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                                 ToricVarieties

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                  A GAP package for handling toric varieties.

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                               Version 2012.12.22

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                                  October 2012

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                               Sebastian Gutsche

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            This  manual  is  best  viewed  as  an HTML document. An

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            offline  version should be included in the documentation

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            subfolder of the package.

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  Sebastian Gutsche

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      Email:    mailto:[email protected]

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      Homepage: http://wwwb.math.rwth-aachen.de/~gutsche

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      Address:  Lehrstuhl  B  für  Mathematik,  RWTH Aachen, Templergraben 64,

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                52056 Aachen, Germany

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  -------------------------------------------------------

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  Copyright

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  © 2011-2012 by Sebastian Gutsche

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  This  package  may  be distributed under the terms and conditions of the GNU

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  Public License Version 2.

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  -------------------------------------------------------

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  Acknowledgements

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  -------------------------------------------------------

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  Contents (ToricVarieties)

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  1 Introduction

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    1.1 What is the goal of the ToricVarieties package?

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  2 Installation of the ToricVarieties Package

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  3 Toric varieties

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    3.1 Toric variety: Category and Representations

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      3.1-1 IsToricVariety

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    3.2 Toric varieties: Properties

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      3.2-1 IsNormalVariety

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      3.2-2 IsAffine

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      3.2-3 IsProjective

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      3.2-4 IsComplete

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      3.2-5 IsSmooth

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      3.2-6 HasTorusfactor

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      3.2-7 HasNoTorusfactor

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      3.2-8 IsOrbifold

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    3.3 Toric varieties: Attributes

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      3.3-1 AffineOpenCovering

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      3.3-2 CoxRing

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      3.3-3 ListOfVariablesOfCoxRing

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      3.3-4 ClassGroup

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      3.3-5 PicardGroup

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      3.3-6 TorusInvariantDivisorGroup

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      3.3-7 MapFromCharacterToPrincipalDivisor

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      3.3-8 Dimension

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      3.3-9 DimensionOfTorusfactor

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      3.3-10 CoordinateRingOfTorus

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      3.3-11 IsProductOf

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      3.3-12 CharacterLattice

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      3.3-13 TorusInvariantPrimeDivisors

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      3.3-14 IrrelevantIdeal

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      3.3-15 MorphismFromCoxVariety

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      3.3-16 CoxVariety

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      3.3-17 FanOfVariety

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      3.3-18 CartierTorusInvariantDivisorGroup

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      3.3-19 NameOfVariety

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      3.3-20 twitter

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    3.4 Toric varieties: Methods

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      3.4-1 UnderlyingSheaf

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      3.4-2 CoordinateRingOfTorus

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      3.4-3 \*

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      3.4-4 CharacterToRationalFunction

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      3.4-5 CoxRing

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      3.4-6 WeilDivisorsOfVariety

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      3.4-7 Fan

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    3.5 Toric varieties: Constructors

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      3.5-1 ToricVariety

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    3.6 Toric varieties: Examples

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      3.6-1 The Hirzebruch surface of index 5

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  4 Toric subvarieties

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    4.1 Toric subvarieties: Category and Representations

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      4.1-1 IsToricSubvariety

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    4.2 Toric subvarieties: Properties

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      4.2-1 IsClosed

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      4.2-2 IsOpen

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      4.2-3 IsWholeVariety

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    4.3 Toric subvarieties: Attributes

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      4.3-1 UnderlyingToricVariety

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      4.3-2 InclusionMorphism

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      4.3-3 AmbientToricVariety

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    4.4 Toric subvarieties: Methods

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      4.4-1 ClosureOfTorusOrbitOfCone

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    4.5 Toric subvarieties: Constructors

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      4.5-1 ToricSubvariety

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  5 Affine toric varieties

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    5.1 Affine toric varieties: Category and Representations

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      5.1-1 IsAffineToricVariety

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    5.2 Affine toric varieties: Properties

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    5.3 Affine toric varieties: Attributes

