[ToricSchemes] Add strict and total transform on polynomials, fix FTh…

…eoryTools bug (#4503)

docs/src/AlgebraicGeometry/ToricVarieties/BlowupMorphisms.md

@@ -91,7 +91,7 @@ We can compute the total and strict transforms of homogeneous ideals in Cox ring
 ```@docs
 strict_transform(f::ToricBlowupMorphism, I::MPolyIdeal)
 strict_transform_with_index(f::ToricBlowupMorphism, I::MPolyIdeal)
-total_transform(f::ToricBlowupMorphism, I::MPolyIdeal)
+total_transform(f::ToricBlowupMorphism, I::Union{MPolyIdeal, MPolyDecRingElem})
 ```
 The above functions are implemented using a $\mathbb{C}$-module
 homomorphism between the Cox rings, considered as $\mathbb{C}$-modules,

experimental/FTheoryTools/src/AbstractFTheoryModels/methods.jl

@@ -157,7 +157,8 @@ 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 = cox_ring_module_homomorphism(bd, tate_polynomial(m))
+      f = tate_polynomial(m)
+      new_tate_polynomial = strict_transform(bd, f)
       model = GlobalTateModel(explicit_model_sections(m), defining_section_parametrization(m), new_tate_polynomial, base_space(m), new_ambient_space)
     else
       if bd isa ToricBlowupMorphism
@@ -169,7 +170,8 @@ function blow_up(m::AbstractFTheoryModel, I::AbsIdealSheaf; coordinate_name::Str
     end
   else
     if isdefined(m, :weierstrass_polynomial) && new_ambient_space isa NormalToricVariety
-      new_weierstrass_polynomial = cox_ring_module_homomorphism(bd, weierstrass_polynomial(m))
+      f = weierstrass_polynomial(m)
+      new_weierstrass_polynomial = strict_transform(bd, f)
       model = WeierstrassModel(explicit_model_sections(m), defining_section_parametrization(m), new_weierstrass_polynomial, base_space(m), new_ambient_space)
     else
       if bd isa ToricBlowupMorphism

experimental/Schemes/src/ToricBlowups/methods.jl

@@ -1,20 +1,27 @@
 @doc raw"""
     strict_transform(f::ToricBlowupMorphism, I::MPolyIdeal) -> MPolyIdeal
+    strict_transform(f::ToricBlowupMorphism, g::MPolyDecRingElem) -> MPolyDecRingElem
 
 Let $f\colon Y \to X$ be the toric blowup corresponding to a star
 subdivision along a ray. Let $R$ and $S$ be the Cox rings of $X$ and
 $Y$, respectively.
 Here "strict transform" means the "scheme-theoretic closure of the
 complement of the exceptional divisor in the scheme-theoretic inverse
 image".
-This function returns a homogeneous ideal in $S$ corresponding to the
-strict transform under $f$ of the closed subscheme of $X$ defined by the
-homogeneous ideal $I$ in $R$.
+This function returns a homogeneous ideal (or a polynomial generating a
+principal ideal) in $S$ corresponding to the strict transform under $f$
+of the closed subscheme of $X$ defined by the homogeneous ideal $I$ (or
+by the homogeneous principal ideal generated by $g$) in $R$.
 
 This is implemented under the following assumptions:
   * the variety $X$ has no torus factors (meaning the rays span
     $N_{\mathbb{R}}$).
 
