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6 Projective toric varieties
 6.1 Projective toric varieties: Examples
 6.2 The GAP category
 6.3 Attribute
 6.4 Properties
 6.5 Methods
 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> 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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