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      5.3-1 CoordinateRing

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      5.3-2 ListOfVariablesOfCoordinateRing

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      5.3-3 MorphismFromCoordinateRingToCoordinateRingOfTorus

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      5.3-4 ConeOfVariety

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    5.4 Affine toric varieties: Methods

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      5.4-1 CoordinateRing

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      5.4-2 Cone

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    5.5 Affine toric varieties: Constructors

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    5.6 Affine toric Varieties: Examples

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      5.6-1 Affine space

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  6 Projective toric varieties

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    6.1 Projective toric varieties: Category and Representations

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      6.1-1 IsProjectiveToricVariety

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    6.2 Projective toric varieties: Properties

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    6.3 Projective toric varieties: Attributes

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      6.3-1 AffineCone

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      6.3-2 PolytopeOfVariety

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      6.3-3 ProjectiveEmbedding

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    6.4 Projective toric varieties: Methods

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      6.4-1 Polytope

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    6.5 Projective toric varieties: Constructors

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    6.6 Projective toric varieties: Examples

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      6.6-1 PxP1 created by a polytope

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  7 Toric morphisms

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    7.1 Toric morphisms: Category and Representations

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      7.1-1 IsToricMorphism

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    7.2 Toric morphisms: Properties

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      7.2-1 IsMorphism

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      7.2-2 IsProper

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    7.3 Toric morphisms: Attributes

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      7.3-1 SourceObject

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      7.3-2 UnderlyingGridMorphism

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      7.3-3 ToricImageObject

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      7.3-4 RangeObject

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      7.3-5 MorphismOnWeilDivisorGroup

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      7.3-6 ClassGroup

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      7.3-7 MorphismOnCartierDivisorGroup

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      7.3-8 PicardGroup

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    7.4 Toric morphisms: Methods

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      7.4-1 UnderlyingListList

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    7.5 Toric morphisms: Constructors

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      7.5-1 ToricMorphism

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      7.5-2 ToricMorphism

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    7.6 Toric morphisms: Examples

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      7.6-1 Morphism between toric varieties and their class groups

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  8 Toric divisors

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    8.1 Toric divisors: Category and Representations

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      8.1-1 IsToricDivisor

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    8.2 Toric divisors: Properties

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      8.2-1 IsCartier

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      8.2-2 IsPrincipal

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      8.2-3 IsPrimedivisor

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      8.2-4 IsBasepointFree

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      8.2-5 IsAmple

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      8.2-6 IsVeryAmple

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    8.3 Toric divisors: Attributes

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      8.3-1 CartierData

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      8.3-2 CharacterOfPrincipalDivisor

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      8.3-3 ToricVarietyOfDivisor

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      8.3-4 ClassOfDivisor

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      8.3-5 PolytopeOfDivisor

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      8.3-6 BasisOfGlobalSections

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      8.3-7 IntegerForWhichIsSureVeryAmple

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      8.3-8 AmbientToricVariety

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      8.3-9 UnderlyingGroupElement

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      8.3-10 UnderlyingToricVariety

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      8.3-11 DegreeOfDivisor

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      8.3-12 MonomsOfCoxRingOfDegree

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      8.3-13 CoxRingOfTargetOfDivisorMorphism

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      8.3-14 RingMorphismOfDivisor

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    8.4 Toric divisors: Methods

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      8.4-1 VeryAmpleMultiple

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      8.4-2 CharactersForClosedEmbedding

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      8.4-3 MonomsOfCoxRingOfDegree

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      8.4-4 DivisorOfGivenClass

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      8.4-5 AddDivisorToItsAmbientVariety

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      8.4-6 Polytope

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      8.4-7 +

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      8.4-8 -

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      8.4-9 *

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    8.5 Toric divisors: Constructors

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      8.5-1 DivisorOfCharacter

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      8.5-2 DivisorOfCharacter

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      8.5-3 CreateDivisor

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      8.5-4 CreateDivisor

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    8.6 Toric divisors: Examples

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      8.6-1 Divisors on a toric variety

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