+!!! note
+    Computing `strict_transform(f, g)` is faster than computing
+    `strict_transform(f, ideal(g))[1]` since it avoids creating an
+    ideal.
+
 # Examples
 ```jldoctest
 julia> X = affine_space(NormalToricVariety, 2)
@@ -49,22 +56,37 @@ function strict_transform(f::ToricBlowupMorphism, I::MPolyIdeal)
   return saturation(J, ideal(S, exceptional_var))
 end
 
+function strict_transform(f::ToricBlowupMorphism, g::MPolyDecRingElem)
+  X = codomain(f)
+  @req !has_torusfactor(X) "Only implemented when there are no torus factors"
+  S = cox_ring(domain(f))
+  exceptional_var = S[index_of_exceptional_ray(f)]
+  h = cox_ring_module_homomorphism(f, g)
+  return remove(h, exceptional_var)[2]
+end
+
 @doc raw"""
     total_transform(f::ToricBlowupMorphism, I::MPolyIdeal) -> MPolyIdeal
+    total_transform(f::ToricBlowupMorphism, g::MPolyDecRingElem) -> MPolyDecRingElem
 
 Let $f\colon Y \to X$ be the toric blowup corresponding to a star
 subdivision along a ray. Let $R$ and $S$ be the Cox rings of $X$ and
 $Y$, respectively.
-This function returns a homogeneous ideal in $S$ corresponding to the
-total transform (meaning the scheme-theoretic inverse image) under $f$
-of the closed subscheme of $X$ defined by the homogeneous ideal $I$ in
-$R$.
+Here "total transform" means the "scheme-theoretic inverse image".
+This function returns a homogeneous ideal (or a polynomial generating a
+principal ideal) in $S$ corresponding to the total transform under $f$
+of the closed subscheme of $X$ defined by the homogeneous ideal $I$ (or
+by the homogeneous principal ideal generated by $g$) in $R$.
 
 This is implemented under the following assumptions:
   * the variety $X$ has no torus factors (meaning the rays span
     $N_{\mathbb{R}}$), and
   * the variety $X$ is an orbifold (meaning its fan is simplicial).
 
+!!! note
+    Computing `total_transform(f, g)` is faster than computing
+    `total_transform(f, ideal(g))[1]` since it avoids creating an ideal.
+
 # Examples
 ```jldoctest
 julia> X = affine_space(NormalToricVariety, 2)
@@ -90,7 +112,7 @@ Ideal generated by
   x1*e^2 + x2*e^3
 ```
 """
-function total_transform(f::ToricBlowupMorphism, I::MPolyIdeal)
+function total_transform(f::ToricBlowupMorphism, I::Union{MPolyIdeal, MPolyDecRingElem})
   X = codomain(f)
   @req !has_torusfactor(X) "Only implemented when there are no torus factors"
   @req is_orbifold(X) "Only implemented when the fan is simplicial"
@@ -99,10 +121,11 @@ end
 
 @doc raw"""
     strict_transform_with_index(f::ToricBlowupMorphism, I::MPolyIdeal) -> (MPolyIdeal, Int)
+    strict_transform_with_index(f::ToricBlowupMorphism, g::MPolyDecRingElem) -> (MPolyDecRingElem, Int)
 
-Returns the pair $(J, k)$, where $J$ coincides with `strict_transform(f, I)`
-and where $k$ is the multiplicity of the total transform along the
-exceptional prime divisor.
+Returns the pair $(J, k)$, where $J$ coincides with `strict_transform(f,
+I)` (or with `strict_transform(f, g)`) and where $k$ is the multiplicity
+of the total transform along the exceptional prime divisor.
 
 This is implemented under the following assumptions:
   * the variety $X$ has no torus factors (meaning the rays span
@@ -113,6 +136,11 @@ This is implemented under the following assumptions:
     If the multiplicity $k$ is not needed, we recommend to use
     `strict_transform(f, I)` which is typically faster.
 
+!!! note
+    Computing `strict_transform_with_index(f, g)` is faster than
+    computing `strict_transform_with_index(f, ideal(g))` since it avoids
+    creating an ideal.
+
 # Examples
 ```jldoctest
 julia> X = affine_space(NormalToricVariety, 2)
@@ -147,6 +175,17 @@ function strict_transform_with_index(f::ToricBlowupMorphism, I::MPolyIdeal)
   return saturation_with_index(J, ideal(exceptional_var))
 end
 
+function strict_transform_with_index(f::ToricBlowupMorphism, g::MPolyDecRingElem)
+  X = codomain(f)
+  @req !has_torusfactor(X) "Only implemented when there are no torus factors"
+  @req is_smooth(X) "Only implemented when the fan is smooth"
+  S = cox_ring(domain(f))
+  exceptional_var = S[index_of_exceptional_ray(f)]
+  h = cox_ring_module_homomorphism(f, g)
+  pair = remove(h, exceptional_var)
+  return pair[2], pair[1]
+end
+
 @doc raw"""
     cox_ring_module_homomorphism(f::ToricBlowupMorphism, g::MPolyDecRingElem) -> MPolyDecRingElem
     cox_ring_module_homomorphism(f::ToricBlowupMorphism, I::MPolyIdeal) -> MPolyIdeal

experimental/Schemes/test/runtests.jl

@@ -44,6 +44,10 @@ end
   J, k = strict_transform_with_index(f, I)
   @test J == ideal(S, [x_^3 + y_^3*u^3])
   @test k == 6
+  g = x^3 + y^3
+  h, k_poly = strict_transform_with_index(f, g)
+  @test ideal(h) == J
+  @test k == k_poly
 
   # 1/2(1, 1) quotient singularity, (1/2, 1/2)-blowup
   ray_generators = [[2, -1], [0, 1]]
@@ -69,6 +73,11 @@ end
   J_total = total_transform(f, I)
   @test J_total == ideal(S, [x_*u + y_^3*u^2])
   @test ideal_sheaf(Y, J_total) == total_transform(f, ideal_sheaf(X, I))
+  g = x + y^3
+  h_strict = strict_transform(f, g)
+  @test ideal(h_strict) == J_strict
+  h_total = total_transform(f, g)
+  @test ideal(h_total) == J_total
 
   ## Subscheme is a zero-dimensional scheme, topologically three points
   I = ideal(R, [x - y^3, x - y^5])