GAP (ToricVarieties) - Chapter 6: Projective toric varieties

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6 Projective toric varieties
6.1 Projective toric varieties: Examples
6.1-1 P1xP1 created by a polytope
6.1-2 P1xP1 from product of P1s
6.2 The GAP category
6.2-1 IsProjectiveToricVariety
6.3 Attribute
6.3-1 PolytopeOfVariety
6.3-2 AffineCone
6.3-3 ProjectiveEmbedding
6.4 Properties
6.4-1 IsIsomorphicToProjectiveSpace
6.4-2 IsDirectProductOfPNs
6.5 Methods
6.5-1 Polytope
6.5-2 AmpleDivisor
6.6 Constructors

### 6 Projective toric varieties

#### 6.1 Projective toric varieties: Examples

##### 6.1-1 P1xP1 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_]/( x1*x1_-1, x2*x2_-1 )
    gap> IsVeryAmple( Polytope( P1P1 ) );
    true
    gap> ProjectiveEmbedding( P1P1 );
    [ |[ x1_*x2_ ]|, |[ x1_ ]|, |[ x1_*x2 ]|, |[ x2_ ]|,
    |[ 1 ]|, |[ x2 ]|, |[ x1*x2_ ]|, |[ x1 ]|, |[ x1*x2 ]| ]
    gap> Length( ProjectiveEmbedding( P1P1 ) );
    9
    gap> CoxRing( P1P1 );
    Q[x_1,x_2,x_3,x_4]
    (weights: [ ( 0, 1 ), ( 1, 0 ), ( 1, 0 ), ( 0, 1 ) ])
    gap> Display( SRIdeal( P1P1 ) );
    x_1*x_4,
    x_2*x_3

    A (left) ideal generated by the 2 entries of the above matrix

    (graded, degrees of generators: [ ( 0, 2 ), ( 2, 0 ) ])
    gap> Display( IrrelevantIdeal( P1P1 ) );
    x_1*x_2,
    x_1*x_3,
    x_2*x_4,
    x_3*x_4

    A (left) ideal generated by the 4 entries of the above matrix

    (graded, degrees of generators: [ ( 1, 1 ), ( 1, 1 ), ( 1, 1 ), ( 1, 1 ) ])

##### 6.1-2 P1xP1 from product of P1s

    gap> P1 := ProjectiveSpace( 1 );
    <A projective toric variety of dimension 1>
    gap> IsComplete( P1 );
    true
    gap> IsSmooth( P1 );
    true
    gap> Dimension( P1 );
    1
    gap> CoxRing( P1, "q" );
    Q[q_1,q_2]
    (weights: [ 1, 1 ])
    gap> P1xP1 := P1*P1;
    <A projective smooth toric variety of dimension 2 which is a product
    of 2 toric varieties>
    gap> ByASmallerPresentation( ClassGroup( P1xP1 ) );
    <A free left module of rank 2 on free generators>
    gap> CoxRing( P1xP1, "x1,y1,y2,x2" );
    Q[x1,y1,y2,x2]
    (weights: [ ( 0, 1 ), ( 1, 0 ), ( 1, 0 ), ( 0, 1 ) ])
    gap> Display( SRIdeal( P1xP1 ) );
    x1*x2,
    y1*y2

    A (left) ideal generated by the 2 entries of the above matrix

    (graded, degrees of generators: [ ( 0, 2 ), ( 2, 0 ) ])
    gap> Display( IrrelevantIdeal( P1xP1 ) );
    x1*y1,
    x1*y2,
    y1*x2,
    y2*x2

    A (left) ideal generated by the 4 entries of the above matrix

    (graded, degrees of generators: [ ( 1, 1 ), ( 1, 1 ), ( 1, 1 ), ( 1, 1 ) ])

#### 6.2 The GAP category

##### 6.2-1 IsProjectiveToricVariety

`‣ IsProjectiveToricVariety`( M ) | ( filter )

Returns: true or false

The GAP category of a projective toric variety.

#### 6.3 Attribute

##### 6.3-1 PolytopeOfVariety

`‣ PolytopeOfVariety`( vari ) | ( attribute )

Returns: a polytope

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

##### 6.3-2 AffineCone

`‣ AffineCone`( vari ) | ( attribute )

Returns: a cone

Returns the affine cone of the projective toric variety vari.

##### 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 Properties

##### 6.4-1 IsIsomorphicToProjectiveSpace

`‣ IsIsomorphicToProjectiveSpace`( vari ) | ( property )

Returns: true or false

Checks if the given toric variety vari is a projective space.

##### 6.4-2 IsDirectProductOfPNs

`‣ IsDirectProductOfPNs`( vari ) | ( property )

Returns: true or false

Checks if the given toric variety vari is a direct product of projective spaces.

#### 6.5 Methods

##### 6.5-1 Polytope

`‣ Polytope`( vari ) | ( operation )

Returns: a polytope

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

##### 6.5-2 AmpleDivisor

`‣ AmpleDivisor`( vari ) | ( operation )

Returns: an ample divisor

Given a projective toric variety vari constructed from a polytope, this method computes the toric divisor associated to this polytope. By general theory (see Cox-Schenk-Little) this divisor is known to be ample. Thus this method computes an ample divisor on the given toric variety.

#### 6.6 Constructors

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

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