  
  [1X3 [33X[0;0YToric Varieties[133X[101X
  
  
  [1X3.1 [33X[0;0YToric Varieties: Examples[133X[101X
  
  
  [1X3.1-1 [33X[0;0YThe Hirzebruch surface of index 5[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XH5 := Fan( [[-1,5],[0,1],[1,0],[0,-1]],[[1,2],[2,3],[3,4],[4,1]] );[127X[104X
    [4X[28X<A fan in |R^2>[128X[104X
    [4X[25Xgap>[125X [27XH5 := ToricVariety( H5 );[127X[104X
    [4X[28X<A toric variety of dimension 2>[128X[104X
    [4X[25Xgap>[125X [27XIsComplete( H5 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsAffine( H5 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsOrbifold( H5 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsProjective( H5 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XTorusInvariantPrimeDivisors( H5 );[127X[104X
    [4X[28X[ <A prime divisor of a toric variety with coordinates ( 1, 0, 0, 0 )>,[128X[104X
    [4X[28X  <A prime divisor of a toric variety with coordinates ( 0, 1, 0, 0 )>,[128X[104X
    [4X[28X  <A prime divisor of a toric variety with coordinates ( 0, 0, 1, 0 )>,[128X[104X
    [4X[28X  <A prime divisor of a toric variety with coordinates ( 0, 0, 0, 1 )> ][128X[104X
    [4X[25Xgap>[125X [27XP := TorusInvariantPrimeDivisors( H5 );[127X[104X
    [4X[28X[ <A prime divisor of a toric variety with coordinates ( 1, 0, 0, 0 )>,[128X[104X
    [4X[28X  <A prime divisor of a toric variety with coordinates ( 0, 1, 0, 0 )>,[128X[104X
    [4X[28X  <A prime divisor of a toric variety with coordinates ( 0, 0, 1, 0 )>,[128X[104X
    [4X[28X  <A prime divisor of a toric variety with coordinates ( 0, 0, 0, 1 )> ][128X[104X
    [4X[25Xgap>[125X [27XA := P[ 1 ] - P[ 2 ] + 4*P[ 3 ];[127X[104X
    [4X[28X<A divisor of a toric variety with coordinates ( 1, -1, 4, 0 )>[128X[104X
    [4X[25Xgap>[125X [27XA;[127X[104X
    [4X[28X<A divisor of a toric variety with coordinates ( 1, -1, 4, 0 )>[128X[104X
    [4X[25Xgap>[125X [27XIsAmple( A );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XCoordinateRingOfTorus( H5,"x" );[127X[104X
    [4X[28XQ[x1,x1_,x2,x2_]/( x1*x1_-1, x2*x2_-1 )[128X[104X
    [4X[25Xgap>[125X [27XD:=CreateDivisor( [ 0,0,0,0 ],H5 );[127X[104X
    [4X[28X<A divisor of a toric variety with coordinates 0>[128X[104X
    [4X[25Xgap>[125X [27XBasisOfGlobalSections( D );[127X[104X
    [4X[28X[ |[ 1 ]| ][128X[104X
    [4X[25Xgap>[125X [27XD:=Sum( P );[127X[104X
    [4X[28X<A divisor of a toric variety with coordinates ( 1, 1, 1, 1 )>[128X[104X
    [4X[25Xgap>[125X [27XBasisOfGlobalSections(D);[127X[104X
    [4X[28X[ |[ x1_ ]|, |[ x1_*x2 ]|, |[ 1 ]|, |[ x2 ]|,[128X[104X
    [4X[28X  |[ x1 ]|, |[ x1*x2 ]|, |[ x1^2*x2 ]|, [128X[104X
    [4X[28X  |[ x1^3*x2 ]|, |[ x1^4*x2 ]|, |[ x1^5*x2 ]|, [128X[104X
    [4X[28X  |[ x1^6*x2 ]| ][128X[104X
    [4X[25Xgap>[125X [27Xdivi := DivisorOfCharacter( [ 1,2 ],H5 );[127X[104X
    [4X[28X<A principal divisor of a toric variety with coordinates ( 9, -2, 2, 1 )>[128X[104X
    [4X[25Xgap>[125X [27XBasisOfGlobalSections( divi );[127X[104X
    [4X[28X[ |[ x1_*x2_^2 ]| ][128X[104X
  [4X[32X[104X
  
