From f58e9f698fe324f5630a20a940bd80a7f4126e2d Mon Sep 17 00:00:00 2001
From: Martin Bies <MBies87@googlemail.com>
Date: Thu, 7 Sep 2023 14:10:10 +0200
Subject: [PATCH] [ToricSchemes] Optimize and extend tests

---
 .../ToricVarieties/toric_schemes.jl           | 41 +++++++++++++------
 1 file changed, 28 insertions(+), 13 deletions(-)

diff --git a/test/AlgebraicGeometry/ToricVarieties/toric_schemes.jl b/test/AlgebraicGeometry/ToricVarieties/toric_schemes.jl
index dde3b7b14ecf..c52f16e1a181 100644
--- a/test/AlgebraicGeometry/ToricVarieties/toric_schemes.jl
+++ b/test/AlgebraicGeometry/ToricVarieties/toric_schemes.jl
@@ -21,6 +21,7 @@ using Test
   end
   
   IP1 = projective_space(NormalToricVariety, 1; set_attributes = set_attributes)
+  set_coordinate_names(IP1, ["x", "y"])
   Y = IP1*IP1
   
   @testset "Product of projective spaces" begin
@@ -51,23 +52,37 @@ using Test
     II = IdealSheaf(IP3, I)
   end
 
-  @testset "conversion of toric morphisms to morphisms of covered schemes" begin
-    bl = blow_up(IP2, [1, 1])
-    X = domain(bl)
-    bl_cov = covering_morphism(bl)
-    S = cox_ring(IP2)
-    (x, y, z) = gens(S)
-    I = IdealSheaf(IP2, ideal(S, [z*(x-y)]))
+  S = cox_ring(IP2)
+  (x, y, z) = gens(S)
+  I = IdealSheaf(IP2, ideal(S, [z*(x-y)]))
+  J = IdealSheaf(IP2, ideal(S, [x, y]))
+  bl = blow_up(IP2, [1, 1])
+  pb_I = pullback(bl, I)
+  pb_J = pullback(bl, J)
+
+  @testset "toric blowdown morphism as morphism of covered schemes" begin
     @test scheme(I) === IP2
     @test length(Oscar.maximal_associated_points(I)) == 2
-    g = underlying_morphism(bl)
-    pb_I = pullback(g, I)
     @test length(Oscar.maximal_associated_points(pb_I)) == 3
-
-    J = IdealSheaf(IP2, ideal(S, [x, y]))
     @test dim(J) == 0
-    pb_J = pullback(bl, J)
-    #pb_J = pullback(g, J)
     @test dim(pb_J) == 1
   end
+
+  F2 = hirzebruch_surface(NormalToricVariety, 2; set_attributes = set_attributes)
+  f = toric_morphism(F2, matrix(ZZ, [[1], [0]]), IP1; check = true)
+  S = cox_ring(IP1)
+  (x, y) = gens(S)
+  K = IdealSheaf(IP1, ideal(S, [x]))
+  pb_K = pullback(f, K)
+  # The support of pb_K should be nothing but P1, it is the fiber
+  # of the P1-fibration that defines the Hirebruch surface over the
+  # point defined by K.
+  # TODO: Add a test that verifies that indeed pb_K is P1.
+  @testset "Hirzebruch surface as P1 fibration over P1" begin
+    @test length(Oscar.maximal_associated_points(K)) == 1
+    @test dim(K) == 0
+    @test length(Oscar.maximal_associated_points(pb_K)) == 1
+    @test dim(pb_K) == 1
+  end
+
 end