From 1971b4d67e7f5599bdac2d8f1a7cc18b879e2e09 Mon Sep 17 00:00:00 2001
From: Erik Paemurru <computeralgebra@runbox.com>
Date: Thu, 26 Sep 2024 17:25:11 +0200
Subject: [PATCH] Strict transform of an ideal in Cox ring

TODO: tests, docstrings, documentation
---
 .../Schemes/src/ToricBlowups/methods.jl       | 57 +++++++++++++++++++
 1 file changed, 57 insertions(+)

diff --git a/experimental/Schemes/src/ToricBlowups/methods.jl b/experimental/Schemes/src/ToricBlowups/methods.jl
index 44d68edb7c4f..9f7324837735 100644
--- a/experimental/Schemes/src/ToricBlowups/methods.jl
+++ b/experimental/Schemes/src/ToricBlowups/methods.jl
@@ -44,3 +44,60 @@ end
 function total_transform(f::AbsBlowupMorphism, II::AbsIdealSheaf)
   return pullback(f, II)
 end
+
+function _cox_ring_homomorphism(f::ToricBlowupMorphism)
+  Y = domain(f)
+  @assert grading_group(cox_ring(Y)) === class_group(Y)
+  G = class_group(Y)
+  n = number_of_generators(G)
+  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