  
  [1X3.1-2 [33X[0;0YA smooth, complete toric variety which is not projective[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrays := [ [1,0,0], [-1,0,0], [0,1,0], [0,-1,0], [0,0,1], [0,0,-1],[127X[104X
    [4X[25X>[125X [27X          [2,1,1], [1,2,1], [1,1,2], [1,1,1] ];[127X[104X
    [4X[28X[ [ 1, 0, 0 ], [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, -1, 0 ], [ 0, 0, 1 ], [ 0, 0, -1 ], [128X[104X
    [4X[28X[ 2, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 2 ], [ 1, 1, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xcones := [ [1,3,6], [1,4,6], [1,4,5], [2,3,6], [2,4,6], [2,3,5], [2,4,5],[127X[104X
    [4X[25X>[125X [27X           [1,5,9], [3,5,8], [1,3,7], [1,7,9], [5,8,9], [3,7,8],[127X[104X
    [4X[25X>[125X [27X           [7,9,10], [8,9,10], [7,8,10] ];[127X[104X
    [4X[28X[ [ 1, 3, 6 ], [ 1, 4, 6 ], [ 1, 4, 5 ], [ 2, 3, 6 ], [ 2, 4, 6 ], [ 2, 3, 5 ],[128X[104X
    [4X[28X  [ 2, 4, 5 ], [ 1, 5, 9 ], [ 3, 5, 8 ], [ 1, 3, 7 ], [ 1, 7, 9 ], [ 5, 8, 9 ], [128X[104X
    [4X[28X  [ 3, 7, 8 ], [ 7, 9, 10 ], [ 8, 9, 10 ], [ 7, 8, 10 ] ][128X[104X
    [4X[25Xgap>[125X [27XF := Fan( rays, cones );[127X[104X
    [4X[28X<A fan in |R^3>[128X[104X
    [4X[25Xgap>[125X [27XT := ToricVariety( F );[127X[104X
    [4X[28X<A toric variety of dimension 3>[128X[104X
    [4X[25Xgap>[125X [27X[ IsSmooth( T ), IsComplete( T ), IsProjective( T ) ];[127X[104X
    [4X[28X[ true, true, false ][128X[104X
    [4X[25Xgap>[125X [27XSRIdeal( T );[127X[104X
    [4X[28X<A graded torsion-free (left) ideal given by 23 generators>[128X[104X
  [4X[32X[104X
  
  
  [1X3.1-3 [33X[0;0YConvenient construction of toric varieties[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrays := [ [1,0],[-1,0],[0,1],[0,-1] ];[127X[104X
    [4X[28X[ [ 1, 0 ], [ -1, 0 ], [ 0, 1 ], [ 0, -1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xcones := [ [1,3],[1,4],[2,3],[2,4] ];[127X[104X
    [4X[28X[ [1,3],[1,4],[2,3],[2,4] ][128X[104X
    [4X[25Xgap>[125X [27Xweights := [ [1,0],[1,0],[0,1],[0,1] ];[127X[104X
    [4X[28X[ [1,0],[1,0],[0,1],[0,1] ][128X[104X
    [4X[25Xgap>[125X [27Xweights2 := [ [1,1],[1,1],[1,2],[1,2] ];[127X[104X
    [4X[28X[ [1,1],[1,1],[1,2],[1,2] ][128X[104X
    [4X[25Xgap>[125X [27Xtor1 := ToricVariety( rays, cones, weights, "x1,x2,y1,y2" );[127X[104X
    [4X[28X<A toric variety of dimension 2>[128X[104X
    [4X[25Xgap>[125X [27XCoxRing( tor1 );[127X[104X
    [4X[28XQ[x2,y2,y1,x1][128X[104X
    [4X[28X(weights: [ ( 1, 0 ), ( 0, 1 ), ( 0, 1 ), ( 1, 0 ) ])[128X[104X
    [4X[25Xgap>[125X [27Xtor2:= ToricVariety( rays, cones, weights, "q" );[127X[104X
    [4X[28X<A toric variety of dimension 2>[128X[104X
    [4X[25Xgap>[125X [27XCoxRing( tor2 );[127X[104X
    [4X[28XQ[q_2,q_4,q_3,q_1][128X[104X
    [4X[28X(weights: [ ( 1, 0 ), ( 0, 1 ), ( 0, 1 ), ( 1, 0 ) ])[128X[104X
    [4X[25Xgap>[125X [27Xtor3:= ToricVariety( rays, cones, weights );[127X[104X
    [4X[28X<A toric variety of dimension 2>[128X[104X
    [4X[25Xgap>[125X [27XCoxRing( tor3 );[127X[104X
    [4X[28XQ[x_2,x_4,x_3,x_1][128X[104X
    [4X[28X(weights: [ ( 1, 0 ), ( 0, 1 ), ( 0, 1 ), ( 1, 0 ) ])[128X[104X
    [4X[25Xgap>[125X [27Xtor4:= ToricVariety( rays, cones, weights2, "x1,x2,z1,z2" );[127X[104X
    [4X[28X<A toric variety of dimension 2>[128X[104X
    [4X[25Xgap>[125X [27XCoxRing( tor4 );[127X[104X
    [4X[28XQ[x2,z2,z1,x1][128X[104X
    [4X[28X(weights: [ ( 1, 1 ), ( 1, 2 ), ( 1, 2 ), ( 1, 1 ) ])[128X[104X
  [4X[32X[104X
  
  
  [1X3.1-4 [33X[0;0YToric varieties from gradings[133X[101X
  
  [33X[0;0YThe  following example shows how to create the projective space [23X\mathbb{P}^2[123X
  from  the grading of its Cox ring. Note that this functionality requires the
  package TopcomInterface.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg := [[1,1,1]];[127X[104X
    [4X[28X[ [ 1,1,1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xv1 := ToricVarietyFromGrading( g );[127X[104X
    [4X[28X<A toric variety of dimension 2>[128X[104X
    [4X[25Xgap>[125X [27XCoxRing( v1 );[127X[104X
    [4X[28XQ[x_1,x_2,x_3][128X[104X
    [4X[28X(weights: [ 1, 1, 1 ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  following example shows how to create the resolved conifold(s) from the
  grading of its Cox ring.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg2 := [[1,1,-1,-1]];[127X[104X
    [4X[28X[ [ 1,1,-1,-1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xv2 := ToricVarietiesFromGrading( g2 );[127X[104X
    [4X[28X[ <A toric variety of dimension 3>, <A toric variety of dimension 3> ][128X[104X
    [4X[25Xgap>[125X [27XCoxRing( v2[ 1 ] );[127X[104X
    [4X[28XQ[x_1,x_2,x_3,x_4][128X[104X
    [4X[28X(weights: [ 1, -1, -1, 1 ])[128X[104X
    [4X[25Xgap>[125X [27XDisplay( SRIdeal( v2[ 1 ] ) );[127X[104X
    [4X[28Xx_2*x_3[128X[104X
    [4X[28X[128X[104X
    [4X[28XA (left) ideal generated by the entry of the above matrix[128X[104X
    [4X[28X[128X[104X
    [4X[28X(graded, degree of generator: -2)[128X[104X
    [4X[25Xgap>[125X [27XDisplay( SRIdeal( v2[ 2 ] ) );[127X[104X
    [4X[28Xx_1*x_4[128X[104X
    [4X[28X[128X[104X
    [4X[28XA (left) ideal generated by the entry of the above matrix[128X[104X
    [4X[28X[128X[104X
    [4X[28X(graded, degree of generator: 2)[128X[104X
  [4X[32X[104X
  
