Gorenstein and Picard index

docs/oscar_references.bib

@@ -976,6 +976,17 @@ @Book{HEO05
   url           = {https://doi.org/10.1201/9781420035216}
 }
 
+@Article{HHS11,
+  author        = {Hausen, J\"{u}rgen and Herppich, Elaine and S\"{u}ss, Hendrik},
+  title         = {Multigraded factorial rings and {F}ano varieties with torus action},
+  mrnumber      = {2804508},
+  journal       = {Doc. Math.},
+  fjournal      = {Documenta Mathematica},
+  volume        = {16},
+  pages         = {71--109},
+  year          = {2011}
+}
+
 @Book{HP89,
   author        = {Holt, Derek F. and Plesken, W.},
   title         = {Perfect groups},

docs/src/AlgebraicGeometry/ToricVarieties/NormalToricVarieties.md

@@ -140,6 +140,13 @@ torusinvariant_weil_divisor_group(v::AbstractNormalToricVariety)
 torusinvariant_prime_divisors(v::AbstractNormalToricVariety)
 ```
 
+### Gorenstein and Picard index
+
+```@docs
+gorenstein_index(v::AbstractNormalToricVariety)
+picard_index(v::AbstractNormalToricVariety)
+```
+
 ### Cones and Fans
 
 ```@docs

experimental/ToricSchemes/src/NormalToricSchemes/Attributes.jl

@@ -158,18 +158,22 @@ cox_ring(X::ToricCoveredScheme) = cox_ring(X.ntv)
 dim(X::ToricCoveredScheme) = dim(X.ntv)
 dim_of_torusfactor(X::ToricCoveredScheme) = dim_of_torusfactor(X.ntv)
 euler_characteristic(X::ToricCoveredScheme) = euler_characteristic(X.ntv)
+gorenstein_index(X::ToricCoveredScheme) = gorenstein_index(X.ntv)
 polyhedral_fan(X::ToricCoveredScheme) = polyhedral_fan(X.ntv)
 ideal_of_linear_relations(R::MPolyRing, X::ToricCoveredScheme) = ideal_of_linear_relations(R, X.ntv)
 ideal_of_linear_relations(X::ToricCoveredScheme) = ideal_of_linear_relations(X.ntv)
 irrelevant_ideal(R::MPolyRing, X::ToricCoveredScheme) = irrelevant_ideal(R, X.ntv)
 irrelevant_ideal(X::ToricCoveredScheme) = irrelevant_ideal(X.ntv)
 map_from_character_lattice_to_torusinvariant_weil_divisor_group(X::ToricCoveredScheme) = map_from_character_lattice_to_torusinvariant_weil_divisor_group(X.ntv)
+map_from_torusinvariant_cartier_divisor_group_to_class_group(X::ToricCoveredScheme) = map_from_torusinvariant_cartier_divisor_group_to_class_group(X.ntv)
 map_from_torusinvariant_cartier_divisor_group_to_picard_group(X::ToricCoveredScheme) = map_from_torusinvariant_cartier_divisor_group_to_picard_group(X.ntv)
 map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(X::ToricCoveredScheme) = map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(X.ntv)
 map_from_torusinvariant_weil_divisor_group_to_class_group(X::ToricCoveredScheme) = map_from_torusinvariant_weil_divisor_group_to_class_group(X.ntv)
+map_from_picard_group_to_class_group(X::ToricCoveredScheme) = map_from_picard_group_to_class_group(X.ntv)
 mori_cone(X::ToricCoveredScheme) = mori_cone(X.ntv)
 nef_cone(X::ToricCoveredScheme) = nef_cone(X.ntv)
 picard_group(X::ToricCoveredScheme) = picard_group(X.ntv)
+picard_index(X::ToricCoveredScheme) = picard_index(X.ntv)
 set_coordinate_names(X::ToricCoveredScheme) = set_coordinate_names(X.ntv)
 set_coordinate_names_of_torus(X::ToricCoveredScheme) = set_coordinate_names_of_torus(X.ntv)
 stanley_reisner_ideal(R::MPolyRing, X::ToricCoveredScheme) = stanley_reisner_ideal(R, X.ntv)
@@ -199,17 +203,21 @@ cox_ring(X::ToricSpec) = cox_ring(X.antv)
 dim(X::ToricSpec) = dim(X.antv)
 dim_of_torusfactor(X::ToricSpec) = dim_of_torusfactor(X.antv)
 euler_characteristic(X::ToricSpec) = euler_characteristic(X.antv)
+gorenstein_index(X::ToricSpec) = gorenstein_index(X.antv)
 polyhedral_fan(X::ToricSpec) = polyhedral_fan(X.antv)
 ideal_of_linear_relations(R::MPolyRing, X::ToricSpec) = ideal_of_linear_relations(R, X.antv)
 ideal_of_linear_relations(X::ToricSpec) = ideal_of_linear_relations(X.antv)
 irrelevant_ideal(R::MPolyRing, X::ToricSpec) = irrelevant_ideal(R, X.antv)
 irrelevant_ideal(X::ToricSpec) = irrelevant_ideal(X.antv)
 map_from_character_lattice_to_torusinvariant_weil_divisor_group(X::ToricSpec) = map_from_character_lattice_to_torusinvariant_weil_divisor_group(X.antv)
+map_from_torusinvariant_cartier_divisor_group_to_class_group(X::ToricSpec) = map_from_torusinvariant_cartier_divisor_group_to_class_group(X.antv)
 map_from_torusinvariant_cartier_divisor_group_to_picard_group(X::ToricSpec) = map_from_torusinvariant_cartier_divisor_group_to_picard_group(X.antv)
 map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(X::ToricSpec) = map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(X.antv)
 map_from_torusinvariant_weil_divisor_group_to_class_group(X::ToricSpec) = map_from_torusinvariant_weil_divisor_group_to_class_group(X.antv)
+map_from_picard_group_to_class_group(X::ToricSpec) = map_from_picard_group_to_class_group(X.antv)
 mori_cone(X::ToricSpec) = mori_cone(X.antv)
 nef_cone(X::ToricSpec) = nef_cone(X.antv)
+picard_index(X::ToricSpec) = picard_index(X.antv)
 picard_group(X::ToricSpec) = picard_group(X.antv)
 set_coordinate_names(X::ToricSpec) = set_coordinate_names(X.antv)
 set_coordinate_names_of_torus(X::ToricSpec) = set_coordinate_names_of_torus(X.antv)

