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 ) ])
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 ) ])
‣ IsProjectiveToricVariety ( M ) | ( filter ) |
Returns: true or false
The GAP category of a projective toric variety.
‣ PolytopeOfVariety ( vari ) | ( attribute ) |
Returns: a polytope
Returns the polytope corresponding to the projective toric variety vari, if it exists.
‣ AffineCone ( vari ) | ( attribute ) |
Returns: a cone
Returns the affine cone of the projective toric variety vari.
‣ ProjectiveEmbedding ( vari ) | ( attribute ) |
Returns: a list
Returns characters for a closed embedding in an projective space for the projective toric variety vari.
‣ IsIsomorphicToProjectiveSpace ( vari ) | ( property ) |
Returns: true or false
Checks if the given toric variety vari is a projective space.
‣ IsDirectProductOfPNs ( vari ) | ( property ) |
Returns: true or false
Checks if the given toric variety vari is a direct product of projective spaces.
‣ Polytope ( vari ) | ( operation ) |
Returns: a polytope
Returns the polytope of the variety vari. Another name for PolytopeOfVariety for compatibility and shortness.
‣ 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.
The constructors are the same as for toric varieties. Calling them with a polytope will result in a projective variety.
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