  
  [1X3.1-5 [33X[0;0YBlowups of toric varieties by star subdivisions of fans[133X[101X
  
  [33X[0;0YThe following code exemplifies blowups of the 3-dimensional affine space.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrays := [ [1,0,0], [0,1,0], [0,0,1] ];[127X[104X
    [4X[28X[ [1,0,0], [0,1,0], [0,0,1] ][128X[104X
    [4X[25Xgap>[125X [27Xmax_cones := [ [1,2,3] ];[127X[104X
    [4X[28X[ [1,2,3] ][128X[104X
    [4X[25Xgap>[125X [27Xfan := Fan( rays, max_cones );[127X[104X
    [4X[28X<A fan in |R^3>[128X[104X
    [4X[25Xgap>[125X [27XC3 := ToricVariety( rays, max_cones, [[0],[0],[0]], "x1,x2,x3" );[127X[104X
    [4X[28X<A toric variety of dimension 3>[128X[104X
    [4X[25Xgap>[125X [27XB1C3 := BlowupOfToricVariety( C3, "x1,x2,x3", "u0" );[127X[104X
    [4X[28X<A toric variety of dimension 3>[128X[104X
    [4X[25Xgap>[125X [27X[ IsComplete( B1C3 ), IsOrbifold( B1C3 ), IsSmooth( B1C3 ) ];[127X[104X
    [4X[28X[ false, true, true ][128X[104X
    [4X[25Xgap>[125X [27XB2C3 := BlowupOfToricVariety( B1C3, "x1,u0", "u1" );[127X[104X
    [4X[28X<A toric variety of dimension 3>[128X[104X
    [4X[25Xgap>[125X [27XRank( ClassGroup( B2C3 ) );[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XB3C3 := BlowupOfToricVariety( B2C3, "x1,u1", "u2" );[127X[104X
    [4X[28X<A toric variety of dimension 3>[128X[104X
    [4X[25Xgap>[125X [27XCoxRing( B3C3 );[127X[104X
    [4X[28XQ[x3,x2,x1,u0,u1,u2][128X[104X
    [4X[28X(weights: [ ( 0, 1, 0, 0 ), ( 0, 1, 0, 0 ), ( 0, 1, 1, 1 ), [128X[104X
    [4X[28X( 0, -1, 1, 0 ), ( 0, 0, -1, 1 ), ( 0, 0, 0, -1 ) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YLikewise, we can also perform blowups of the 3-dimensional projective space.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrays := [ [1,0,0], [0,1,0], [0,0,1], [-1,-1,-1] ];[127X[104X
    [4X[28X[ [1,0,0], [0,1,0], [0,0,1], [-1,-1,-1] ][128X[104X
    [4X[25Xgap>[125X [27Xmax_cones := [ [1,2,3], [1,2,4], [1,3,4], [2,3,4] ];[127X[104X
    [4X[28X[ [1,2,3], [1,2,4], [1,3,4], [2,3,4] ][128X[104X
    [4X[25Xgap>[125X [27Xfan := Fan( rays, max_cones );[127X[104X
    [4X[28X<A fan in |R^3>[128X[104X
    [4X[25Xgap>[125X [27XP3 := ToricVariety( rays, max_cones, [[1],[1],[1],[1]], "x1,x2,x3,x4" );[127X[104X
    [4X[28X<A toric variety of dimension 3>[128X[104X
    [4X[25Xgap>[125X [27XB1P3 := BlowupOfToricVariety( P3, "x1,x2,x3", "u0" );[127X[104X
    [4X[28X<A toric variety of dimension 3>[128X[104X
    [4X[25Xgap>[125X [27X[ IsComplete( B1P3 ), IsOrbifold( B1P3 ), IsSmooth( B1P3 ) ];[127X[104X
    [4X[28X[ true, true, true ][128X[104X
    [4X[25Xgap>[125X [27XB2P3 := BlowupOfToricVariety( B1P3, "x1,u0", "u1" );[127X[104X
    [4X[28X<A toric variety of dimension 3>[128X[104X
    [4X[25Xgap>[125X [27XRank( ClassGroup( B2C3 ) );[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XB3P3 := BlowupOfToricVariety( B2P3, "x1,u1", "u2" );[127X[104X
    [4X[28X<A toric variety of dimension 3>[128X[104X
    [4X[25Xgap>[125X [27XCoxRing( B3P3 );[127X[104X
    [4X[28XQ[x4,x3,x2,x1,u0,u1,u2][128X[104X
    [4X[28X(weights: [ ( 1, 0, 0, 0 ), ( 1, 1, 0, 0 ), ( 1, 1, 0, 0 ), [128X[104X
    [4X[28X( 1, 1, 1, 1 ), ( 0, -1, 1, 0 ), ( 0, 0, -1, 1 ), ( 0, 0, 0, -1 ) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlso, we can perform blowups of a generalized Hirzebruch 3-fold.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xvars := "u,s,v,t,r";[127X[104X
    [4X[28X"u,s,v,t,r"[128X[104X
    [4X[25Xgap>[125X [27Xrays := [ [0,0,-1],[1,0,0],[0,1,0],[-1,-1,-17],[0,0,1] ];[127X[104X
    [4X[28X[ [0,0,-1],[1,0,0],[0,1,0],[-1,-1,-17],[0,0,1] ][128X[104X
    [4X[25Xgap>[125X [27Xcones := [ [1,2,3], [1,2,4], [1,3,4], [2,3,5], [2,4,5], [3,4,5] ];[127X[104X
    [4X[28X[ [1,2,3], [1,2,4], [1,3,4], [2,3,5], [2,4,5], [3,4,5] ][128X[104X
    [4X[25Xgap>[125X [27Xweights := [ [1,-17], [0,1], [0,1], [0,1], [1,0] ];[127X[104X
    [4X[28X[ [1,-17], [0,1], [0,1], [0,1], [1,0] ][128X[104X
    [4X[25Xgap>[125X [27XH3fold := ToricVariety( rays, cones, weights, vars );[127X[104X
    [4X[28X<A toric variety of dimension 3>[128X[104X
    [4X[25Xgap>[125X [27XB1H3fold := BlowupOfToricVariety( H3fold, "u,s", "u1" );[127X[104X
    [4X[28X<A toric variety of dimension 3>[128X[104X
    [4X[25Xgap>[125X [27XCoxRing( B1H3fold );[127X[104X
    [4X[28XQ[t,u,r,v,u1,s][128X[104X
    [4X[28X(weights: [ ( 0, 1, 0 ), ( 1, -17, 1 ), ( 1, 0, 0 ), [128X[104X
    [4X[28X( 0, 1, 0 ), ( 0, 0, -1 ), ( 0, 1, 1 ) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis example easily extends to an entire sequence of blowups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xvars := "u,s,v,t,r,x,y,w";[127X[104X
    [4X[28X"u,s,v,t,r,x,y,w"[128X[104X
    [4X[25Xgap>[125X [27Xrays := [ [0,0,-1,-2,-3], [1,0,0,-2,-3], [0,1,0,-2,-3], [-1,-1,-17,-2,-3], [127X[104X
