From 3a6c0564005b4e5fcae007d5f2cfac4ec925d62c Mon Sep 17 00:00:00 2001
From: Martin Bies <MBies87@googlemail.com>
Date: Thu, 7 Sep 2023 15:18:01 +0200
Subject: [PATCH] [ToricSchemes] Document covering_morphism of a toric morphism

---
 .../ToricVarieties/ToricMorphisms.md          |  1 +
 .../ToricMorphisms/attributes.jl              | 43 +++++++++++++++++++
 2 files changed, 44 insertions(+)

diff --git a/docs/src/AlgebraicGeometry/ToricVarieties/ToricMorphisms.md b/docs/src/AlgebraicGeometry/ToricVarieties/ToricMorphisms.md
index 8dedc444349e..9f51932d90db 100644
--- a/docs/src/AlgebraicGeometry/ToricVarieties/ToricMorphisms.md
+++ b/docs/src/AlgebraicGeometry/ToricVarieties/ToricMorphisms.md
@@ -49,6 +49,7 @@ morphism_on_torusinvariant_weil_divisor_group(tm::ToricMorphism)
 morphism_on_torusinvariant_cartier_divisor_group(tm::ToricMorphism)
 morphism_on_class_group(tm::ToricMorphism)
 morphism_on_picard_group(tm::ToricMorphism)
+covering_morphism(f::ToricMorphism)
 ```
 
 ### Special attributes of toric varieties
diff --git a/src/AlgebraicGeometry/ToricVarieties/ToricMorphisms/attributes.jl b/src/AlgebraicGeometry/ToricVarieties/ToricMorphisms/attributes.jl
index 3908326b67c0..c5f2d0605053 100644
--- a/src/AlgebraicGeometry/ToricVarieties/ToricMorphisms/attributes.jl
+++ b/src/AlgebraicGeometry/ToricVarieties/ToricMorphisms/attributes.jl
@@ -184,9 +184,52 @@ Abelian group with structure: Z^2
     return domain_preinverse * morphism_on_cartier_divisors * codomain_projection
 end
 
+
 ########################################################################
 # Functionality for the associated CoveredSchemeMorphism               #
 ########################################################################
+
+@doc raw"""
+    covering_morphism(f::ToricMorphism)
+
+For a given toric morphism `tm`, we can compute the corresponding
+morphism of covered schemes. The following demonstrates this for the
+blow-down morphism of a blow-up of the projective space.
+
+# Examples
+```jldoctest
+julia> IP2 = projective_space(NormalToricVariety, 2)
+Normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor
+
+julia> bl = blow_up(IP2, [1, 1]);
+
+julia> cov_bl = covering_morphism(bl);
+
+julia> domain(cov_bl)
+Covering
+  described by patches
+    1: normal, affine toric variety
+    2: normal, affine toric variety
+    3: normal, affine toric variety
+    4: normal, affine toric variety
+  in the coordinate(s)
+    1: [x_1_1, x_2_1]
+    2: [x_1_2, x_2_2]
+    3: [x_1_3, x_2_3]
+    4: [x_1_4, x_2_4]
+
+julia> codomain(cov_bl)
+Covering
+  described by patches
+    1: normal, affine toric variety
+    2: normal, affine toric variety
+    3: normal, affine toric variety
+  in the coordinate(s)
+    1: [x_1_1, x_2_1]
+    2: [x_1_2, x_2_2]
+    3: [x_1_3, x_2_3]
+```
+"""
 @attr CoveringMorphism function covering_morphism(f::ToricMorphism)
   # TODO: If f is a blowdown morphism, we can simplify 
   # the matchings of cones below.