  
  [1X3 [33X[0;0YToric varieties[133X[101X
  
  
  [1X3.1 [33X[0;0YToric variety: Category and Representations[133X[101X
  
  [1X3.1-1 IsToricVariety[101X
  
  [29X[2XIsToricVariety[102X( [3XM[103X ) [32X Category
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe [5XGAP[105X category of a toric variety.[133X
  
  
  [1X3.2 [33X[0;0YToric varieties: Properties[133X[101X
  
  [1X3.2-1 IsNormalVariety[101X
  
  [29X[2XIsNormalVariety[102X( [3Xvari[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YChecks if the toric variety [3Xvari[103X is a normal variety.[133X
  
  [1X3.2-2 IsAffine[101X
  
  [29X[2XIsAffine[102X( [3Xvari[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YChecks if the toric variety [3Xvari[103X is an affine variety.[133X
  
  [1X3.2-3 IsProjective[101X
  
  [29X[2XIsProjective[102X( [3Xvari[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YChecks if the toric variety [3Xvari[103X is a projective variety.[133X
  
  [1X3.2-4 IsComplete[101X
  
  [29X[2XIsComplete[102X( [3Xvari[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YChecks if the toric variety [3Xvari[103X is a complete variety.[133X
  
  [1X3.2-5 IsSmooth[101X
  
  [29X[2XIsSmooth[102X( [3Xvari[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YChecks if the toric variety [3Xvari[103X is a smooth variety.[133X
  
  [1X3.2-6 HasTorusfactor[101X
  
  [29X[2XHasTorusfactor[102X( [3Xvari[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YChecks if the toric variety [3Xvari[103X has a torus factor.[133X
  
  [1X3.2-7 HasNoTorusfactor[101X
  
  [29X[2XHasNoTorusfactor[102X( [3Xvari[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YChecks if the toric variety [3Xvari[103X has no torus factor.[133X
  
  [1X3.2-8 IsOrbifold[101X
  
  [29X[2XIsOrbifold[102X( [3Xvari[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[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 [33X[0;0YToric varieties: Attributes[133X[101X
  
  [1X3.3-1 AffineOpenCovering[101X
  
  [29X[2XAffineOpenCovering[102X( [3Xvari[103X ) [32X attribute
  [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
  
  [1X3.3-2 CoxRing[101X
  
  [29X[2XCoxRing[102X( [3Xvari[103X ) [32X attribute
  [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
  
  [1X3.3-3 ListOfVariablesOfCoxRing[101X
  
  [29X[2XListOfVariablesOfCoxRing[102X( [3Xvari[103X ) [32X attribute
  [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
  
  [1X3.3-4 ClassGroup[101X
  
  [29X[2XClassGroup[102X( [3Xvari[103X ) [32X attribute
  [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
  
  [1X3.3-5 PicardGroup[101X
  
  [29X[2XPicardGroup[102X( [3Xvari[103X ) [32X attribute
  [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
  
  [1X3.3-6 TorusInvariantDivisorGroup[101X
  
  [29X[2XTorusInvariantDivisorGroup[102X( [3Xvari[103X ) [32X attribute
  [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
  
  [1X3.3-7 MapFromCharacterToPrincipalDivisor[101X
  
  [29X[2XMapFromCharacterToPrincipalDivisor[102X( [3Xvari[103X ) [32X attribute
  [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 viewn as an help
  method to compute divisor classes.[133X
  
  [1X3.3-8 Dimension[101X
  
  [29X[2XDimension[102X( [3Xvari[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YReturns the dimension of the variety [3Xvari[103X.[133X
  
  [1X3.3-9 DimensionOfTorusfactor[101X
  
  [29X[2XDimensionOfTorusfactor[102X( [3Xvari[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YReturns the dimension of the torus factor of the variety [3Xvari[103X.[133X
  
  [1X3.3-10 CoordinateRingOfTorus[101X
  
  [29X[2XCoordinateRingOfTorus[102X( [3Xvari[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya ring[133X
  
  [33X[0;0YReturns the coordinate ring of the torus of the variety [3Xvari[103X. This method is
  not implemented, you need to call it with a second argument, which is a list
  of strings for the variables of the ring.[133X
  
  [1X3.3-11 IsProductOf[101X
  
  [29X[2XIsProductOf[102X( [3Xvari[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YIf  the  variety  [3Xvari[103X is a product of 2 or more varieties, the list contain
  those  varieties.  If  it  is  not  a product or at least not generated as a
  product, the list only contains the variety itself.[133X
  
