGluing` for gluings of affine schemes
               }(X, Y, U, V, f, g)
   end
 end
-

src/AlgebraicGeometry/Schemes/PrincipalOpenSubset/Objects/Constructors.jl

@@ -4,3 +4,10 @@
 
 ### Conversion of an arbitrary AbsAffineScheme
 PrincipalOpenSubset(X::AbsAffineScheme) = PrincipalOpenSubset(X, one(OO(X)))
+
+function EmptyPrincipalOpenSubset(X::AbsAffineScheme)
+  Ue = subscheme(X, OO(X)(1))
+  U = PrincipalOpenSubset(X, Ue, OO(X)(0))
+  return U
+end 
+

src/AlgebraicGeometry/Schemes/PrincipalOpenSubset/Objects/Types.jl

@@ -28,8 +28,18 @@
     )
     U = spec(R)
     @check U == hypersurface_complement(X, f) "scheme is not isomorphic to the anticipated open subset"
-    return new{base_ring_type(X), ring_type(U), typeof(X)}(X, U, lifted_numerator.(f))
+    n = lifted_numerator(f)
+    return new{base_ring_type(X), ring_type(U), typeof(X)}(X, U, [n], n)
   end
+
+  function PrincipalOpenSubset(X::AbsAffineScheme, U::AffineScheme, f::RingElem;
+      check::Bool=true
+    )
+    @check ((iszero(f) && is_empty(U)) || U == hypersurface_complement(X, f)) "scheme is not isomorphic to the anticipated open subset"
+    n = lifted_numerator(f)
+    return new{base_ring_type(X), ring_type(U), typeof(X)}(X, U, [n], n)
+  end
+
   function PrincipalOpenSubset(X::AbsAffineScheme, R::Ring, f::Vector{T};
       check::Bool=true
     ) where {T<:RingElem}

src/exports.jl

@@ -759,6 +759,7 @@ export is_cyclic, has_is_cyclic, set_is_cyclic
 export is_degenerate
 export is_dense
 export is_dihedral_group, has_is_dihedral_group, set_is_dihedral_group
+export is_disjoint_gluing
 export is_du_val_singularity
 export is_duplicate_table
 export is_effective

test/AlgebraicGeometry/Schemes/FunctionFields.jl

@@ -47,7 +47,7 @@ end
 
   # Join the resulting schemes in a disjoint union with two
   # components
-  C = disjoint_union(Cs, Ct)
+  C = Oscar._disjoint_union_raw(Cs, Ct)
 
   # Manually glue two (dense) patches of the two components
   X = Cs[3]

test/AlgebraicGeometry/Schemes/Gluing.jl

@@ -159,3 +159,14 @@ end
   end
 end
 
+@testset "disjoint gluings" begin 
+  P2a = projective_space(QQ,2)
+  P2b = projective_space(QQ,2)
+  P2ac = covered_scheme(P2a)
+  P2bc = covered_scheme(P2b)
+  D,_ = disjoint_union(P2ac,P2bc)
+  @test is_disjoint_gluing(D[1][1,4]) 
+  @test !is_disjoint_gluing(D[1][1,2]) 
+  @test 12 == number_of_edges(gluing_graph(D))
+end 
+

test/AlgebraicGeometry/Schemes/IdealSheaves.jl

@@ -24,7 +24,7 @@
 
   # Join the resulting schemes in a disjoint union with two
   # components
-  C = disjoint_union(Cs, Ct)
+  C = Oscar._disjoint_union_raw(Cs, Ct)
 
   # Manually glue two (dense) patches of the two components
   X = Cs[3]

test/AlgebraicGeometry/Schemes/WeilDivisor.jl

@@ -24,7 +24,7 @@
 
   # Join the resulting schemes in a disjoint union with two
   # components
-  C = disjoint_union(Cs, Ct)
+  C = Oscar._disjoint_union_raw(Cs, Ct)
 
   # Manually glue two (dense) patches of the two components
   X = Cs[3]