ToricVarieties
  
  
                  A GAP package for handling toric varieties.
  
  
                               Version 2012.12.22
  
  
                                  October 2012
  
  
                               Sebastian Gutsche
  
  
  
            This  manual  is  best  viewed  as  an HTML document. An
            offline  version should be included in the documentation
            subfolder of the package.
  
  
  
  Sebastian Gutsche
      Email:    mailto:sebastian.gutsche@rwth-aachen.de
      Homepage: http://wwwb.math.rwth-aachen.de/~gutsche
      Address:  Lehrstuhl  B  für  Mathematik,  RWTH Aachen, Templergraben 64,
                52056 Aachen, Germany
  
  
  
  -------------------------------------------------------
  Copyright
  © 2011-2012 by Sebastian Gutsche
  
  This  package  may  be distributed under the terms and conditions of the GNU
  Public License Version 2.
  
  
  -------------------------------------------------------
  Acknowledgements
  
  -------------------------------------------------------
  
  
  Contents (ToricVarieties)
  
  1 Introduction
    1.1 What is the goal of the ToricVarieties package?
  2 Installation of the ToricVarieties Package
  3 Toric varieties
    3.1 Toric variety: Category and Representations
      3.1-1 IsToricVariety
    3.2 Toric varieties: Properties
      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 Toric varieties: Attributes
      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 Toric varieties: Methods
      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 Toric varieties: Constructors
      3.5-1 ToricVariety
    3.6 Toric varieties: Examples
      3.6-1 The Hirzebruch surface of index 5
  4 Toric subvarieties
    4.1 Toric subvarieties: Category and Representations
      4.1-1 IsToricSubvariety
    4.2 Toric subvarieties: Properties
      4.2-1 IsClosed
      4.2-2 IsOpen
      4.2-3 IsWholeVariety
    4.3 Toric subvarieties: Attributes
      4.3-1 UnderlyingToricVariety
      4.3-2 InclusionMorphism
      4.3-3 AmbientToricVariety
    4.4 Toric subvarieties: Methods
      4.4-1 ClosureOfTorusOrbitOfCone
    4.5 Toric subvarieties: Constructors
      4.5-1 ToricSubvariety
  5 Affine toric varieties
    5.1 Affine toric varieties: Category and Representations
      5.1-1 IsAffineToricVariety
    5.2 Affine toric varieties: Properties
    5.3 Affine toric varieties: Attributes
      5.3-1 CoordinateRing
      5.3-2 ListOfVariablesOfCoordinateRing
      5.3-3 MorphismFromCoordinateRingToCoordinateRingOfTorus
      5.3-4 ConeOfVariety
    5.4 Affine toric varieties: Methods
      5.4-1 CoordinateRing
      5.4-2 Cone
    5.5 Affine toric varieties: Constructors
    5.6 Affine toric Varieties: Examples
      5.6-1 Affine space
  6 Projective toric varieties
    6.1 Projective toric varieties: Category and Representations
      6.1-1 IsProjectiveToricVariety
    6.2 Projective toric varieties: Properties
    6.3 Projective toric varieties: Attributes
      6.3-1 AffineCone
      6.3-2 PolytopeOfVariety
      6.3-3 ProjectiveEmbedding
    6.4 Projective toric varieties: Methods
      6.4-1 Polytope
    6.5 Projective toric varieties: Constructors
    6.6 Projective toric varieties: Examples
      6.6-1 PxP1 created by a polytope
  7 Toric morphisms
    7.1 Toric morphisms: Category and Representations
      7.1-1 IsToricMorphism
    7.2 Toric morphisms: Properties
      7.2-1 IsMorphism
      7.2-2 IsProper
    7.3 Toric morphisms: Attributes
      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 Toric morphisms: Methods
      7.4-1 UnderlyingListList
    7.5 Toric morphisms: Constructors
      7.5-1 ToricMorphism
      7.5-2 ToricMorphism
    7.6 Toric morphisms: Examples
      7.6-1 Morphism between toric varieties and their class groups
  8 Toric divisors
    8.1 Toric divisors: Category and Representations
      8.1-1 IsToricDivisor
    8.2 Toric divisors: Properties
      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 Toric divisors: Attributes
      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 Toric divisors: Methods
      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 Toric divisors: Constructors
      8.5-1 DivisorOfCharacter
      8.5-2 DivisorOfCharacter
      8.5-3 CreateDivisor
      8.5-4 CreateDivisor
    8.6 Toric divisors: Examples
      8.6-1 Divisors on a toric variety