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### 5 Affine toric varieties

#### 5.1 Affine toric varieties: Category and Representations

##### 5.1-1 IsAffineToricVariety
 `‣ IsAffineToricVariety`( M ) ( category )

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.

#### 5.2 Affine toric varieties: Properties

Affine toric varieties have no additional properties. Remember that affine toric varieties are toric varieties, so every property of a toric variety is a property of an affine toric variety.

#### 5.3 Affine toric varieties: Attributes

##### 5.3-1 CoordinateRing
 `‣ CoordinateRing`( vari ) ( attribute )

Returns: a ring

Returns the coordinate ring of the affine toric variety vari. The computation is mainly done in ToricIdeals package.

##### 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 ring of the affine toric variety vari.

#### 5.4 Affine toric varieties: Methods

##### 5.4-1 CoordinateRing
 `‣ CoordinateRing`( vari, indet ) ( operation )

Returns: a variety

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 Affine toric varieties: Constructors

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

#### 5.6 Affine toric Varieties: Examples

##### 5.6-1 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_]/( x3*x3_-1, x2*x2_-1, x1*x1_-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"
```
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