GAP (ToricVarieties) - Chapter 6: Projective toric varieties

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6 Projective toric varieties
6.1 Projective toric varieties: Category and Representations
6.1-1 IsProjectiveToricVariety
6.2 Projective toric varieties: Properties
6.3 Projective toric varieties: Attributes
6.3-1 AffineCone
6.3-2 PolytopeOfVariety
6.3-3 ProjectiveEmbedding
6.4 Projective toric varieties: Methods
6.4-1 Polytope
6.5 Projective toric varieties: Constructors
6.6 Projective toric varieties: Examples
6.6-1 PxP1 created by a polytope

### 6 Projective toric varieties

#### 6.1 Projective toric varieties: Category and Representations

##### 6.1-1 IsProjectiveToricVariety

`‣ IsProjectiveToricVariety`( M ) | ( category )

Returns: `true` or `false`

The GAP category of a projective toric variety.

#### 6.2 Projective toric varieties: Properties

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

#### 6.3 Projective toric varieties: Attributes

##### 6.3-1 AffineCone

`‣ AffineCone`( vari ) | ( attribute )

Returns: a variety

Returns the affine cone of the projective toric variety vari.

##### 6.3-2 PolytopeOfVariety

`‣ PolytopeOfVariety`( vari ) | ( attribute )

Returns: a polytope

Returns the polytope corresponding to the projective toric variety vari, if it exists.

##### 6.3-3 ProjectiveEmbedding

`‣ ProjectiveEmbedding`( vari ) | ( attribute )

Returns: a list

Returns characters for a closed embedding in an projective space for the projective toric variety vari.

#### 6.4 Projective toric varieties: Methods

##### 6.4-1 Polytope

`‣ Polytope`( vari ) | ( operation )

Returns: a polytope

Returns the polytope of the variety vari. Another name for PolytopeOfVariety for compatibility and shortness.

#### 6.5 Projective toric varieties: Constructors

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

#### 6.6 Projective toric varieties: Examples

##### 6.6-1 PxP1 created by a polytope

    gap> P1P1 := Polytope( [[1,1],[1,-1],[-1,-1],[-1,1]] );
    <A polytope in |R^2>
    gap> P1P1 := ToricVariety( P1P1 );
    <A projective toric variety of dimension 2>
    gap> IsProjective( P1P1 );
    true
    gap> IsComplete( P1P1 );
    true
    gap> CoordinateRingOfTorus( P1P1, "x" );
    Q[x1,x1_,x2,x2_]/( x2*x2_-1, x1*x1_-1 )
    gap> IsVeryAmple( Polytope( P1P1 ) );
    true
    gap> ProjectiveEmbedding( P1P1 );
    [ |[ x1_*x2_ ]|, |[ x1_ ]|, |[ x1_*x2 ]|, |[ x2_ ]|,
    |[ 1 ]|, |[ x2 ]|, |[ x1*x2_ ]|, |[ x1 ]|, |[ x1*x2 ]| ]
    gap> Length( last );
    9

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