Make ideal sheaves on toric varieties work

experimental/Schemes/IdealSheaves.jl

@@ -1216,17 +1216,53 @@ end
 
 Create a sheaf of ideals on a toric variety ``X`` from a homogeneous ideal 
 `I` in its `cox_ring`.
+
+# Examples
+```jldoctest
+julia> P3 = projective_space(NormalToricVariety, 3)
+Normal, non-affine, smooth, projective, gorenstein, fano, 3-dimensional toric variety without torusfactor
+
+julia> (x1,x2,x3,x4) = gens(cox_ring(P3))
+4-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
+ x1
+ x2
+ x3
+ x4
+
+julia> I = ideal([x2,x3])
+ideal(x2, x3)
+
+julia> IdealSheaf(P3, I)
+
+```
 """
 function IdealSheaf(X::NormalToricVariety, I::MPolyIdeal)
   @req base_ring(I) === cox_ring(X) "ideal must live in the cox ring of the variety"
 
+  # We currently only support this provided that the following conditions are met:
+  # 1. All maximal cones are smooth, i.e. the fan is smooth/X is smooth.
+  # 2. The dimension of all maximal cones matches the dimension of the fan.
+  @req is_smooth(X) "Currently, ideal sheaves are only supported for smooth toric varieties"
+  @req all(m -> dim(m) == dim(X), maximal_cones(X)) "Currently, ideal sheaves require that all maximal cones have the dimension of the variety"
+
   # TODO: In the long run we should think about making the creation of the 
   # following dictionary lazy. But this requires partial rewriting of the 
   # ideal sheaves as a whole, so we postpone it for the moment.
 
   ideal_dict = IdDict{AbsSpec, Ideal}()
-  for U in affine_charts(X)
-    ideal_dict[U] = _dehomogenize(I, U) # Method to be written!!!
+  for (k, U) in enumerate(affine_charts(X))
+    R = base_ring(toric_ideal(U))
+    indices = ray_indices(maximal_cones(X))[k,:]
+    imgs = [one(R) for k in 1:nrays(X)]
+    count = 1
+    for l in 1:nrays(X)
+      if indices[l]
+        imgs[l] = gens(R)[count]
+        count += 1
+      end
+    end
+    map = hom(cox_ring(X), R, imgs)
+    ideal_dict[U] = map(I)
   end
 
   return IdealSheaf(X, ideal_dict)