  
  
                                [1X ToricVarieties [101X
  
  
                     [1X A package to handle toric varieties [101X
  
  
                                   2019.12.05
  
  
                                5 December 2019
  
  
                               Sebastian Gutsche
  
                                  Martin Bies
  
  
  
  Sebastian Gutsche
      Email:    [7Xmailto:gutsche@mathematik.uni-siegen.de[107X
      Homepage: [7Xhttps://sebasguts.github.io[107X
      Address:  [33X[0;14YDepartment Mathematik[133X
                [33X[0;14YUniversität Siegen[133X
                [33X[0;14YWalter-Flex-Straße 3[133X
                [33X[0;14Y57072 Siegen[133X
                [33X[0;14YGermany[133X
  
  
  Martin Bies
      Email:    [7Xmailto:martin.bies@alumni.uni-heidelberg.de[107X
      Address:  [33X[0;14YPhysique Théorique et Mathématique[133X
                [33X[0;14YUniversité Libre de Bruxelles[133X
                [33X[0;14YCampus Plaine - CP 231[133X
                [33X[0;14YBuilding NO - Level 6 - Office O.6.111[133X
                [33X[0;14Y1050 Brussels[133X
                [33X[0;14YBelgium[133X
  
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0YThis  package  may  be distributed under the terms and conditions of the GNU
  Public License Version 2 or (at your option) any later version.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (ToricVarieties)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YWhat is the goal of the ToricVarieties package?[133X
  2 [33X[0;0YInstallation of the ToricVarieties Package[133X
  3 [33X[0;0YToric Varieties[133X
    3.1 [33X[0;0YToric Varieties: Examples[133X
      3.1-1 [33X[0;0YThe Hirzebruch surface of index 5[133X
      3.1-2 [33X[0;0YA smooth, complete toric variety which is not projective[133X
      3.1-3 [33X[0;0YConvenient construction of toric varieties[133X
      3.1-4 [33X[0;0YToric varieties from gradings[133X
      3.1-5 [33X[0;0YBlowups of toric varieties by star subdivisions of fans[133X
    3.2 [33X[0;0YToric variety: Category and Representations[133X
      3.2-1 IsToricVariety
      3.2-2 IsCategoryOfToricVarieties
      3.2-3 twitter
    3.3 [33X[0;0YProperties[133X
      3.3-1 IsNormalVariety
      3.3-2 IsAffine
      3.3-3 IsProjective
      3.3-4 IsSmooth
      3.3-5 IsComplete
      3.3-6 HasTorusfactor
      3.3-7 HasNoTorusfactor
      3.3-8 IsOrbifold
      3.3-9 IsSimplicial
    3.4 [33X[0;0YAttributes[133X
      3.4-1 AffineOpenCovering
      3.4-2 CoxRing
      3.4-3 ListOfVariablesOfCoxRing
      3.4-4 ClassGroup
      3.4-5 TorusInvariantDivisorGroup
      3.4-6 MapFromCharacterToPrincipalDivisor
      3.4-7 MapFromWeilDivisorsToClassGroup
      3.4-8 Dimension
      3.4-9 DimensionOfTorusfactor
      3.4-10 CoordinateRingOfTorus
      3.4-11 ListOfVariablesOfCoordinateRingOfTorus
      3.4-12 IsProductOf
      3.4-13 CharacterLattice
      3.4-14 TorusInvariantPrimeDivisors
      3.4-15 IrrelevantIdeal
      3.4-16 SRIdeal
      3.4-17 MorphismFromCoxVariety
      3.4-18 CoxVariety
      3.4-19 FanOfVariety
      3.4-20 CartierTorusInvariantDivisorGroup
      3.4-21 PicardGroup
      3.4-22 NameOfVariety
      3.4-23 ZariskiCotangentSheaf
      3.4-24 CotangentSheaf
      3.4-25 EulerCharacteristic
    3.5 [33X[0;0YMethods[133X
      3.5-1 UnderlyingSheaf
      3.5-2 CoordinateRingOfTorus
      3.5-3 \*
      3.5-4 CharacterToRationalFunction
      3.5-5 CoxRing
      3.5-6 WeilDivisorsOfVariety
      3.5-7 Fan
      3.5-8 Factors
      3.5-9 BlowUpOnIthMinimalTorusOrbit
      3.5-10 ZariskiCotangentSheafViaEulerSequence
      3.5-11 ZariskiCotangentSheafViaPoincareResidueMap
      3.5-12 ithBettiNumber
      3.5-13 NrOfqRationalPoints
    3.6 [33X[0;0YConstructors[133X
      3.6-1 ToricVariety
      3.6-2 ToricVariety
      3.6-3 ToricVariety
      3.6-4 ToricVariety
      3.6-5 ToricVariety
      3.6-6 ToricVarietiesFromGrading
      3.6-7 ToricVarietyFromGrading
  4 [33X[0;0YToric subvarieties[133X