src/AlgebraicGeometry/ToricVarieties/NormalToricVarieties/attributes.jl

@@ -833,6 +833,36 @@ GrpAb: Z^3
     return domain(map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(v))
 end
 
+@doc raw"""
+    map_from_torusinvariant_cartier_divisor_group_to_class_group(v::AbstractNormalToricVariety)
+
+Return the map from the Cartier divisors to the class group
+of an abstract normal toric variety `v`.
+
+# Examples
+```jldoctest
+julia> p2 = projective_space(NormalToricVariety, 2)
+Normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor
+
+julia> map_from_torusinvariant_cartier_divisor_group_to_class_group(p2)
+Map with following data
+Domain:
+=======
+Abelian group with structure: Z^3
+Codomain:
+=========
+Abelian group with structure: Z
+```
+"""
+@attr GrpAbFinGenMap function map_from_torusinvariant_cartier_divisor_group_to_class_group(v::AbstractNormalToricVariety)
+    # check input
+    @req !has_torusfactor(v) "Group of the torus-invariant Cartier divisors can only be computed if the variety has no torus factor"
+
+    f = map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(v)
+    g = map_from_torusinvariant_weil_divisor_group_to_class_group(v)
+    return f * g
+end
+
 
 @doc raw"""
     map_from_torusinvariant_cartier_divisor_group_to_picard_group(v::AbstractNormalToricVariety)
@@ -861,10 +891,8 @@ Abelian group with structure: Z
         throw(ArgumentError("Group of the torus-invariant Cartier divisors can only be computed if the variety has no torus factor"))
     end
 
-    # compute mapping
-    map1 = map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(v)
-    map2 = map_from_torusinvariant_weil_divisor_group_to_class_group(v)
-    return restrict_codomain(map1*map2)
+    f = restrict_codomain(map_from_torusinvariant_cartier_divisor_group_to_class_group(v))
+    return f * inv(snf(codomain(f))[2])
 end
 
 
@@ -886,6 +914,77 @@ GrpAb: Z
     return codomain(map_from_torusinvariant_cartier_divisor_group_to_picard_group(v))
 end
 
