GAP (ToricVarieties) - Chapter 5: Affine toric varieties

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5 Affine toric varieties
5.1 Affine toric varieties: Examples
5.1-1 Affine space
5.2 The GAP category
5.2-1 IsAffineToricVariety
5.3 Attributes
5.3-1 CoordinateRing
5.3-2 ListOfVariablesOfCoordinateRing
5.3-3 MorphismFromCoordinateRingToCoordinateRingOfTorus
5.3-4 ConeOfVariety
5.4 Methods
5.4-1 CoordinateRing
5.4-2 Cone
5.5 Constructors

### 5 Affine toric varieties

#### 5.1 Affine toric varieties: Examples

##### 5.1-1 Affine space

    gap> C:=Cone( [[1,0,0],[0,1,0],[0,0,1]] );
    <A cone in |R^3>
    gap> C3:=ToricVariety(C);
    <An affine normal toric variety of dimension 3>
    gap> Dimension(C3);
    3
    gap> IsOrbifold(C3);
    true
    gap> IsSmooth(C3);
    true
    gap> CoordinateRingOfTorus(C3,"x");
    Q[x1,x1_,x2,x2_,x3,x3_]/( x1*x1_-1, x2*x2_-1, x3*x3_-1 )
    gap> CoordinateRing(C3,"x");
    Q[x_1,x_2,x_3]
    gap> MorphismFromCoordinateRingToCoordinateRingOfTorus( C3 );
    <A monomorphism of rings>
    gap> C3;
    <An affine normal smooth toric variety of dimension 3>
    gap> StructureDescription( C3 );
    "|A^3"

#### 5.2 The GAP category

##### 5.2-1 IsAffineToricVariety

`‣ IsAffineToricVariety`( M ) | ( filter )

Returns: true or false

The GAP category of an affine toric variety. All affine toric varieties are toric varieties, so everything applicable to toric varieties is applicable to affine toric varieties.

#### 5.3 Attributes

##### 5.3-1 CoordinateRing

`‣ CoordinateRing`( vari ) | ( attribute )

Returns: a ring

Returns the coordinate ring of the affine toric variety vari.

##### 5.3-2 ListOfVariablesOfCoordinateRing

`‣ ListOfVariablesOfCoordinateRing`( vari ) | ( attribute )

Returns: a list

Returns a list containing the variables of the CoordinateRing of the variety vari.

##### 5.3-3 MorphismFromCoordinateRingToCoordinateRingOfTorus

`‣ MorphismFromCoordinateRingToCoordinateRingOfTorus`( vari ) | ( attribute )

Returns: a morphism

Returns the morphism between the coordinate ring of the variety vari and the coordinate ring of its torus. This defines the embedding of the torus in the variety.

##### 5.3-4 ConeOfVariety

`‣ ConeOfVariety`( vari ) | ( attribute )

Returns: a cone

Returns the cone of the affine toric variety vari.

#### 5.4 Methods

##### 5.4-1 CoordinateRing

`‣ CoordinateRing`( vari, indet ) | ( operation )

Returns: a ring

Computes the coordinate ring of the affine toric variety vari with indeterminates indet.

##### 5.4-2 Cone

`‣ Cone`( vari ) | ( operation )

Returns: a cone

Returns the cone of the variety vari. Another name for ConeOfVariety for compatibility and shortness.

#### 5.5 Constructors

The constructors are the same as for toric varieties. Calling them with a cone will result in an affine variety.

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