GAP (ToricVarieties) - Chapter 4: Toric subvarieties

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4 Toric subvarieties
4.1 The GAP category
4.1-1 IsToricSubvariety
4.2 Properties
4.2-1 IsClosedSubvariety
4.2-2 IsOpen
4.2-3 IsWholeVariety
4.3 Attributes
4.3-1 UnderlyingToricVariety
4.3-2 InclusionMorphism
4.3-3 AmbientToricVariety
4.4 Methods
4.4-1 ClosureOfTorusOrbitOfCone
4.5 Constructors
4.5-1 ToricSubvariety

### 4 Toric subvarieties

#### 4.1 The GAP category

##### 4.1-1 IsToricSubvariety

`‣ IsToricSubvariety`( M ) | ( filter )

Returns: true or false

The GAP category of a toric subvariety. Every toric subvariety is a toric variety, so every method applicable to toric varieties is also applicable to toric subvarieties.

#### 4.2 Properties

##### 4.2-1 IsClosedSubvariety

`‣ IsClosedSubvariety`( vari ) | ( property )

Returns: true or false

Checks if the subvariety vari is a closed subset of its ambient variety.

##### 4.2-2 IsOpen

`‣ IsOpen`( vari ) | ( property )

Returns: true or false

Checks if a subvariety is a closed subset.

##### 4.2-3 IsWholeVariety

`‣ IsWholeVariety`( vari ) | ( property )

Returns: true or false

Returns true if the subvariety vari is the whole variety.

#### 4.3 Attributes

##### 4.3-1 UnderlyingToricVariety

`‣ UnderlyingToricVariety`( vari ) | ( attribute )

Returns: a variety

Returns the toric variety which is represented by vari. This method implements the forgetful functor subvarieties -> varieties.

##### 4.3-2 InclusionMorphism

`‣ InclusionMorphism`( vari ) | ( attribute )

Returns: a morphism

If the variety vari is an open subvariety, this method returns the inclusion morphism in its ambient variety. If not, it will fail.

##### 4.3-3 AmbientToricVariety

`‣ AmbientToricVariety`( vari ) | ( attribute )

Returns: a variety

Returns the ambient toric variety of the subvariety vari

#### 4.4 Methods

##### 4.4-1 ClosureOfTorusOrbitOfCone

`‣ ClosureOfTorusOrbitOfCone`( vari, cone ) | ( operation )

Returns: a subvariety

The method returns the closure of the orbit of the torus contained in vari which corresponds to the cone cone as a closed subvariety of vari.

#### 4.5 Constructors

##### 4.5-1 ToricSubvariety

`‣ ToricSubvariety`( vari, ambvari ) | ( operation )

Returns: a subvariety

The method returns the closure of the orbit of the torus contained in vari which corresponds to the cone cone as a closed subvariety of vari.

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