    [4X[25X>[125X [27X          [0,0,1,-2,-3], [0, 0, 0, 1, 0], [127X[104X
    [4X[25X>[125X [27X[0, 0, 0, 0, 1], [0, 0, 0, -2, -3] ];[127X[104X
    [4X[28X[ [0,0,-1,-2,-3], [1,0,0,-2,-3], [0,1,0,-2,-3], [-1,-1,-17,-2,-3], [128X[104X
    [4X[28X[0,0,1,-2,-3], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [0, 0, 0, -2, -3] ][128X[104X
    [4X[25Xgap>[125X [27Xcones := [ [1,2,3,6,7], [1,2,3,6,8], [1,2,3,7,8], [1,2,4,6,7], [1,2,4,6,8],[127X[104X
    [4X[25X>[125X [27X           [1,2,4,7,8], [1,3,4,6,7], [1,3,4,6,8], [1,3,4,7,8], [2,3,5,6,7],[127X[104X
    [4X[25X>[125X [27X           [2,3,5,6,8], [2,3,5,7,8], [2,4,5,6,7], [2,4,5,6,8], [2,4,5,7,8],[127X[104X
    [4X[25X>[125X [27X           [3,4,5,6,7], [3,4,5,6,8], [3,4,5,7,8] ];[127X[104X
    [4X[28X[ [ 1, 2, 3, 6, 7 ], [ 1, 2, 3, 6, 8 ], [ 1, 2, 3, 7, 8 ],[128X[104X
    [4X[28X  [ 1, 2, 4, 6, 7 ], [ 1, 2, 4, 6, 8 ], [ 1, 2, 4, 7, 8 ],[128X[104X
    [4X[28X  [ 1, 3, 4, 6, 7 ], [ 1, 3, 4, 6, 8 ], [ 1, 3, 4, 7, 8 ],[128X[104X
    [4X[28X  [ 2, 3, 5, 6, 7 ], [ 2, 3, 5, 6, 8 ], [ 2, 3, 5, 7, 8 ],[128X[104X
    [4X[28X  [ 2, 4, 5, 6, 7 ], [ 2, 4, 5, 6, 8 ], [ 2, 4, 5, 7, 8 ],[128X[104X
    [4X[28X  [ 3, 4, 5, 6, 7 ], [ 3, 4, 5, 6, 8 ], [ 3, 4, 5, 7, 8 ] ][128X[104X
    [4X[25Xgap>[125X [27Xw := [ [1,-17,0], [0,1,0], [0,1,0], [0,1,0], [1,0,0], [0,0,2], [0,0,3], [127X[104X
    [4X[25X>[125X [27X       [-2,14,1] ];[127X[104X
    [4X[28X[ [1,-17,0], [0,1,0], [0,1,0], [0,1,0], [1,0,0], [0,0,2], [0,0,3], [-2,14,1] ][128X[104X
    [4X[25Xgap>[125X [27Xbase := ToricVariety( rays, cones, w, vars );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xb1 := BlowupOfToricVariety( base, "x,y,u", "u1" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xb2 := BlowupOfToricVariety( b1, "x,y,u1", "u2" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xb3 := BlowupOfToricVariety( b2, "y,u1", "u3" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xb4 := BlowupOfToricVariety( b3, "y,u2", "u4" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xb5 := BlowupOfToricVariety( b4, "u2,u3", "u5" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xb6 := BlowupOfToricVariety( b5, "u1,u3", "u6" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xb7 := BlowupOfToricVariety( b6, "u2,u4", "u7" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xb8 := BlowupOfToricVariety( b7, "u3,u4", "u8" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xb9 := BlowupOfToricVariety( b8, "u4,u5", "u9" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xb10 := BlowupOfToricVariety( b9, "u5,u8", "u10" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xb11 := BlowupOfToricVariety( b10, "u4,u8", "u11" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xb12 := BlowupOfToricVariety( b11, "u4,u9", "u12" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xb13 := BlowupOfToricVariety( b12, "u8,u9", "u13" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xb14 := BlowupOfToricVariety( b13, "u9,u11", "u14" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xb15 := BlowupOfToricVariety( b14, "u4,v", "d" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27Xfinal_space := BlowupOfToricVariety( b15, "u3,u5", "u15" );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis sequence of blowups can also be performed with a single command.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfinal_space2 := SequenceOfBlowupsOfToricVariety( base, [127X[104X
    [4X[25X>[125X [27X                    [ ["x,y,u","u1"], [127X[104X
    [4X[25X>[125X [27X                    ["x,y,u1","u2"],[127X[104X
    [4X[25X>[125X [27X                    ["y,u1","u3"],[127X[104X
    [4X[25X>[125X [27X                    ["y,u2","u4"],[127X[104X
    [4X[25X>[125X [27X                    ["u2,u3","u5"],[127X[104X
    [4X[25X>[125X [27X                    ["u1,u3","u6"],[127X[104X
    [4X[25X>[125X [27X                    ["u2,u4","u7"],[127X[104X
    [4X[25X>[125X [27X                    ["u3,u4","u8"],[127X[104X
    [4X[25X>[125X [27X                    ["u4,u5","u9"],[127X[104X
    [4X[25X>[125X [27X                    ["u5,u8","u10"],[127X[104X
    [4X[25X>[125X [27X                    ["u4,u8","u11"],[127X[104X
    [4X[25X>[125X [27X                    ["u4,u9","u12"],[127X[104X
    [4X[25X>[125X [27X                    ["u8,u9","u13"],[127X[104X
    [4X[25X>[125X [27X                    ["u9,u11","u14"],[127X[104X
    [4X[25X>[125X [27X                    ["u4,v","d"],[127X[104X
    [4X[25X>[125X [27X                    ["u3,u5","u15"] ] );[127X[104X
    [4X[28X<A toric variety of dimension 5>[128X[104X
    [4X[25Xgap>[125X [27X[ IsComplete( final_space2 ), IsOrbifold( final_space2 ), [127X[104X
    [4X[25X>[125X [27X  IsSmooth( final_space2 ) ];[127X[104X
    [4X[28X[ true, true, false ][128X[104X
  [4X[32X[104X
  