  [1X3.3-12 CharacterLattice[101X
  
  [29X[2XCharacterLattice[102X( [3Xvari[103X ) [32X attribute
  [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
  
  [1X3.3-13 TorusInvariantPrimeDivisors[101X
  
  [29X[2XTorusInvariantPrimeDivisors[102X( [3Xvari[103X ) [32X attribute
  [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
  
  [1X3.3-14 IrrelevantIdeal[101X
  
  [29X[2XIrrelevantIdeal[102X( [3Xvari[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Yan ideal[133X
  
  [33X[0;0YReturns the irrelevant ideal of the cox ring of the variety [3Xvari[103X.[133X
  
  [1X3.3-15 MorphismFromCoxVariety[101X
  
  [29X[2XMorphismFromCoxVariety[102X( [3Xvari[103X ) [32X attribute
  [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
  
  [1X3.3-16 CoxVariety[101X
  
  [29X[2XCoxVariety[102X( [3Xvari[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya variety[133X
  
  [33X[0;0YThe method returns the Cox variety of the variety [3Xvari[103X.[133X
  
  [1X3.3-17 FanOfVariety[101X
  
  [29X[2XFanOfVariety[102X( [3Xvari[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10Ya fan[133X
  
  [33X[0;0YReturns the fan of the variety [3Xvari[103X. This is set by default.[133X
  
  [1X3.3-18 CartierTorusInvariantDivisorGroup[101X
  
  [29X[2XCartierTorusInvariantDivisorGroup[102X( [3Xvari[103X ) [32X attribute
  [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
  
  [1X3.3-19 NameOfVariety[101X
  
  [29X[2XNameOfVariety[102X( [3Xvari[103X ) [32X attribute
  [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
  
  [1X3.3-20 twitter[101X
  
  [29X[2Xtwitter[102X( [3Xvari[103X ) [32X attribute
  [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
  
  
  [1X3.4 [33X[0;0YToric varieties: Methods[133X[101X
  
  [1X3.4-1 UnderlyingSheaf[101X
  
  [29X[2XUnderlyingSheaf[102X( [3Xvari[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya sheaf[133X
  
  [33X[0;0YThe method returns the underlying sheaf of the variety [3Xvari[103X.[133X
  
  [1X3.4-2 CoordinateRingOfTorus[101X
  
  [29X[2XCoordinateRingOfTorus[102X( [3Xvari[103X, [3Xvars[103X ) [32X operation
  [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
  
  [1X3.4-3 \*[101X
  
  [29X[2X\*[102X( [3Xvari1[103X, [3Xvari2[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya variety[133X
  
  [33X[0;0YComputes the categorial product of the varieties [3Xvari1[103X and [3Xvari2[103X.[133X
  
  [1X3.4-4 CharacterToRationalFunction[101X
  
  [29X[2XCharacterToRationalFunction[102X( [3Xelem[103X, [3Xvari[103X ) [32X operation
  [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. To compute rational functions you
  first need to compute to coordinate ring of the torus of the variety [3Xvari[103X.[133X
  
  [1X3.4-5 CoxRing[101X
  
  [29X[2XCoxRing[102X( [3Xvari[103X, [3Xvars[103X ) [32X operation
  [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
  containing one variable, which will be numbered by the method.[133X
  
  [1X3.4-6 WeilDivisorsOfVariety[101X
  
  [29X[2XWeilDivisorsOfVariety[102X( [3Xvari[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns a list of the currently defined Divisors of the toric variety.[133X
  
  [1X3.4-7 Fan[101X
  
  [29X[2XFan[102X( [3Xvari[103X ) [32X operation
  [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 [33X[0;0YToric varieties: Constructors[133X[101X
  
  [1X3.5-1 ToricVariety[101X
  
  [29X[2XToricVariety[102X( [3Xconv[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ya ring[133X
  
  [33X[0;0YCreates a toric variety out of the convex object [3Xconv[103X.[133X
  
  
  [1X3.6 [33X[0;0YToric varieties: Examples[133X[101X
  
  
  [1X3.6-1 [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_]/( x2*x2_-1, x1*x1_-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 [27XDivisorOfCharacter([1,2],H5);[127X[104X
    [4X[28X<A principal divisor of a toric variety with coordinates [ 9, 2, 1, -2 ]>[128X[104X
    [4X[25Xgap>[125X [27XBasisOfGlobalSections(last);[127X[104X
    [4X[28X[ |[ x1_*x2_^2 ]| ][128X[104X
  [4X[32X[104X
  
