Subvarieties

Introduction

We focus on simplicial toric varieties. Then, any closed subvariety is given as the vanishing set of a homogeneous ideal in the Cox ring of the toric variety in question (cf. proposition 5.2.4 in [CLS11]). As of now, we provide elementary support for closed subvarieties of simplicial toric varieties.

Constructors

General constructors

closed_subvariety_of_toric_variety — Method

closed_subvariety_of_toric_variety(toric_variety::NormalToricVarietyType, defining_polynomials::Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}})

Construct the closed subvariety of a simplicial normal toric variety. The defining data for the closed subvariety is a list of homogeneous polynomials, all of which must be elements of the Cox ring of the toric variety in question. The common vanishing locus of these polynomials defines the closed subvariety in question. By proposition 5.2.4 in [CLS11] every closed subvariety of a simplicial toric variety arises in this way.

Examples

julia> f2 = hirzebruch_surface(NormalToricVariety, 2);

julia> (t1, x1, t2, x2) = gens(cox_ring(f2));

julia> closed_subvariety_of_toric_variety(f2, [t1])
Closed subvariety of a normal toric variety

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closed_subvariety_of_toric_variety — Method

closed_subvariety_of_toric_variety(toric_variety::NormalToricVarietyType, defining_ideal::MPolyIdeal)

Construct the closed subvariety of a simplicial normal toric variety. The defining data for the closed subvariety is an ideal of the Cox ring of the toric variety in question. By proposition 5.2.4 in [CLS11] every closed subvariety of a simplicial toric variety arises in this way.

Examples

julia> f2 = hirzebruch_surface(NormalToricVariety, 2);

julia> (t1, x1, t2, x2) = gens(cox_ring(f2));

julia> closed_subvariety_of_toric_variety(f2, ideal([t1]))
Closed subvariety of a normal toric variety

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Properties

is_empty — Method

is_empty(c::ClosedSubvarietyOfToricVariety)

Checks if a closed subvariety of a toric variety is empty. This check uses proposition 5.2.6 in [CLS11].

Examples

julia> f2 = hirzebruch_surface(NormalToricVariety, 2);

julia> (t1, x1, t2, x2) = gens(cox_ring(f2));

julia> c = closed_subvariety_of_toric_variety(f2, [t1])
Closed subvariety of a normal toric variety

julia> is_empty(c)
false

julia> c2 = closed_subvariety_of_toric_variety(f2, [x1,x2])
Closed subvariety of a normal toric variety

julia> is_empty(c2)
true

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Attributes

toric_variety — Method

toric_variety(c::ClosedSubvarietyOfToricVariety)

When constructing a closed subvariety, a toric variety must be provided in which the closed subvariety is contained. This method returns this initially provided toric supervariety.

Note however that perse, a closed subvariety can be contained in different non-isomorphic toric varieties.

Examples

julia> f2 = hirzebruch_surface(NormalToricVariety, 2);

julia> (t1, x1, t2, x2) = gens(cox_ring(f2));

julia> c = closed_subvariety_of_toric_variety(f2, [t1])
Closed subvariety of a normal toric variety

julia> toric_variety(c) == f2
true

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defining_ideal — Method

defining_ideal(c::ClosedSubvarietyOfToricVariety)

When constructing a closed subvariety, an ideal in the Cox ring of a normal toric variety must be provided. This method returns this initially provided ideal.

Examples

julia> f2 = hirzebruch_surface(NormalToricVariety, 2);

julia> (t1, x1, t2, x2) = gens(cox_ring(f2));

julia> c = closed_subvariety_of_toric_variety(f2, [t1])
Closed subvariety of a normal toric variety

julia> defining_ideal(c) == ideal([t1])
true

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radical — Method

radical(c::ClosedSubvarietyOfToricVariety)

When constructing a closed subvariety, an ideal in the Cox ring of a normal toric variety must be provided. This method returns the radical of this initially provided ideal.

Examples

julia> f2 = hirzebruch_surface(NormalToricVariety, 2);

julia> (t1, x1, t2, x2) = gens(cox_ring(f2));

julia> c = closed_subvariety_of_toric_variety(f2, [t1])
Closed subvariety of a normal toric variety

julia> radical(c) == ideal([t1])
true

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