  
  [1X6 [33X[0;0YProjective toric varieties[133X[101X
  
  
  [1X6.1 [33X[0;0YProjective toric varieties: Examples[133X[101X
  
  
  [1X6.1-1 [33X[0;0YP1xP1 created by a polytope[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XP1P1 := Polytope( [[1,1],[1,-1],[-1,-1],[-1,1]] );[127X[104X
    [4X[28X<A polytope in |R^2>[128X[104X
    [4X[25Xgap>[125X [27XP1P1 := ToricVariety( P1P1 );[127X[104X
    [4X[28X<A projective toric variety of dimension 2>[128X[104X
    [4X[25Xgap>[125X [27XIsProjective( P1P1 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsComplete( P1P1 );[127X[104X
    [4X[28Xtrue [128X[104X
    [4X[25Xgap>[125X [27XCoordinateRingOfTorus( P1P1, "x" );[127X[104X
    [4X[28XQ[x1,x1_,x2,x2_]/( x1*x1_-1, x2*x2_-1 )[128X[104X
    [4X[25Xgap>[125X [27XIsVeryAmple( Polytope( P1P1 ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XProjectiveEmbedding( P1P1 );[127X[104X
    [4X[28X[ |[ x1_*x2_ ]|, |[ x1_ ]|, |[ x1_*x2 ]|, |[ x2_ ]|,[128X[104X
    [4X[28X|[ 1 ]|, |[ x2 ]|, |[ x1*x2_ ]|, |[ x1 ]|, |[ x1*x2 ]| ][128X[104X
    [4X[25Xgap>[125X [27XLength( ProjectiveEmbedding( P1P1 ) );[127X[104X
    [4X[28X9[128X[104X
    [4X[25Xgap>[125X [27XCoxRing( P1P1 );[127X[104X
    [4X[28XQ[x_1,x_2,x_3,x_4][128X[104X
    [4X[28X(weights: [ ( 0, 1 ), ( 1, 0 ), ( 1, 0 ), ( 0, 1 ) ])[128X[104X
    [4X[25Xgap>[125X [27XDisplay( SRIdeal( P1P1 ) );[127X[104X
    [4X[28Xx_1*x_4,[128X[104X
    [4X[28Xx_2*x_3 [128X[104X
    [4X[28X[128X[104X
    [4X[28XA (left) ideal generated by the 2 entries of the above matrix[128X[104X
    [4X[28X[128X[104X
    [4X[28X(graded, degrees of generators: [ ( 0, 2 ), ( 2, 0 ) ])[128X[104X
    [4X[25Xgap>[125X [27XDisplay( IrrelevantIdeal( P1P1 ) );[127X[104X
    [4X[28Xx_1*x_2,[128X[104X
    [4X[28Xx_1*x_3,[128X[104X
    [4X[28Xx_2*x_4,[128X[104X
    [4X[28Xx_3*x_4 [128X[104X
    [4X[28X[128X[104X
    [4X[28XA (left) ideal generated by the 4 entries of the above matrix[128X[104X
    [4X[28X[128X[104X
    [4X[28X(graded, degrees of generators: [ ( 1, 1 ), ( 1, 1 ), ( 1, 1 ), ( 1, 1 ) ])[128X[104X
  [4X[32X[104X
  