  
  [1X3.2 [33X[0;0YToric variety: Category and Representations[133X[101X
  
  [1X3.2-1 IsToricVariety[101X
  
  [33X[1;0Y[29X[2XIsToricVariety[102X( [3XM[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YChecks if an object is a toric variety.[133X
  
  [1X3.2-2 IsCategoryOfToricVarieties[101X
  
  [33X[1;0Y[29X[2XIsCategoryOfToricVarieties[102X( [3Xobject[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe [3XGAP[103X category of toric varieties.[133X
  
  [1X3.2-3 twitter[101X
  
  [33X[1;0Y[29X[2Xtwitter[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya ring[133X
  
  [33X[0;0YThis  is  a dummy to get immediate methods triggered at some times. It never
  has a value.[133X
  
  
  [1X3.3 [33X[0;0YProperties[133X[101X
  
  [1X3.3-1 IsNormalVariety[101X
  
  [33X[1;0Y[29X[2XIsNormalVariety[102X( [3Xvari[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YChecks if the toric variety [3Xvari[103X is a normal variety.[133X
  
  [1X3.3-2 IsAffine[101X
  
  [33X[1;0Y[29X[2XIsAffine[102X( [3Xvari[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YChecks if the toric variety [3Xvari[103X is an affine variety.[133X
  
  [1X3.3-3 IsProjective[101X
  
  [33X[1;0Y[29X[2XIsProjective[102X( [3Xvari[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YChecks if the toric variety [3Xvari[103X is a projective variety.[133X
  
  [1X3.3-4 IsSmooth[101X
  
  [33X[1;0Y[29X[2XIsSmooth[102X( [3Xvari[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YChecks if the toric variety [3Xvari[103X is smooth.[133X
  
  [1X3.3-5 IsComplete[101X
  
  [33X[1;0Y[29X[2XIsComplete[102X( [3Xvari[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YChecks if the toric variety [3Xvari[103X is complete.[133X
  
  [1X3.3-6 HasTorusfactor[101X
  
  [33X[1;0Y[29X[2XHasTorusfactor[102X( [3Xvari[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YChecks if the toric variety [3Xvari[103X has a torus factor.[133X
  