+@doc raw"""
+    map_from_picard_group_to_class_group(v::AbstractNormalToricVariety)
+
+Return the embedding of the Picard group into the class group of an abstract normal toric variety `v`.
+
+# Examples
+```jldoctest
+julia> p2 = projective_space(NormalToricVariety, 2)
+Normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor
+
+julia> map_from_picard_group_to_class_group(p2)
+Map with following data
+Domain:
+=======
+Abelian group with structure: Z
+Codomain:
+=========
+Abelian group with structure: Z
+```
+"""
+@attr GrpAbFinGenMap function map_from_picard_group_to_class_group(v::AbstractNormalToricVariety)
+    f = image(map_from_torusinvariant_cartier_divisor_group_to_class_group(v))[2]
+    g = snf(domain(f))[2] * f
+    return hom(picard_group(v), class_group(v), matrix(g))
+end
+
+
+############################
+# Gorenstein and Picard Index
+############################
+
+@doc raw"""
+    gorenstein_index(v::AbstractNormalToricVariety)
+
+Return the Gorenstein index of a $\mathbb{Q}$-Gorenstein normal toric variety `v`. 
+This is the smallest positive integer $l$ such that $-l K$ is Cartier, where $K$
+is a canonical divisor on `v`. See exercise 8.3.10 and 8.3.11 in [CLS11](@cite) for more details.
+
+# Examples
+```jldoctest
+julia> gorenstein_index(weighted_projective_space(NormalToricVariety, [2,3,5]))
+3
+```
+"""
+@attr ZZRingElem function gorenstein_index(v::AbstractNormalToricVariety)
+    @req is_q_gorenstein(v) "gorenstein index can only be computed for Q-gorenstein varieties"
+    c = divisor_class(canonical_divisor_class(v))
+    f = cokernel(map_from_picard_group_to_class_group(v))[2]
+    order(f(c))
+end
+
+
+@doc raw"""
+    picard_index(v::AbstractNormalToricVariety)
+
+Return the index of the Picard group in the class group of a simplicial normal 
+toric variety `v`. Here, the Picard group embeds as the group of Cartier divisor
+classes into the class group via `map_from_picard_group_to_class_group`. See 
+[HHS11](@cite) for more details.
+
+# Examples
+```jldoctest
+julia> picard_index(weighted_projective_space(NormalToricVariety, [2,3,5]))
+30
+```
+"""
+@attr ZZRingElem function picard_index(v::AbstractNormalToricVariety) 
+    @req is_simplicial(v) "picard index can only be computed for simplicial varieties"
+    return order(cokernel(map_from_picard_group_to_class_group(v))[1]) 
+end
+
 
 ############################
 # Cones and fans

src/exports.jl

@@ -573,6 +573,7 @@ export glueing_domains
 export glueing_graph
 export glueing_morphisms
 export glueings
+export gorenstein_index
 export goldfarb_cube
 export goldfarb_sit_cube
 export grade
@@ -913,6 +914,8 @@ export low_index_subgroup_reps
 export lower_triangular_matrix
 export map
 export map_from_character_lattice_to_torusinvariant_weil_divisor_group
+export map_from_picard_group_to_class_group
+export map_from_torusinvariant_cartier_divisor_group_to_class_group
 export map_from_torusinvariant_cartier_divisor_group_to_picard_group
 export map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group
 export map_from_torusinvariant_weil_divisor_group_to_class_group
@@ -1106,6 +1109,7 @@ export permutation_of_terms
 export permuted
 export picard_class
 export picard_group
+export picard_index
 export pile_polytope
 export pitman_stanley_polytope
 export platonic_solid

test/AlgebraicGeometry/ToricVarieties/affine_normal_varieties.jl

@@ -26,6 +26,8 @@
         @test elementary_divisors(class_group(antv)) == [ 2 ]
         @test ngens(cox_ring(antv)) == 2
         @test length(torusinvariant_prime_divisors(antv)) == 2
+        @test gorenstein_index(antv) == 1
+        @test picard_index(antv) == 2
     end
 
