Line bundle cohomology with cohomCalg

We employ the cohomCalg algorithm [BJRR10*1] to compute the dimension of line bundle cohomologies as well as vanishing sets.

Dimensions of line bundle cohomology

all_cohomologies — Method

all_cohomologies(l::ToricLineBundle)

Computes the dimension of all sheaf cohomologies of the toric line bundle l by use of the cohomCalg algorithm [BJRR10], [BJRR10*1] (see also [RR10], [Jow11] and [BJRR12]).

Examples

julia> dP3 = del_pezzo_surface(NormalToricVariety, 3)
Normal toric variety

julia> all_cohomologies(toric_line_bundle(dP3, [1, 2, 3, 4]))
3-element Vector{ZZRingElem}:
 0
 16
 0

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cohomology — Method

cohomology(l::ToricLineBundle, i::Int)

Computes the dimension of the i-th sheaf cohomology of the toric line bundle l by use of the cohomCalg algorithm [BJRR10], [BJRR10*1] (see also [RR10], [Jow11] and [BJRR12]).

Examples

julia> dP3 = del_pezzo_surface(NormalToricVariety, 3)
Normal toric variety

julia> cohomology(toric_line_bundle(dP3, [4, 1, 1, 1]), 0)
12

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Toric vanishing sets

Vanishing sets describe subsets of the Picard group of toric varieties. Their computations is based on [BJRR10*1], i.e. this functionality is only available if the toric variety in question is either smooth and complete or alternatively, simplicial and projective. This approach to identify vanishing sets on toric varieties was originally introduced in [Bie18]. As described there, on a technical level, a vanishing set is the complement of a finite family of polyhedra.

For a toric variety, all vanishing sets are computed as follows:

vanishing_sets — Method

vanishing_sets(variety::NormalToricVarietyType)

Compute the vanishing sets of an abstract toric variety v by use of the cohomCalg algorithm.

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The return value is a vector of vanishing sets. This vector has length one larger than the dimension of the variety in question. The first vanishing set in this vector describes all line bundles for which the zero-th sheaf cohomology vanishes. More generally, if a line bundle is contained in the n-th vanishing set, then its n-1-th sheaf cohomology vanishes. The following method checks if a line bundle is contained in a vanishing set:

contains — Method

contains(tvs::ToricVanishingSet, l::ToricLineBundle)

Checks if the toric line bundle l is contained in the toric vanishing set tvs.

Examples

julia> dP1 = del_pezzo_surface(NormalToricVariety, 1)
Normal toric variety

julia> l = toric_line_bundle(dP1, [3, 2])
Toric line bundle on a normal toric variety

julia> all_cohomologies(l)
3-element Vector{ZZRingElem}:
 7
 0
 0

julia> vs = vanishing_sets(dP1)
3-element Vector{ToricVanishingSet}:
 Toric vanishing set for cohomology indices [0]
 Toric vanishing set for cohomology indices [1]
 Toric vanishing set for cohomology indices [2]

julia> contains(vs[1], l)
false

julia> contains(vs[2], l)
true

julia> contains(vs[3], l)
true

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A vanishing set can in principle cover the entire Picard group. This can be checked with isfull. This methods returns true if the vanishing set is the entire Picard group and false otherwise. Beyond this, we support the following attributes for vanishing sets:

toric_variety — Method

toric_variety(tvs::ToricVanishingSet)

Return the toric variety of the vanishing set tvs.

Examples

julia> dP1 = del_pezzo_surface(NormalToricVariety, 1)
Normal toric variety

julia> vs = vanishing_sets(dP1)
3-element Vector{ToricVanishingSet}:
 Toric vanishing set for cohomology indices [0]
 Toric vanishing set for cohomology indices [1]
 Toric vanishing set for cohomology indices [2]

julia> toric_variety(vs[3])
Normal, 2-dimensional toric variety without torusfactor

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polyhedra — Method

polyhedra(tvs::ToricVanishingSet)

Return the vector of the polyhedra whose complement defines the vanishing set tvs.

Examples

julia> dP1 = del_pezzo_surface(NormalToricVariety, 1)
Normal toric variety

julia> vs = vanishing_sets(dP1)
3-element Vector{ToricVanishingSet}:
 Toric vanishing set for cohomology indices [0]
 Toric vanishing set for cohomology indices [1]
 Toric vanishing set for cohomology indices [2]

julia> polyhedra(vs[3])
1-element Vector{Polyhedron{QQFieldElem}}:
 Polyhedron in ambient dimension 2

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cohomology_indices — Method

cohomology_indices(tvs::ToricVanishingSet)

Return the cohomology indices of the toric vanishing set tvs.

Examples

julia> dP1 = del_pezzo_surface(NormalToricVariety, 1)
Normal toric variety

julia> vs = vanishing_sets(dP1)
3-element Vector{ToricVanishingSet}:
 Toric vanishing set for cohomology indices [0]
 Toric vanishing set for cohomology indices [1]
 Toric vanishing set for cohomology indices [2]

julia> cohomology_indices(vs[3])
1-element Vector{Int64}:
 2

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Certainly, this also allows to compute the immaculate line bundles:

immaculate_line_bundles — Method

immaculate_line_bundles(variety::NormalToricVarietyType)

Computes all immaculate line bundles as a toric vanishing set by intersecting the vanishing sets for all cohomology indices.

Examples

julia> dP1 = del_pezzo_surface(NormalToricVariety, 1)
Normal toric variety

julia> ilb = immaculate_line_bundles(dP1)
Toric vanishing set for cohomology indices [0, 1, 2]

julia> polyhedra(ilb)
4-element Vector{Polyhedron{QQFieldElem}}:
 Polyhedron in ambient dimension 2
 Polyhedron in ambient dimension 2
 Polyhedron in ambient dimension 2
 Polyhedron in ambient dimension 2

julia> print_constraints(polyhedra(ilb)[1])
-x_1 <= 0
-x_1 + x_2 <= 0

julia> print_constraints(polyhedra(ilb)[2])
-x_1 + x_2 <= 0
x_2 <= -2

julia> print_constraints(polyhedra(ilb)[3])
-x_2 <= -1
x_1 - x_2 <= -2

julia> print_constraints(polyhedra(ilb)[4])
x_1 - x_2 <= -2
x_1 <= -3

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