Compute strict transform, not total transform

Fixes a bug in FTheoryTools instroduced by
oscar-system#4484
Namely, it previously computed the total transform, not the strict
transform.

Even with the speed improvements of
oscar-system#4485
computing the strict transform with ideals is slower than operating on
polynomials. Therefore, I added a strict transform function on
polynomials.

Author: paemurru

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 +1,26 @@
 @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]`.
+
 # Examples
 ```jldoctest
 julia> X = affine_space(NormalToricVariety, 2)
@@ -40,6 +46,8 @@ Ideal generated by
   x1 + x2*e
 ```
 """
+strict_transform(f::ToricBlowupMorphism, I::Union{MPolyIdeal, MPolyDecRingElem})
+
 function strict_transform(f::ToricBlowupMorphism, I::MPolyIdeal)
   X = codomain(f)
   @req !has_torusfactor(X) "Only implemented when there are no torus factors"
@@ -49,6 +57,15 @@ 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