6 Projective toric varieties

6.1 Projective toric varieties: Category and Representations

6.1-1 IsProjectiveToricVariety

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

Returns: a variety

Returns the affine cone of the projective toric variety vari.

6.3-2 PolytopeOfVariety

Returns: a polytope

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

6.3-3 ProjectiveEmbedding

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

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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