     @testset "Cast to affine" begin

test/AlgebraicGeometry/ToricVarieties/del_pezzo_surfaces.jl

@@ -32,6 +32,12 @@
         @test codomain(map_from_torusinvariant_cartier_divisor_group_to_picard_group(dP1)) == picard_group(dP1)
         @test domain(map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(dP1)) == torusinvariant_cartier_divisor_group(dP1)
         @test codomain(map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(dP1)) == torusinvariant_weil_divisor_group(dP1)
+        @test domain(map_from_torusinvariant_cartier_divisor_group_to_class_group(dP1)) == torusinvariant_cartier_divisor_group(dP1)
+        @test codomain(map_from_torusinvariant_cartier_divisor_group_to_class_group(dP1)) == class_group(dP1)
+        @test domain(map_from_picard_group_to_class_group(dP1)) == picard_group(dP1)
+        @test codomain(map_from_picard_group_to_class_group(dP1)) == class_group(dP1)
+        @test gorenstein_index(dP1) == 1
+        @test picard_index(dP1) == 1
         if set_attributes
             @test matrix(map_from_torusinvariant_weil_divisor_group_to_class_group(dP1)) == matrix(ZZ, [[1, 1], [1, 1], [1, 0], [0, -1]])
             @test coordinate_names(dP1) == ["x1", "x2", "x3", "e1"]
@@ -49,6 +55,12 @@
         @test codomain(map_from_torusinvariant_cartier_divisor_group_to_picard_group(dP2)) == picard_group(dP2)
         @test domain(map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(dP2)) == torusinvariant_cartier_divisor_group(dP2)
         @test codomain(map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(dP2)) == torusinvariant_weil_divisor_group(dP2)
+        @test domain(map_from_torusinvariant_cartier_divisor_group_to_class_group(dP2)) == torusinvariant_cartier_divisor_group(dP2)
+        @test codomain(map_from_torusinvariant_cartier_divisor_group_to_class_group(dP2)) == class_group(dP2)
+        @test domain(map_from_picard_group_to_class_group(dP2)) == picard_group(dP2)
+        @test codomain(map_from_picard_group_to_class_group(dP2)) == class_group(dP2)
+        @test gorenstein_index(dP2) == 1
+        @test picard_index(dP2) == 1
         if set_attributes
             @test matrix(map_from_torusinvariant_weil_divisor_group_to_class_group(dP2)) == matrix(ZZ, [[1, 1, 1], [1, 1, 0], [1, 0, 1], [0, -1, 0], [0, 0, -1]])
             @test coordinate_names(dP2) == ["x1", "x2", "x3", "e1", "e2"]
@@ -66,6 +78,12 @@
         @test codomain(map_from_torusinvariant_cartier_divisor_group_to_picard_group(dP3)) == picard_group(dP3)
         @test domain(map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(dP3)) == torusinvariant_cartier_divisor_group(dP3)
         @test codomain(map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(dP3)) == torusinvariant_weil_divisor_group(dP3)
+        @test domain(map_from_torusinvariant_cartier_divisor_group_to_class_group(dP3)) == torusinvariant_cartier_divisor_group(dP3)
+        @test codomain(map_from_torusinvariant_cartier_divisor_group_to_class_group(dP3)) == class_group(dP3)
+        @test domain(map_from_picard_group_to_class_group(dP3)) == picard_group(dP3)
+        @test codomain(map_from_picard_group_to_class_group(dP3)) == class_group(dP3)
+        @test gorenstein_index(dP3) == 1
+        @test picard_index(dP3) == 1
         if set_attributes
             @test matrix(map_from_torusinvariant_weil_divisor_group_to_class_group(dP3)) == matrix(ZZ, [[1, 1, 1, 0], [1, 1, 0, 1], [1, 0, 1, 1], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, -1]])
             @test coordinate_names(dP3) == ["x1", "x2", "x3", "e1", "e2", "e3"]

test/AlgebraicGeometry/ToricVarieties/hirzebruch_surfaces.jl

@@ -53,6 +53,12 @@
         @test domain(map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(F5)) == torusinvariant_cartier_divisor_group(F5)
         @test codomain(map_from_torusinvariant_cartier_divisor_group_to_torusinvariant_weil_divisor_group(F5)) == torusinvariant_weil_divisor_group(F5)
         @test matrix(map_from_torusinvariant_weil_divisor_group_to_class_group(F5)) == matrix(ZZ, [[1, 0], [0, 1], [1, 0], [5, 1]])
+        @test domain(map_from_torusinvariant_cartier_divisor_group_to_class_group(F5)) == torusinvariant_cartier_divisor_group(F5)
+        @test codomain(map_from_torusinvariant_cartier_divisor_group_to_class_group(F5)) == class_group(F5)
+        @test domain(map_from_picard_group_to_class_group(F5)) == picard_group(F5)
+        @test codomain(map_from_picard_group_to_class_group(F5)) == class_group(F5)
+        @test gorenstein_index(F5) == 1
+        @test picard_index(F5) == 1
         if set_attributes
             @test coordinate_names(F5) == ["t1", "x1", "t2", "x2"]
         end