  [1X3.3-7 HasNoTorusfactor[101X
  
  [33X[1;0Y[29X[2XHasNoTorusfactor[102X( [3Xvari[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YChecks if the toric variety [3Xvari[103X has no torus factor.[133X
  
  [1X3.3-8 IsOrbifold[101X
  
  [33X[1;0Y[29X[2XIsOrbifold[102X( [3Xvari[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YChecks  if  the  toric  variety [3Xvari[103X has an orbifold, which is, in the toric
  case, equivalent to the simpliciality of the fan.[133X
  
  [1X3.3-9 IsSimplicial[101X
  
  [33X[1;0Y[29X[2XIsSimplicial[102X( [3Xvari[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YChecks if the toric variety [3Xvari[103X is simplicial. This is a convenience method
  equivalent to IsOrbifold.[133X
  
  
  [1X3.4 [33X[0;0YAttributes[133X[101X
  
  [1X3.4-1 AffineOpenCovering[101X
  
  [33X[1;0Y[29X[2XAffineOpenCovering[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns  a  torus  invariant  affine  open covering of the variety [3Xvari[103X. The
  affine open cover is computed out of the cones of the fan.[133X
  
  [1X3.4-2 CoxRing[101X
  
  [33X[1;0Y[29X[2XCoxRing[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya ring[133X
  
  [33X[0;0YReturns  the  Cox  ring  of  the  variety [3Xvari[103X. The actual method requires a
  string  with  a  name for the variables. A method for computing the Cox ring
  without a variable given is not implemented. You will get an error.[133X
  
  [1X3.4-3 ListOfVariablesOfCoxRing[101X
  
  [33X[1;0Y[29X[2XListOfVariablesOfCoxRing[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns a list of the variables of the cox ring of the variety [3Xvari[103X.[133X
  
  [1X3.4-4 ClassGroup[101X
  
  [33X[1;0Y[29X[2XClassGroup[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya module[133X
  
  [33X[0;0YReturns the class group of the variety [3Xvari[103X as factor of a free module.[133X
  
  [1X3.4-5 TorusInvariantDivisorGroup[101X
  
  [33X[1;0Y[29X[2XTorusInvariantDivisorGroup[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya module[133X
  
  [33X[0;0YReturns the subgroup of the Weil divisor group of the variety [3Xvari[103X generated
  by  the  torus invariant prime divisors. This is always a finitely generated
  free module over the integers.[133X
  
  [1X3.4-6 MapFromCharacterToPrincipalDivisor[101X
  
  [33X[1;0Y[29X[2XMapFromCharacterToPrincipalDivisor[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism[133X
  
  [33X[0;0YReturns  a  map  which maps an element of the character group into the torus
  invariant  Weil  group  of the variety [3Xvari[103X. This has to be viewed as a help
  method to compute divisor classes.[133X
  
  [1X3.4-7 MapFromWeilDivisorsToClassGroup[101X
  
  [33X[1;0Y[29X[2XMapFromWeilDivisorsToClassGroup[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism[133X
  
  [33X[0;0YReturns a map which maps a Weil divisor into the class group.[133X
  
  [1X3.4-8 Dimension[101X
  
  [33X[1;0Y[29X[2XDimension[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YReturns the dimension of the variety [3Xvari[103X.[133X
  
  [1X3.4-9 DimensionOfTorusfactor[101X
  
  [33X[1;0Y[29X[2XDimensionOfTorusfactor[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YReturns the dimension of the torus factor of the variety [3Xvari[103X.[133X
  
  [1X3.4-10 CoordinateRingOfTorus[101X
  
  [33X[1;0Y[29X[2XCoordinateRingOfTorus[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya ring[133X
  
  [33X[0;0YReturns  the  coordinate  ring  of the torus of the variety [3Xvari[103X. This is by
  default  done  with  the  variables [3Xx1[103X to [3Xxn[103X where [3Xn[103X is the dimension of the
  variety.  To  use  a  different  set  of  variables, a convenience method is
  provided and described in the [3Xmethods[103X section.[133X
  
  [1X3.4-11 ListOfVariablesOfCoordinateRingOfTorus[101X
  
  [33X[1;0Y[29X[2XListOfVariablesOfCoordinateRingOfTorus[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns  the  list  of  variables in the coordinate ring of the torus of the
  variety [3Xvari[103X.[133X
  
  [1X3.4-12 IsProductOf[101X
  
  [33X[1;0Y[29X[2XIsProductOf[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YIf  the  variety [3Xvari[103X is a product of 2 or more varieties, the list contains
  those  varieties.  If  it  is  not  a product or at least not generated as a
  product, the list only contains the variety itself.[133X
  
  [1X3.4-13 CharacterLattice[101X
  
  [33X[1;0Y[29X[2XCharacterLattice[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya module[133X
  
  [33X[0;0YThe  method  returns  the character lattice of the variety [3Xvari[103X, computed as
  the containing grid of the underlying convex object, if it exists.[133X
  
  [1X3.4-14 TorusInvariantPrimeDivisors[101X
  
  [33X[1;0Y[29X[2XTorusInvariantPrimeDivisors[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YThe  method  returns  a  list  of  the torus invariant prime divisors of the
  variety [3Xvari[103X.[133X
  
  [1X3.4-15 IrrelevantIdeal[101X
  
  [33X[1;0Y[29X[2XIrrelevantIdeal[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan ideal[133X
  
  [33X[0;0YReturns the irrelevant ideal of the Cox ring of the variety [3Xvari[103X.[133X
  
