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### 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> 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 );
Q[x_1,x_2,x_3,x_4]
(weights: [ ( 0, 1 ), ( 1, 0 ), ( 1, 0 ), ( 0, 1 ) ])
gap> Display( SRIdeal( P1xP1 ) );
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( P1xP1 ) );
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 ) ])


#### 6.2 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 IsProjectiveSpace
 ‣ IsProjectiveSpace( 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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