Elliptic fibrations and pushforward of Weil Divisors (#3609)

* Reroute computation of dimensions for localized ideals at rational points.

* Add some type stability.

* Avoid expensive is_subset computation.

* Customize kernel computations.

* Work on pushforward of divisors
* Remove deprecated simplify method for ideal sheaves.

* Rework the fibration projections.

* Rework simplify_light for powers of elements and move types to an extra file.

* Switch to use of simplify_light for hypersurface complements.

* Some cleanup.

* Deprecate morphism_of_projective_schemes.

* Add some hasserts.

* Introduce keyword for small_generating_set.

---------

Co-authored-by: Simon Brandhorst <[email protected]>
Co-authored-by: Johannes Schmitt <[email protected]>

docs/src/AlgebraicGeometry/Schemes/MorphismsOfProjectiveSchemes.md

@@ -22,9 +22,9 @@ out.
 
 ## Constructors
 ```@docs
-    morphism_of_projective_schemes(P::AbsProjectiveScheme, Q::AbsProjectiveScheme, f::Map; check::Bool=true )
-    morphism_of_projective_schemes(P::AbsProjectiveScheme, Q::AbsProjectiveScheme, f::Map, h::SchemeMor; check::Bool=true )
-    morphism_of_projective_schemes(X::AbsProjectiveScheme, Y::AbsProjectiveScheme, a::Vector{<:RingElem})
+    morphism(P::AbsProjectiveScheme, Q::AbsProjectiveScheme, f::Map; check::Bool=true )
+    morphism(P::AbsProjectiveScheme, Q::AbsProjectiveScheme, f::Map, h::SchemeMor; check::Bool=true )
+    morphism(X::AbsProjectiveScheme, Y::AbsProjectiveScheme, a::Vector{<:RingElem})
 ```
 ## Attributes
 As every instance of `Map`, a morphism of projective schemes can be asked for its (co-)domain:
@@ -41,6 +41,6 @@ Moreover, we provide getters for the associated morphisms of rings:
 ```
 ## Methods
 ```@docs
-    covered_scheme_morphism(f::ProjectiveSchemeMor)
+    covered_scheme_morphism(f::AbsProjectiveSchemeMorphism)
 ```
 

docs/src/CommutativeAlgebra/affine_algebras.md

@@ -350,7 +350,7 @@ the type of both `R` and `S`  is a subtype of `Union{MPolyRing{T}, MPolyQuoRing{
 ### Data Associated to Homomorphisms of Affine Algebras
 
 ```@docs
-preimage(F::AffAlgHom, I::MPolyIdeal)
+preimage(F::MPolyAnyMap, I::Ideal)
 kernel(F::AffAlgHom)
 ```
 

experimental/Experimental.jl

@@ -49,7 +49,6 @@ for pkg in exppkgs
   include(joinpath(expdir, pkg, "src", "$pkg.jl"))
 end
 
-
 # Force some structure for `oldexppkgs`
 for pkg in oldexppkgs
   if !isfile(joinpath(expdir, pkg, "$pkg.jl"))

experimental/Schemes/CoveredProjectiveSchemes.jl

@@ -144,8 +144,8 @@ mutable struct ProjectiveGluing{
     (fb, gb) = gluing_morphisms(G)
     (PX, QY) = (codomain(incP), codomain(incQ))
     (PU, QV) = (domain(incP), domain(incQ))
-    (base_scheme(PX) == X && base_scheme(QY) == Y) || error("base gluing is incompatible with the projective schemes")
-    domain(f) == codomain(g) == PU && domain(g) == codomain(f) == QV || error("maps are not compatible")
+    (base_scheme(PX) === X && base_scheme(QY) === Y) || error("base gluing is incompatible with the projective schemes")
+    domain(f) === codomain(g) === PU && domain(g) === codomain(f) === QV || error("maps are not compatible")
     SPU = homogeneous_coordinate_ring(domain(f))
     SQV = homogeneous_coordinate_ring(codomain(f))
     @check begin
@@ -290,10 +290,10 @@ function Base.show(io::IO, ::MIME"text/plain", CPS::CoveredProjectiveScheme)
   print(io, Dedent(), "covered with ", ItemQuantity(n, "projective patch"))
   print(io, Indent())
   l = ndigits(n)
-  for i in 1:n
+  for (i, ppp) in enumerate(pp)
     li = ndigits(i)
     println(io)
-    print(io, " "^(l-li)*"$(i): ", Lowercase(), CPS[patches(base_covering(CPS))[i]])
+    print(io, " "^(l-li)*"$(i): ", Lowercase(), ppp)
   end
   print(io, Dedent())
 end
@@ -1000,7 +1000,7 @@ end
     covered_scheme(P)
   end
   covering_pr = get_attribute(P, :covering_projection_to_base)::CoveringMorphism
-  return CoveredSchemeMorphism(covered_scheme(P), base_scheme(P), covering_pr)
+  return CoveredSchemeMorphism(covered_scheme(P), base_scheme(P), covering_pr; check=false)
 end
 
 

experimental/Schemes/CoveredScheme.jl

@@ -507,7 +507,7 @@ _compose_along_path(X::CoveredScheme, p::Vector{Int}) = _compose_along_path(X, [
              MorphismType<:ClosedEmbedding,
              BaseMorType
             }
-    ff = CoveredSchemeMorphism(X, Y, f)
+    ff = CoveredSchemeMorphism(X, Y, f; check)
     if has_decomposition_info(codomain(f))
       for U in patches(domain(f))
         floc = f[U]

experimental/Schemes/FunctionFields.jl

@@ -75,7 +75,11 @@ represented by
   patch 1: 1
 ```
 """
-@attr VarietyFunctionField function_field(X::AbsCoveredScheme) = VarietyFunctionField(X)
+function function_field(X::AbsCoveredScheme; check::Bool=true)
+  return get_attribute!(X, :function_field) do
+    VarietyFunctionField(X, check=check)
+  end
+end
 
 ########################################################################
 # Methods for VarietyFunctionFieldElem                                 #
@@ -272,13 +276,16 @@ function move_representative(
   iszero(pbb) && error("pullback of denominator is zero")
   # in the next line, A is either a AffineSchemeOpenSubscheme or a PrincipalOpenSubset
   h_generic = generic_fraction(pba, A)//generic_fraction(pbb, A)
+  return h_generic
+
+  # It turned out that the following is too expensive in general
   if domain(f) isa PrincipalOpenSubset
     fac = factor(lifted_numerator(complement_equation(domain(f))))
     p = OO(U)(numerator(h_generic))
     q = OO(U)(denominator(h_generic))
     for (a, e) in fac
       aa = OO(U)(a)
-      k_num, _ = _minimal_power_such_that(aa, x->divides(p, x)[1])
+      k_num, _ = _minimal_power_such_that(aa, x->divides(p, x)[1]) # This division takes ages for big polynomials.
       k_den, _ = _minimal_power_such_that(aa, x->divides(q, x)[1])
       k = minimum([k_num, k_den])
       aa = aa^k