[ToricVarieties] Extend blow_up method to use schemes as fallback if …

…needed

experimental/Schemes/IdealSheaves.jl

@@ -1305,3 +1305,10 @@ function _my_dot(v::PointVector{ZZRingElem}, u::ZZMatrix)
   return sum(v[i]*u[1,i] for i in 1:n; init=zero(QQ))
 end
 
+function _generic_blow_up(v::Any, I::Any)
+  error("Not yet supported")
+end
+
+function _generic_blow_up(v::NormalToricVarietyType, I::MPolyIdeal)
+  return blow_up(IdealSheaf(v, I))
+end

src/AlgebraicGeometry/ToricVarieties/NormalToricVarieties/standard_constructions.jl

@@ -321,19 +321,55 @@ Multivariate polynomial ring in 5 variables over QQ graded by
   x3 -> [0 1]
   x4 -> [1 0]
   e -> [1 -1]
+
+julia> I2 = ideal([x2 * x3])
+ideal(x2 * x3)
+
+julia> b2P3 = blow_up(P3, I2)
+Blow up
+  of scheme over QQ covered with 4 patches
+    1b: [x_1_1, x_2_1, x_3_1]   normal, affine toric variety
+    2b: [x_1_2, x_2_2, x_3_2]   normal, affine toric variety
+    3b: [x_1_3, x_2_3, x_3_3]   normal, affine toric variety
+    4b: [x_1_4, x_2_4, x_3_4]   normal, affine toric variety
+  in sheaf of ideals with restrictions
+    1b: ideal(x_2_1*x_3_1)
+    2b: ideal(x_2_2*x_3_2)
+    3b: ideal(x_3_3)
+    4b: ideal(x_2_4)
+with domain
+  scheme over QQ covered with 4 patches
+    1a: [x_1_1, x_2_1, x_3_1]   spec of quotient of multivariate polynomial ring
+    2a: [x_1_2, x_2_2, x_3_2]   spec of quotient of multivariate polynomial ring
+    3a: [x_1_3, x_2_3, x_3_3]   spec of quotient of multivariate polynomial ring
+    4a: [x_1_4, x_2_4, x_3_4]   spec of quotient of multivariate polynomial ring
+and exceptional divisor
+  effective cartier divisor defined by
+    sheaf of ideals with restrictions
+      1a: ideal(x_2_1*x_3_1)
+      2a: ideal(x_2_2*x_3_2)
+      3a: ideal(x_3_3)
+      4a: ideal(x_2_4)
+
 ```
 """
 function blow_up(v::NormalToricVarietyType, I::MPolyIdeal; coordinate_name::String = "e", set_attributes::Bool = true)
     @req base_ring(I) == cox_ring(v) "The ideal must be contained in the cox ring of the toric variety"
     indices = [findfirst(y -> y == x, gens(cox_ring(v))) for x in gens(I)]
-    @req length(indices) == ngens(I) "All generators must be indeterminates of the cox ring of the toric variety"
-    rs = matrix(ZZ, rays(v))
-    new_ray = vec(sum([rs[i,:] for i in indices]))
-    new_ray = new_ray ./ gcd(new_ray)
-    return blow_up(v, new_ray; coordinate_name = coordinate_name, set_attributes = set_attributes)
+    if length(indices) == ngens(I) && (nothing in indices) == false
+      # We perform this blowup with toric techniques.
+      rs = matrix(ZZ, rays(v))
+      new_ray = vec(sum([rs[i,:] for i in indices]))
+      new_ray = new_ray ./ gcd(new_ray)
+      return blow_up(v, new_ray; coordinate_name = coordinate_name, set_attributes = set_attributes)
+    else
+      # We rely on advanced techniques to conduct this blowup (if available).
+      return _generic_blow_up(v, I)
+    end
 end
 
 
+
 @doc raw"""
     blow_up(v::NormalToricVarietyType, new_ray::AbstractVector{<:IntegerUnion}; coordinate_name::String = "e", set_attributes::Bool = true)