  [1X3.4-16 SRIdeal[101X
  
  [33X[1;0Y[29X[2XSRIdeal[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan ideal[133X
  
  [33X[0;0YReturns the Stanley-Reißner ideal of the Cox ring of the variety [3Xvari[103X.[133X
  
  [1X3.4-17 MorphismFromCoxVariety[101X
  
  [33X[1;0Y[29X[2XMorphismFromCoxVariety[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism[133X
  
  [33X[0;0YThe method returns the quotient morphism from the variety of the Cox ring to
  the variety [3Xvari[103X.[133X
  
  [1X3.4-18 CoxVariety[101X
  
  [33X[1;0Y[29X[2XCoxVariety[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya variety[133X
  
  [33X[0;0YThe method returns the Cox variety of the variety [3Xvari[103X.[133X
  
  [1X3.4-19 FanOfVariety[101X
  
  [33X[1;0Y[29X[2XFanOfVariety[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya fan[133X
  
  [33X[0;0YReturns the fan of the variety [3Xvari[103X. This is set by default.[133X
  
  [1X3.4-20 CartierTorusInvariantDivisorGroup[101X
  
  [33X[1;0Y[29X[2XCartierTorusInvariantDivisorGroup[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya module[133X
  
  [33X[0;0YReturns  the the group of Cartier divisors of the variety [3Xvari[103X as a subgroup
  of the divisor group.[133X
  
  [1X3.4-21 PicardGroup[101X
  
  [33X[1;0Y[29X[2XPicardGroup[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya module[133X
  
  [33X[0;0YReturns the Picard group of the variety [3Xvari[103X as factor of a free module.[133X
  
  [1X3.4-22 NameOfVariety[101X
  
  [33X[1;0Y[29X[2XNameOfVariety[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya string[133X
  
  [33X[0;0YReturns the name of the variety [3Xvari[103X if it has one and it is known or can be
  computed.[133X
  
  [1X3.4-23 ZariskiCotangentSheaf[101X
  
  [33X[1;0Y[29X[2XZariskiCotangentSheaf[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya f.p. graded [3XS[103X-module[133X
  
  [33X[0;0YThis  method  returns  a  f. p. graded [3XS[103X-module ([3XS[103X being the Cox ring of the
  variety),  such  that  the  sheafification  of  this  module  is the Zariski
  cotangent sheaf of [3Xvari[103X.[133X
  
  [1X3.4-24 CotangentSheaf[101X
  
  [33X[1;0Y[29X[2XCotangentSheaf[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya f.p. graded [3XS[103X-module[133X
  
  [33X[0;0YThis  method  returns  a  f. p. graded [3XS[103X-module ([3XS[103X being the Cox ring of the
  variety), such that the sheafification of this module is the cotangent sheaf
  of [3Xvari[103X.[133X
  
  [1X3.4-25 EulerCharacteristic[101X
  
  [33X[1;0Y[29X[2XEulerCharacteristic[102X( [3Xvari[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya non-negative integer[133X
  
  [33X[0;0YThis method computes the Euler characteristic of [3Xvari[103X.[133X
  
  
  [1X3.5 [33X[0;0YMethods[133X[101X
  
  [1X3.5-1 UnderlyingSheaf[101X
  
  [33X[1;0Y[29X[2XUnderlyingSheaf[102X( [3Xvari[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya sheaf[133X
  
  [33X[0;0YThe method returns the underlying sheaf of the variety [3Xvari[103X.[133X
  
  [1X3.5-2 CoordinateRingOfTorus[101X
  
  [33X[1;0Y[29X[2XCoordinateRingOfTorus[102X( [3Xvari[103X, [3Xvars[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya ring[133X
  
  [33X[0;0YComputes  the  coordinate  ring  of  the  torus of the variety [3Xvari[103X with the
  variables  [3Xvars[103X.  The argument [3Xvars[103X need to be a list of strings with length
  dimension or two times dimension.[133X
  
  [1X3.5-3 \*[101X
  
  [33X[1;0Y[29X[2X\*[102X( [3Xvari1[103X, [3Xvari2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya variety[133X
  
  [33X[0;0YComputes the categorial product of the varieties [3Xvari1[103X and [3Xvari2[103X.[133X
  
  [1X3.5-4 CharacterToRationalFunction[101X
  
  [33X[1;0Y[29X[2XCharacterToRationalFunction[102X( [3Xelem[103X, [3Xvari[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya homalg element[133X
  
  [33X[0;0YComputes  the  rational function corresponding to the character grid element
  [3Xelem[103X  or  to  the  list of integers [3Xelem[103X. This computation needs to know the
  coordinate  ring  of  the torus of the variety [3Xvari[103X. By default this ring is
  introduced  with variables [3Xx1[103X to [3Xxn[103X where [3Xn[103X is the dimension of the variety.
  If  different variables should be used, then [3XCoordinateRingOfTorus[103X has to be
  set accordingly before calling this method.[133X
  
  [1X3.5-5 CoxRing[101X
  
  [33X[1;0Y[29X[2XCoxRing[102X( [3Xvari[103X, [3Xvars[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya ring[133X
  
  [33X[0;0YComputes  the  Cox  ring  of the variety [3Xvari[103X. [3Xvars[103X needs to be a string. We
  allow  for  two  different  formats.  Either,  it is a string which does not
  contain  ",".  Then  this string will be index and the resulting strings are
  then used as names for the variables of the Cox ring. Alternatively, one can
  also  use  a  string  containing  ",".  In this case, a "," is considered as
  separator  and one can provide individual names for all variables of the Cox
  ring.[133X
  
