  
  
                                 [1X[5XToricVarieties[105X[101X
  
  
                  [1XA [5XGAP[105X package for handling toric varieties.[101X
  
  
                               Version 2012.12.22
  
  
                                  October 2012
  
  
                               Sebastian Gutsche
  
  
  
            [33X[0;10YThis  manual  is  best  viewed  as  an [12XHTML[112X document. An
            [12Xoffline[112X  version should be included in the documentation
            subfolder of the package.[133X
  
  
  
  Sebastian Gutsche
      Email:    [7Xmailto:sebastian.gutsche@rwth-aachen.de[107X
      Homepage: [7Xhttp://wwwb.math.rwth-aachen.de/~gutsche[107X
      Address:  [33X[0;14YLehrstuhl  B  für  Mathematik,  RWTH Aachen, Templergraben 64,
                52056 Aachen, Germany[133X
  
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2011-2012 by Sebastian Gutsche[133X
  
  [33X[0;0YThis  package  may  be distributed under the terms and conditions of the GNU
  Public License Version 2.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  
  -------------------------------------------------------
  
  
  [1XContents (ToricVarieties)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YWhat is the goal of the [5XToricVarieties[105X package?[133X
  2 [33X[0;0YInstallation of the [5XToricVarieties[105X Package[133X
  3 [33X[0;0YToric varieties[133X
    3.1 [33X[0;0YToric variety: Category and Representations[133X
      3.1-1 IsToricVariety
    3.2 [33X[0;0YToric varieties: Properties[133X
      3.2-1 IsNormalVariety
      3.2-2 IsAffine
      3.2-3 IsProjective
      3.2-4 IsComplete
      3.2-5 IsSmooth
      3.2-6 HasTorusfactor
      3.2-7 HasNoTorusfactor
      3.2-8 IsOrbifold
    3.3 [33X[0;0YToric varieties: Attributes[133X
      3.3-1 AffineOpenCovering
      3.3-2 CoxRing
      3.3-3 ListOfVariablesOfCoxRing
      3.3-4 ClassGroup
      3.3-5 PicardGroup
      3.3-6 TorusInvariantDivisorGroup
      3.3-7 MapFromCharacterToPrincipalDivisor
      3.3-8 Dimension
      3.3-9 DimensionOfTorusfactor
      3.3-10 CoordinateRingOfTorus
      3.3-11 IsProductOf
      3.3-12 CharacterLattice
      3.3-13 TorusInvariantPrimeDivisors
      3.3-14 IrrelevantIdeal
      3.3-15 MorphismFromCoxVariety
      3.3-16 CoxVariety
      3.3-17 FanOfVariety
      3.3-18 CartierTorusInvariantDivisorGroup
      3.3-19 NameOfVariety
      3.3-20 twitter
    3.4 [33X[0;0YToric varieties: Methods[133X
      3.4-1 UnderlyingSheaf
      3.4-2 CoordinateRingOfTorus
      3.4-3 \*
      3.4-4 CharacterToRationalFunction
      3.4-5 CoxRing
      3.4-6 WeilDivisorsOfVariety
      3.4-7 Fan
    3.5 [33X[0;0YToric varieties: Constructors[133X
      3.5-1 ToricVariety
    3.6 [33X[0;0YToric varieties: Examples[133X
      3.6-1 [33X[0;0YThe Hirzebruch surface of index 5[133X
  4 [33X[0;0YToric subvarieties[133X
    4.1 [33X[0;0YToric subvarieties: Category and Representations[133X
      4.1-1 IsToricSubvariety
    4.2 [33X[0;0YToric subvarieties: Properties[133X
      4.2-1 IsClosed
      4.2-2 IsOpen
      4.2-3 IsWholeVariety
    4.3 [33X[0;0YToric subvarieties: Attributes[133X
      4.3-1 UnderlyingToricVariety
      4.3-2 InclusionMorphism
      4.3-3 AmbientToricVariety
    4.4 [33X[0;0YToric subvarieties: Methods[133X
      4.4-1 ClosureOfTorusOrbitOfCone
    4.5 [33X[0;0YToric subvarieties: Constructors[133X
      4.5-1 ToricSubvariety
  5 [33X[0;0YAffine toric varieties[133X