    4.1 [33X[0;0YThe GAP category[133X
      4.1-1 IsToricSubvariety
    4.2 [33X[0;0YProperties[133X
      4.2-1 IsClosedSubvariety
      4.2-2 IsOpen
      4.2-3 IsWholeVariety
    4.3 [33X[0;0YAttributes[133X
      4.3-1 UnderlyingToricVariety
      4.3-2 InclusionMorphism
      4.3-3 AmbientToricVariety
    4.4 [33X[0;0YMethods[133X
      4.4-1 ClosureOfTorusOrbitOfCone
    4.5 [33X[0;0YConstructors[133X
      4.5-1 ToricSubvariety
  5 [33X[0;0YAffine toric varieties[133X
    5.1 [33X[0;0YAffine toric varieties: Examples[133X
      5.1-1 [33X[0;0YAffine space[133X
    5.2 [33X[0;0YThe GAP category[133X
      5.2-1 IsAffineToricVariety
    5.3 [33X[0;0YAttributes[133X
      5.3-1 CoordinateRing
      5.3-2 ListOfVariablesOfCoordinateRing
      5.3-3 MorphismFromCoordinateRingToCoordinateRingOfTorus
      5.3-4 ConeOfVariety
    5.4 [33X[0;0YMethods[133X
      5.4-1 CoordinateRing
      5.4-2 Cone
    5.5 [33X[0;0YConstructors[133X
  6 [33X[0;0YProjective toric varieties[133X
    6.1 [33X[0;0YProjective toric varieties: Examples[133X
      6.1-1 [33X[0;0YP1xP1 created by a polytope[133X
      6.1-2 [33X[0;0YP1xP1 from product of P1s[133X
    6.2 [33X[0;0YThe GAP category[133X
      6.2-1 IsProjectiveToricVariety
    6.3 [33X[0;0YAttribute[133X
      6.3-1 PolytopeOfVariety
      6.3-2 AffineCone
      6.3-3 ProjectiveEmbedding
    6.4 [33X[0;0YProperties[133X
      6.4-1 IsIsomorphicToProjectiveSpace
      6.4-2 IsDirectProductOfPNs
    6.5 [33X[0;0YMethods[133X
      6.5-1 Polytope
      6.5-2 AmpleDivisor
    6.6 [33X[0;0YConstructors[133X
  7 [33X[0;0YToric morphisms[133X
    7.1 [33X[0;0YToric morphisms: Examples[133X
      7.1-1 [33X[0;0YMorphism between toric varieties and their class groups[133X
    7.2 [33X[0;0YThe GAP category[133X
      7.2-1 IsToricMorphism
    7.3 [33X[0;0YProperties[133X
      7.3-1 IsMorphism
      7.3-2 IsProper
    7.4 [33X[0;0YAttributes[133X
      7.4-1 SourceObject
      7.4-2 UnderlyingGridMorphism
      7.4-3 ToricImageObject
      7.4-4 RangeObject
      7.4-5 MorphismOnWeilDivisorGroup
      7.4-6 ClassGroup
      7.4-7 MorphismOnCartierDivisorGroup
      7.4-8 PicardGroup
      7.4-9 Source
      7.4-10 Range
      7.4-11 MorphismOnIthFactor
    7.5 [33X[0;0YMethods[133X
      7.5-1 UnderlyingListList
    7.6 [33X[0;0YConstructors[133X
      7.6-1 ToricMorphism
      7.6-2 ToricMorphism
  8 [33X[0;0YToric divisors[133X
    8.1 [33X[0;0YToric divisors: Examples[133X
      8.1-1 [33X[0;0YDivisors on a toric variety[133X
      8.1-2 [33X[0;0YPolytope of toric divisors[133X
    8.2 [33X[0;0YThe GAP category[133X
      8.2-1 IsToricDivisor
      8.2-2 twitter
    8.3 [33X[0;0YProperties[133X
      8.3-1 IsCartier
      8.3-2 IsPrincipal
      8.3-3 IsPrimedivisor
      8.3-4 IsBasepointFree
      8.3-5 IsAmple
      8.3-6 IsVeryAmple
      8.3-7 IsNumericallyEffective
    8.4 [33X[0;0YAttributes[133X
      8.4-1 CartierData
      8.4-2 CharacterOfPrincipalDivisor
      8.4-3 ClassOfDivisor
      8.4-4 PolytopeOfDivisor
      8.4-5 BasisOfGlobalSections
      8.4-6 IntegerForWhichIsSureVeryAmple
      8.4-7 AmbientToricVariety
      8.4-8 UnderlyingGroupElement
      8.4-9 UnderlyingToricVariety
      8.4-10 DegreeOfDivisor
      8.4-11 VarietyOfDivisorpolytope
      8.4-12 MonomsOfCoxRingOfDegree
      8.4-13 CoxRingOfTargetOfDivisorMorphism
      8.4-14 RingMorphismOfDivisor
    8.5 [33X[0;0YMethods[133X
      8.5-1 VeryAmpleMultiple
      8.5-2 CharactersForClosedEmbedding
      8.5-3 \+
      8.5-4 \-
      8.5-5 \*
      8.5-6 MonomsOfCoxRingOfDegree
      8.5-7 DivisorOfGivenClass
      8.5-8 AddDivisorToItsAmbientVariety
      8.5-9 Polytope
      8.5-10 CoxRingOfTargetOfDivisorMorphism
    8.6 [33X[0;0YConstructors[133X
      8.6-1 DivisorOfCharacter
      8.6-2 DivisorOfCharacter
      8.6-3 CreateDivisor
      8.6-4 CreateDivisor
  9 [33X[0;0YBlowups of toric varieties[133X
    9.1 [33X[0;0YConstructors[133X
      9.1-1 BlowupOfToricVariety
      9.1-2 SequenceOfBlowupsOfToricVariety
  
  
  [32X