  [1X3.5-6 WeilDivisorsOfVariety[101X
  
  [33X[1;0Y[29X[2XWeilDivisorsOfVariety[102X( [3Xvari[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns a list of the currently defined Divisors of the toric variety.[133X
  
  [1X3.5-7 Fan[101X
  
  [33X[1;0Y[29X[2XFan[102X( [3Xvari[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya fan[133X
  
  [33X[0;0YReturns the fan of the variety [3Xvari[103X. This is a rename for FanOfVariety.[133X
  
  [1X3.5-8 Factors[101X
  
  [33X[1;0Y[29X[2XFactors[102X( [3Xvari[103X ) [32X operation[133X
  
  [1X3.5-9 BlowUpOnIthMinimalTorusOrbit[101X
  
  [33X[1;0Y[29X[2XBlowUpOnIthMinimalTorusOrbit[102X( [3Xvari[103X, [3Xp[103X ) [32X operation[133X
  
  [1X3.5-10 ZariskiCotangentSheafViaEulerSequence[101X
  
  [33X[1;0Y[29X[2XZariskiCotangentSheafViaEulerSequence[102X( [3Xarg[103X ) [32X function[133X
  
  [1X3.5-11 ZariskiCotangentSheafViaPoincareResidueMap[101X
  
  [33X[1;0Y[29X[2XZariskiCotangentSheafViaPoincareResidueMap[102X( [3Xarg[103X ) [32X function[133X
  
  [1X3.5-12 ithBettiNumber[101X
  
  [33X[1;0Y[29X[2XithBettiNumber[102X( [3Xvari[103X, [3Xp[103X ) [32X operation[133X
  
  [1X3.5-13 NrOfqRationalPoints[101X
  
  [33X[1;0Y[29X[2XNrOfqRationalPoints[102X( [3Xvari[103X, [3Xp[103X ) [32X operation[133X
  
  
  [1X3.6 [33X[0;0YConstructors[133X[101X
  
  [1X3.6-1 ToricVariety[101X
  
  [33X[1;0Y[29X[2XToricVariety[102X( [3Xvari[103X ) [32X operation[133X
  
  [1X3.6-2 ToricVariety[101X
  
  [33X[1;0Y[29X[2XToricVariety[102X( [3Xvari[103X ) [32X operation[133X
  
  [1X3.6-3 ToricVariety[101X
  
  [33X[1;0Y[29X[2XToricVariety[102X( [3Xconv[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya variety[133X
  
  [33X[0;0YCreates a toric variety out of the convex object [3Xconv[103X.[133X
  
  [1X3.6-4 ToricVariety[101X
  
  [33X[1;0Y[29X[2XToricVariety[102X( [3Xrays[103X, [3Xcones[103X, [3Xdegree_list[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya variety[133X
  
  [33X[0;0YCreates  a toric variety from a list [3Xrays[103X of ray generators and cones [3Xcones[103X.
  Beyond  the  functionality  of the other methods, this constructor allows to
  assign  specific gradings to the homogeneous variables of the Cox ring. With
  respect  to  the  order in which the rays appear in the list [3Xrays[103X, we assign
  gradings  as  provided  by  the third argument [3Xdegree_list[103X . The latter is a
  list of integers.[133X
  
  [1X3.6-5 ToricVariety[101X
  
  [33X[1;0Y[29X[2XToricVariety[102X( [3Xrays[103X, [3Xcones[103X, [3Xdegree_list[103X, [3Xvar_list[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya variety[133X
  
  [33X[0;0YCreates  a toric variety from a list [3Xrays[103X of ray generators and cones [3Xcones[103X.
  Beyond  the  functionality  of the other methods, this constructor allows to
  assign   specific  gradings  and  homogeneous  variable  names  to  the  ray
  generators  of  this  toric  variety. With respect to the order in which the
  rays  appear  in  the  list  [3Xrays[103X,  we assign gradings and variable names as
  provided  by  the  third and fourth argument. These are the list of gradings
  [3Xdegree_list[103X  and  the list of variables names [3Xvar_list[103X. The former is a list
  of integers and the latter a list of strings.[133X
  
  [1X3.6-6 ToricVarietiesFromGrading[101X
  
  [33X[1;0Y[29X[2XToricVarietiesFromGrading[102X( [3Xa[103X, [3Xlist[103X, [3Xof[103X, [3Xlists[103X, [3Xof[103X, [3Xintegers[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list of toric varieties[133X
  
  [33X[0;0YGiven  a [23X\mathbb{Z}^n[123X-grading of a polynomial ring, this method computes all
  toric  varieties,  which  are  normal and have no-torus factor and whose Cox
  ring obeys the given [23X\mathbb{Z}^n[123X-grading.[133X
  
  [1X3.6-7 ToricVarietyFromGrading[101X
  
  [33X[1;0Y[29X[2XToricVarietyFromGrading[102X( [3Xa[103X, [3Xlist[103X, [3Xof[103X, [3Xlists[103X, [3Xof[103X, [3Xintegers[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya toric variety[133X
  
  [33X[0;0YGiven  a  [23X\mathbb{Z}^n[123X-grading  of a polynomial ring, this method computes a
  toric  variety,  which  is normal and has no-torus factor and whose Cox ring
  obeys the given [23X\mathbb{Z}^n[123X-grading.[133X
  
