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ToricVarieties

A GAP package for handling toric varieties.

Version 2012.12.22

October 2012

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: [email protected]
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

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

3.4 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

6.5 Projective toric varieties: Constructors

6.6 Projective toric varieties: Examples

6.6-1 PxP1 created by a polytope

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

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