Fast strict transform under toric blowups

experimental/FTheoryTools/src/AbstractFTheoryModels/methods.jl

@@ -157,15 +157,15 @@ function blow_up(m::AbstractFTheoryModel, I::AbsIdealSheaf; coordinate_name::Str
   # Construct the new model
   if m isa GlobalTateModel
     if isdefined(m, :tate_polynomial) && new_ambient_space isa NormalToricVariety
-      new_tate_polynomial = _strict_transform(bd, tate_polynomial(m))
+      new_tate_polynomial = cox_ring_module_homomorphism(bd, tate_polynomial(m))
       model = GlobalTateModel(explicit_model_sections(m), defining_section_parametrization(m), new_tate_polynomial, base_space(m), new_ambient_space)
     else
       new_tate_ideal_sheaf = _strict_transform(bd, tate_ideal_sheaf(m))
       model = GlobalTateModel(explicit_model_sections(m), defining_section_parametrization(m), new_tate_ideal_sheaf, base_space(m), new_ambient_space)
     end
   else
     if isdefined(m, :weierstrass_polynomial) && new_ambient_space isa NormalToricVariety
-      new_weierstrass_polynomial = _strict_transform(bd, weierstrass_polynomial(m))
+      new_weierstrass_polynomial = cox_ring_module_homomorphism(bd, weierstrass_polynomial(m))
       model = WeierstrassModel(explicit_model_sections(m), defining_section_parametrization(m), new_weierstrass_polynomial, base_space(m), new_ambient_space)
     else
       new_weierstrass_ideal_sheaf = _strict_transform(bd, weierstrass_ideal_sheaf(m))

experimental/FTheoryTools/src/auxiliary.jl

@@ -463,43 +463,6 @@
 _strict_transform(bd::AbsCoveredSchemeMorphism, II::AbsIdealSheaf) = strict_transform(bd, II)
 
 function _strict_transform(bd::ToricBlowupMorphism, II::ToricIdealSheafFromCoxRingIdeal)
-  center_ideal = ideal_in_cox_ring(center_unnormalized(bd))
-  if (ngens(ideal_in_cox_ring(II)) != 1) || (all(in(gens(base_ring(center_ideal))), gens(center_ideal)) == false)
-    return strict_transform(bd, II)
-  end
-  S = cox_ring(domain(bd))
-  _e = gen(S, index_of_exceptional_ray(bd))
-  images = MPolyRingElem[]
-  g_list = gens(S)
-  g_center = [string(k) for k in symbols(ideal_in_cox_ring(center_unnormalized(bd)))]
-  for v in g_list
-    v == _e && continue
-    if string(v) in g_center
-      push!(images, v * _e)
-    else
-      push!(images, v)
-    end
-  end
-  ring_map = hom(cox_ring(codomain(bd)), S, images)
-  total_transform = ring_map(ideal_in_cox_ring(II))
-  exceptional_ideal = total_transform + ideal([_e])
-  strict_transform, exceptional_factor = saturation_with_index(total_transform, exceptional_ideal)
-  return ideal_sheaf(domain(bd), strict_transform)
-end
-
-function _strict_transform(bd::ToricBlowupMorphism, tate_poly::MPolyRingElem)
-  S = cox_ring(domain(bd))
-  _e = gen(S, index_of_exceptional_ray(bd))
-  g_list = string.(symbols(S))
-  g_center = [string(k) for k in gens(ideal_in_cox_ring(center_unnormalized(bd)))]
-  position_of_center_variables = [findfirst(==(g), g_list) for g in g_center]
-  pos_of_e = findfirst(==(string(_e)), g_list)
-  C = MPolyBuildCtx(S)
-  for m in terms(tate_poly)
-    exps = collect(exponents(m))[1]
-    insert!(exps, pos_of_e, sum(exps[position_of_center_variables]))
-    push_term!(C, collect(coefficients(m))[1], exps)
-  end
-  f = finish(C)
-  return remove(f, _e)[2]
+  transf = strict_transform(bd, ideal_in_cox_ring(II))
+  return ideal_sheaf(domain(bd), transf)
 end

experimental/Schemes/src/ToricBlowups/methods.jl

@@ -193,20 +193,25 @@ function cox_ring_module_homomorphism(f::ToricBlowupMorphism, g::MPolyDecRingEle
   @req parent(g) === cox_ring(codomain(f)) "g must be an element of the Cox ring of the codomain of f"
   R = cox_ring(codomain(f))
   S = cox_ring(domain(f))
-
-  # Assuming the i-th variable of `R` is the i-th variable of `S`
-  S_vars = elem_type(S)[S[i] for i in 1:nvars(S)]
-
-  nvars(R) == nvars(S) && return hom(R, S, S_vars)(g)
+  nvars(R) == nvars(S) && return evaluate(g, gens(S))
   ps = minimal_supercone_coordinates_of_exceptional_ray(f)
+  if lcm(denominator.(ps)) == 1
+    ps_fast = Vector{Int64}(numerator.(ps))
+  else
+    ps_fast = Vector{Rational{Int64}}(ps)
+  end
   exceptional_var = S[index_of_exceptional_ray(f)]
-  make_S_term(c, exps) = c*prod(map(^, S_vars, exps))
-  coeff_exps = collect(zip(coefficients(g), exponents(g)))
-  h = S(0)
-  for (c, exps) in coeff_exps
-    exceptional_exp = ceil(ZZRingElem, sum(ps.*exps))
-    h += exceptional_var^exceptional_exp * make_S_term(c, exps)
+  C = MPolyBuildCtx(S)
+
+  # Core loop
+  for m in terms(g)
+    exps = first(exponents(m))
+    exceptional_exp = ceil(Int64, sum(ps_fast.*exps))
+    exps = [exps; exceptional_exp]
+    push_term!(C, first(coefficients(m)), exps)
   end
+
+  h = finish(C)
   return h
 end
 

experimental/Schemes/test/runtests.jl

@@ -97,4 +97,23 @@ end
   J_strict = strict_transform(f, I)
   @test J_strict == ideal(S, [x_ + y_*z_^2*u])
   @test ideal_sheaf(Y, J_strict) == strict_transform(f, ideal_sheaf(X, I))
+
+  # Quadratic cone, blowup along non-Cartier divisor
+  ray_generators = [[0, 0, 1], [1, 0, 1], [1, 1, 1], [0, 1, 1]]
+  max_cones = IncidenceMatrix([[1, 2, 3, 4]])
+  PF = polyhedral_fan(max_cones, ray_generators)
+  X = normal_toric_variety(PF)
+  set_coordinate_names(X, ["x", "y", "z", "w"])
+  R = cox_ring(X)
+  x, y, z, w = gens(R)
+  f = blow_up_along_minimal_supercone_coordinates(X, [1, 0, 0, 0]; coordinate_name="u")
+  Y = domain(f)
+  S = cox_ring(Y)
+  x_, y_, z_, w_ = gens(S)
+
+  ## Subscheme is a curve
+  I = ideal(R, [x + z^2*y])
+  J_strict = strict_transform(f, I)
+  @test J_strict == ideal(S, [x_ + y_*z_^2])
+  @test ideal_sheaf(Y, J_strict) == strict_transform(f, ideal_sheaf(X, I))
 end