    5.1 [33X[0;0YAffine toric varieties: Category and Representations[133X
      5.1-1 IsAffineToricVariety
    5.2 [33X[0;0YAffine toric varieties: Properties[133X
    5.3 [33X[0;0YAffine toric varieties: Attributes[133X
      5.3-1 CoordinateRing
      5.3-2 ListOfVariablesOfCoordinateRing
      5.3-3 MorphismFromCoordinateRingToCoordinateRingOfTorus
      5.3-4 ConeOfVariety
    5.4 [33X[0;0YAffine toric varieties: Methods[133X
      5.4-1 CoordinateRing
      5.4-2 Cone
    5.5 [33X[0;0YAffine toric varieties: Constructors[133X
    5.6 [33X[0;0YAffine toric Varieties: Examples[133X
      5.6-1 [33X[0;0YAffine space[133X
  6 [33X[0;0YProjective toric varieties[133X
    6.1 [33X[0;0YProjective toric varieties: Category and Representations[133X
      6.1-1 IsProjectiveToricVariety
    6.2 [33X[0;0YProjective toric varieties: Properties[133X
    6.3 [33X[0;0YProjective toric varieties: Attributes[133X
      6.3-1 AffineCone
      6.3-2 PolytopeOfVariety
      6.3-3 ProjectiveEmbedding
    6.4 [33X[0;0YProjective toric varieties: Methods[133X
      6.4-1 Polytope
    6.5 [33X[0;0YProjective toric varieties: Constructors[133X
    6.6 [33X[0;0YProjective toric varieties: Examples[133X
      6.6-1 [33X[0;0YPxP1 created by a polytope[133X
  7 [33X[0;0YToric morphisms[133X
    7.1 [33X[0;0YToric morphisms: Category and Representations[133X
      7.1-1 IsToricMorphism
    7.2 [33X[0;0YToric morphisms: Properties[133X
      7.2-1 IsMorphism
      7.2-2 IsProper
    7.3 [33X[0;0YToric morphisms: Attributes[133X
      7.3-1 SourceObject
      7.3-2 UnderlyingGridMorphism
      7.3-3 ToricImageObject
      7.3-4 RangeObject
      7.3-5 MorphismOnWeilDivisorGroup
      7.3-6 ClassGroup
      7.3-7 MorphismOnCartierDivisorGroup
      7.3-8 PicardGroup
    7.4 [33X[0;0YToric morphisms: Methods[133X
      7.4-1 UnderlyingListList
    7.5 [33X[0;0YToric morphisms: Constructors[133X
      7.5-1 ToricMorphism
      7.5-2 ToricMorphism
    7.6 [33X[0;0YToric morphisms: Examples[133X
      7.6-1 [33X[0;0YMorphism between toric varieties and their class groups[133X
  8 [33X[0;0YToric divisors[133X
    8.1 [33X[0;0YToric divisors: Category and Representations[133X
      8.1-1 IsToricDivisor
    8.2 [33X[0;0YToric divisors: Properties[133X
      8.2-1 IsCartier
      8.2-2 IsPrincipal
      8.2-3 IsPrimedivisor
      8.2-4 IsBasepointFree
      8.2-5 IsAmple
      8.2-6 IsVeryAmple
    8.3 [33X[0;0YToric divisors: Attributes[133X
      8.3-1 CartierData
      8.3-2 CharacterOfPrincipalDivisor
      8.3-3 ToricVarietyOfDivisor
      8.3-4 ClassOfDivisor
      8.3-5 PolytopeOfDivisor
      8.3-6 BasisOfGlobalSections
      8.3-7 IntegerForWhichIsSureVeryAmple
      8.3-8 AmbientToricVariety
      8.3-9 UnderlyingGroupElement
      8.3-10 UnderlyingToricVariety
      8.3-11 DegreeOfDivisor
      8.3-12 MonomsOfCoxRingOfDegree
      8.3-13 CoxRingOfTargetOfDivisorMorphism
      8.3-14 RingMorphismOfDivisor
    8.4 [33X[0;0YToric divisors: Methods[133X
      8.4-1 VeryAmpleMultiple
      8.4-2 CharactersForClosedEmbedding
      8.4-3 MonomsOfCoxRingOfDegree
      8.4-4 DivisorOfGivenClass
      8.4-5 AddDivisorToItsAmbientVariety
      8.4-6 Polytope
      8.4-7 +
      8.4-8 -
      8.4-9 *
    8.5 [33X[0;0YToric divisors: Constructors[133X
      8.5-1 DivisorOfCharacter
      8.5-2 DivisorOfCharacter
      8.5-3 CreateDivisor
      8.5-4 CreateDivisor
    8.6 [33X[0;0YToric divisors: Examples[133X
      8.6-1 [33X[0;0YDivisors on a toric variety[133X
  
  
  [32X
