Strict transform of an ideal in Cox ring

experimental/Schemes/src/ToricBlowups/methods.jl

@@ -44,3 +44,119 @@ end
 function total_transform(f::AbsBlowupMorphism, II::AbsIdealSheaf)
   return pullback(f, II)
 end
+
+@doc raw"""
+    _minimal_supercone(X::NormalToricVariety, r::AbstractVector{<:IntegerUnion})
+
+Given a ray $r$ inside the support of the simplicial fan $Σ$ of a normal toric variety~$X$, not coinciding with any ray in $Σ(1)$, return the unique cone $σ$ in $Σ$ such that $r$ is in the relative interior of $σ$.
+
+# Examples
+```jldoctest
+julia> X = projective_space(NormalToricVariety, 3)
+Normal toric variety
+
+julia> r = [1, 1, 0]
+2-element Vector{Int64}:
+ 1
+ 1
+
+julia> Oscar._minimal_supercone(X, r)
+Polyhedral cone in ambient dimension 3
+
+julia> rays(c)
+2-element SubObjectIterator{RayVector{QQFieldElem}}:
+ [1, 0, 0]
+ [0, 1, 0]
+```
+"""
+function _minimal_supercone(X::NormalToricVariety, r::AbstractVector{<:Union{IntegerUnion, QQFieldElem}})
+  @assert is_orbifold(X) "Only implemented when fan is simplicial"
+  contained_in_support_of_fan = false
+  mcone = maximal_cones(X)[1]  # initialize `mcone`, fixing its type
+
+  mcones = maximal_cones(X)
+  while true
+    contained_in_subcone = false
+    for c in mcones
+      if r in c
+        contained_in_support_of_fan = true
+        contained_in_subcone = true
+        mcone = c
+      end
+    end
+    @assert contained_in_support_of_fan "Ray not contained in the support of the fan"
+    !contained_in_subcone && return mcone
+    n_rays(mcone) == 1 && return mcone
+
+    # set mcones to be the facets of mcone
+    mcones = [cone(setdiff(collect(rays(mcone)), [ray])) for ray in rays(mcone)]
+  end
+end
+
+function _cox_ring_homomorphism(f::ToricBlowupMorphism)
+  @assert is_normal(codomain(f))
+  @assert is_orbifold(codomain(f)) "Only implemented when fan is simplicial"
+  Y = domain(f)
+  @assert !has_torusfactor(codomain(f)) "Only implemented when there are no torus factors"
+  @assert n_rays(Y) == n_rays(codomain(f)) + 1 "Only implemented when the blowup adds a ray"
+  @assert grading_group(cox_ring(Y)) === class_group(Y)
+  G = class_group(Y)
+  n = number_of_generators(G)
+  new_ray = rays(domain(f))[index_of_new_ray(f)]
+  c = Oscar._minimal_supercone(Y, new_ray)
+  U = affine_normal_toric_variety(c)
+  d = order(class_group(U))
+  # TODO: finish this...
+
+
+  new_var = cox_ring(Y)[index_of_new_ray(f)]
+  M = generator_degrees(cox_ring(Y))
+  @assert M[index_of_new_ray(f)][n] < 0 "Assuming the blowup adds a new column vector to `Oscar.generator_degrees(cox_ring(codomain(f)))`, the entry of that vector corresponding to the new ray should be negative. If not, multiply by -1."
+  ind = Int64(abs(M[index_of_new_ray(f)][n]))
+  for i in 1:n_rays(Y)
+    i == index_of_new_ray(f) || @assert M[i][n] >= 0 "Assuming the blowup adds a new column vector to `Oscar.generator_degrees(cox_ring(codomain(f)))` for which the entry corresponding to the new ray is negative, the entries corresponding to all the other rays should be nonnegative. If not, add integer multiples of the other columns until it is."
+  end
+  if ind == 1
+    S = cox_ring(Y)
+    inj = hom(S, S, gens(S))
+  elseif ind > 1
+    # Change grading so that the negative entry would be -1
+    xs = ZZRingElem[]
+    for i in 1:n-1
+      push!(xs, M[index_of_new_ray(f)][i])
+    end
+    push!(xs, ZZRingElem(-1))
+    M_new = deepcopy(M)
+    M_new[index_of_new_ray(f)] = G(xs)
+    S_ungraded = forget_grading(cox_ring(Y))
+    S = grade(S_ungraded, M_new)[1]
+    inj = hom(cox_ring(Y), S, gens(S))
+  end
+  F = hom(cox_ring(codomain(f)), S, [inj(new_var^(M[i][n])*S[i]) for i in 1:n_rays(Y) if i != index_of_new_ray(f)])
+  return (F, ind)
+end
+
+function strict_transform(f::ToricBlowupMorphism, I::MPolyIdeal)
+  F, ind = _cox_ring_homomorphism(f)
+  Y = domain(f)
+  S = cox_ring(Y)
+  new_var = S[index_of_new_ray(f)]
+  J = saturation(F(I), ideal(new_var))
+  if ind == 1
+    return J
+  elseif ind > 1
+    xs = elem_type(S)[]
+    for gen in gens(J)
+      new_exponent_vectors = Vector{Int64}[]
+      for i in 1:length(terms(gen))
+        exp_vec = exponent_vector(gen, i)
+        bool, result = divides(exp_vec[index_of_new_ray(f)], ind)
+        @assert bool "Given the finite set of generators of `J`, the exponent of `new_var` in every monomial of every member should be divisible by `ind`. If not, first make sure that none of the generators of `J` is divisible by `new_var`."
+        exp_vec[index_of_new_ray(f)] = result
+        push!(new_exponent_vectors, exp_vec)
+      end
+      push!(xs, S(collect(coefficients(gen)), new_exponent_vectors))
+    end
+    return ideal(cox_ring(Y), xs)
+  end
+end