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ToricVarieties

A package to handle toric varieties

2022.07.13

13 July 2022

Sebastian Gutsche
Email: gutsche@mathematik.uni-siegen.de
Homepage: https://sebasguts.github.io
Address:
Department Mathematik
Universität Siegen
Walter-Flex-Straße 3
57072 Siegen
Germany

Martin Bies
Email: martin.bies@alumni.uni-heidelberg.de
Homepage: https://martinbies.github.io/
Address:
Department of Mathematics
University of Pennsylvania
David Rittenhouse Laboratory
209 S 33rd St
Philadelphia
PA 19104

Copyright

This package may be distributed under the terms and conditions of the GNU Public License Version 2 or (at your option) any later version.

Contents

1.1 What is the goal of the ToricVarieties package?

2 Installation of the ToricVarieties Package

3 Toric Varieties

3.1 Toric Varieties: Examples

  3.1-1 The Hirzebruch surface of index 5
  3.1-2 A smooth, complete toric variety which is not projective
  3.1-3 Convenient construction of toric varieties
  3.1-4 Toric varieties from gradings
  3.1-5 Blowups of toric varieties by star subdivisions of fans

3.2 Toric variety: Category and Representations

  3.2-1 IsToricVariety
  3.2-2 IsCategoryOfToricVarieties
  3.2-3 twitter

  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

  3.5-1 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 Constructors

  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 Toric subvarieties

4.1 The GAP category

4.1-1 IsToricSubvariety

  4.2-1 IsClosedSubvariety
  4.2-2 IsOpen
  4.2-3 IsWholeVariety

  4.3-1 UnderlyingToricVariety
  4.3-2 InclusionMorphism
  4.3-3 AmbientToricVariety

4.4-1 ClosureOfTorusOrbitOfCone

4.5 Constructors

4.5-1 ToricSubvariety

5 Affine toric varieties

5.1 Affine toric varieties: Examples

5.1-1 Affine space

5.2 The GAP category

5.2-1 IsAffineToricVariety

  5.3-1 CoordinateRing
  5.3-2 ListOfVariablesOfCoordinateRing
  5.3-3 MorphismFromCoordinateRingToCoordinateRingOfTorus
  5.3-4 ConeOfVariety

5.4-1 CoordinateRing
5.4-2 Cone

5.5 Constructors

6 Projective toric varieties

6.1 Projective toric varieties: Examples

6.1-1 P1xP1 created by a polytope
6.1-2 P1xP1 from product of P1s

6.2 The GAP category

6.2-1 IsProjectiveToricVariety

  6.3-1 PolytopeOfVariety
  6.3-2 AffineCone
  6.3-3 ProjectiveEmbedding

6.4-1 IsIsomorphicToProjectiveSpace
6.4-2 IsDirectProductOfPNs

6.5-1 Polytope
6.5-2 AmpleDivisor

6.6 Constructors

8 Toric divisors

8.1 Toric divisors: Examples

8.1-1 Divisors on a toric variety
8.1-2 Polytope of toric divisors

8.2 The GAP category

8.2-1 IsToricDivisor
8.2-2 twitter

  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

  8.4-1 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

  8.5-1 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 Constructors

  8.6-1 DivisorOfCharacter
  8.6-2 DivisorOfCharacter
  8.6-3 CreateDivisor
  8.6-4 CreateDivisor

9 Blowups of toric varieties

9.1 Constructors

9.1-1 BlowupOfToricVariety
9.1-2 SequenceOfBlowupsOfToricVariety

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