  
  [1X6.1-2 [33X[0;0YP1xP1 from product of P1s[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XP1 := ProjectiveSpace( 1 );[127X[104X
    [4X[28X<A projective toric variety of dimension 1>[128X[104X
    [4X[25Xgap>[125X [27XIsComplete( P1 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSmooth( P1 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDimension( P1 );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XCoxRing( P1, "q" );[127X[104X
    [4X[28XQ[q_1,q_2][128X[104X
    [4X[28X(weights: [ 1, 1 ])[128X[104X
    [4X[25Xgap>[125X [27XP1xP1 := P1*P1;[127X[104X
    [4X[28X<A projective smooth toric variety of dimension 2 which is a product [128X[104X
    [4X[28Xof 2 toric varieties>[128X[104X
    [4X[25Xgap>[125X [27XByASmallerPresentation( ClassGroup( P1xP1 ) );[127X[104X
    [4X[28X<A free left module of rank 2 on free generators>[128X[104X
    [4X[25Xgap>[125X [27XCoxRing( P1xP1, "x1,y1,y2,x2" );[127X[104X
    [4X[28XQ[x1,y1,y2,x2][128X[104X
    [4X[28X(weights: [ ( 0, 1 ), ( 1, 0 ), ( 1, 0 ), ( 0, 1 ) ])[128X[104X
    [4X[25Xgap>[125X [27XDisplay( SRIdeal( P1xP1 ) );[127X[104X
    [4X[28Xx1*x2,[128X[104X
    [4X[28Xy1*y2[128X[104X
    [4X[28X[128X[104X
    [4X[28XA (left) ideal generated by the 2 entries of the above matrix[128X[104X
    [4X[28X[128X[104X
    [4X[28X(graded, degrees of generators: [ ( 0, 2 ), ( 2, 0 ) ])[128X[104X
    [4X[25Xgap>[125X [27XDisplay( IrrelevantIdeal( P1xP1 ) );[127X[104X
    [4X[28Xx1*y1,[128X[104X
    [4X[28Xx1*y2,[128X[104X
    [4X[28Xy1*x2,[128X[104X
    [4X[28Xy2*x2[128X[104X
    [4X[28X[128X[104X
    [4X[28XA (left) ideal generated by the 4 entries of the above matrix[128X[104X
    [4X[28X[128X[104X
    [4X[28X(graded, degrees of generators: [ ( 1, 1 ), ( 1, 1 ), ( 1, 1 ), ( 1, 1 ) ])[128X[104X
  [4X[32X[104X
  
  
  [1X6.2 [33X[0;0YThe GAP category[133X[101X
  
  [1X6.2-1 IsProjectiveToricVariety[101X
  
  [33X[1;0Y[29X[2XIsProjectiveToricVariety[102X( [3XM[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YThe [3XGAP[103X category of a projective toric variety.[133X
  
  
  [1X6.3 [33X[0;0YAttribute[133X[101X
  
  [1X6.3-1 PolytopeOfVariety[101X
  
  [33X[1;0Y[29X[2XPolytopeOfVariety[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya polytope[133X
  
  [33X[0;0YReturns  the polytope corresponding to the projective toric variety [3Xvari[103X, if
  it exists.[133X
  
  [1X6.3-2 AffineCone[101X
  
  [33X[1;0Y[29X[2XAffineCone[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya cone[133X
  
  [33X[0;0YReturns the affine cone of the projective toric variety [3Xvari[103X.[133X
  
  [1X6.3-3 ProjectiveEmbedding[101X
  
  [33X[1;0Y[29X[2XProjectiveEmbedding[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns  characters  for  a  closed embedding in an projective space for the
  projective toric variety [3Xvari[103X.[133X
  
  
  [1X6.4 [33X[0;0YProperties[133X[101X
  
  [1X6.4-1 IsIsomorphicToProjectiveSpace[101X
  
  [33X[1;0Y[29X[2XIsIsomorphicToProjectiveSpace[102X( [3Xvari[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YChecks if the given toric variety [3Xvari[103X is a projective space.[133X
  
  [1X6.4-2 IsDirectProductOfPNs[101X
  
  [33X[1;0Y[29X[2XIsDirectProductOfPNs[102X( [3Xvari[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YChecks  if  the  given  toric variety [3Xvari[103X is a direct product of projective
  spaces.[133X
  
  
  [1X6.5 [33X[0;0YMethods[133X[101X
  
  [1X6.5-1 Polytope[101X
  
  [33X[1;0Y[29X[2XPolytope[102X( [3Xvari[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya polytope[133X
  
  [33X[0;0YReturns the polytope of the variety [3Xvari[103X. Another name for PolytopeOfVariety
  for compatibility and shortness.[133X
  
  [1X6.5-2 AmpleDivisor[101X
  
  [33X[1;0Y[29X[2XAmpleDivisor[102X( [3Xvari[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan ample divisor[133X
  
  [33X[0;0YGiven  a  projective  toric  variety  [3Xvari[103X 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.[133X
  
  
  [1X6.6 [33X[0;0YConstructors[133X[101X
  
  [33X[0;0YThe  constructors  are  the same as for toric varieties. Calling them with a
  polytope will result in a projective variety.[133X
  
