NormalToricVarieties -- working with normal toric varieties and related objects

Description

A toric variety is an integral scheme such that an algebraic torus forms a Zariski open subscheme and the natural action this torus on itself extends to an action on the entire scheme. Normal toric varieties correspond to strongly convex rational polyhedral fans. This makes the theory of normal toric varieties very explicit and computable.

This Macaulay2 package is designed to manipulate normal toric varieties and related geometric objects. An introduction to the theory of normal toric varieties can be found in the following textbooks:

David A. Cox, John B. Little, Hal Schenck, Toric varieties, Graduate Studies in Mathematics 124. American Mathematical Society, Providence RI, 2011. ISBN: 978-0-8218-4817-7
Günter Ewald, Combinatorial convexity and algebraic geometry, Graduate Texts in Mathematics 168. Springer-Verlag, New York, 1996. ISBN: 0-387-94755-8
William Fulton, Introduction to toric varieties, Annals of Mathematics Studies 131, Princeton University Press, Princeton, NJ, 1993. ISBN: 0-691-00049-2
Tadao Oda, Convex bodies and algebraic geometry, an introduction to the theory of toric varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 15, Springer-Verlag, Berlin, 1988. ISBN: 3-540-17600-4

Contributors

The following people have generously contributed code, improved existing code, or enhanced the documentation: Paul Aspinwall, Christine Berkesch, René Birkner, Justin Chen, Chris Eur, Matthew Faust, Corey Harris, Michael Loper, Diane Maclagan, Maryam Nowroozi, Erika Pirnes, Ritvik Ramkumar, Julie Rana, Mahrud Sayrafi, Alexandra Seceleanu, Mike Stillman, Sameera Vemulapalli, Elise Walker, Weikun Wang, Rachel Webb, Thomas Yahl, and Jay Yang.

Author

Gregory G. Smith <[email protected]>

Version

This documentation describes version 1.9 of NormalToricVarieties.

Source code

The source code from which this documentation is derived is in the file NormalToricVarieties.m2. The auxiliary files accompanying it are in the directory NormalToricVarieties/.

Exports

Types
- NormalToricVariety -- the class of all normal toric varieties
- ToricDivisor -- the class of all torus-invariant Weil divisors
- ToricMap -- the class of all torus-equivariant maps between normal toric varieties
Functions and commands
- affineSpace -- see affineSpace(ZZ) -- make an affine space as a normal toric variety
- cartesianProduct -- see cartesianProduct(Sequence) -- make the Cartesian product of normal toric varieties
- cartierDivisorGroup -- see cartierDivisorGroup(NormalToricVariety) -- compute the group of torus-invariant Cartier divisors
- classGroup -- see classGroup(NormalToricVariety) -- make the class group
- diagonalToricMap -- make a diagonal map into a Cartesian product
- fromCDivToPic -- see fromCDivToPic(NormalToricVariety) -- get the map from Cartier divisors to the Picard group
- fromCDivToWDiv -- see fromCDivToWDiv(NormalToricVariety) -- get the map from Cartier divisors to Weil divisors
- fromPicToCl -- see fromPicToCl(NormalToricVariety) -- get the map from Picard group to class group
- fromWDivToCl -- see fromWDivToCl(NormalToricVariety) -- get the map from the group of Weil divisors to the class group
- hirzebruchSurface -- see hirzebruchSurface(ZZ) -- make any Hirzebruch surface
- isAmple -- see isAmple(ToricDivisor) -- whether a torus-invariant Weil divisor is ample
- isCartier -- see isCartier(ToricDivisor) -- whether a torus-invariant Weil divisor is Cartier
- isDegenerate -- see isDegenerate(NormalToricVariety) -- whether a toric variety is degenerate
- isDominant -- see isDominant(ToricMap) -- whether a toric map is dominant
- isEffective -- see isEffective(ToricDivisor) -- whether a torus-invariant Weil divisor is effective
- isFano -- see isFano(NormalToricVariety) -- whether a normal toric variety is Fano
- isFibration -- see isFibration(ToricMap) -- whether a toric map is a fibration
- isNef -- see isNef(ToricDivisor) -- whether a torus-invariant Weil divisor is nef
- isProper -- see isProper(ToricMap) -- whether a toric map is proper
- isQQCartier -- see isQQCartier(ToricDivisor) -- whether a torus-invariant Weil divisor is QQ-Cartier
- kleinschmidt -- see kleinschmidt(ZZ,List) -- make any smooth normal toric variety having Picard rank two
- makeSimplicial -- see makeSimplicial(NormalToricVariety) -- make a birational simplicial toric variety
- makeSmooth -- see makeSmooth(NormalToricVariety) -- make a birational smooth toric variety
- normalToricVariety -- see normalToricVariety(List,List) -- make a normal toric variety
- orbits -- see orbits(NormalToricVariety) -- make a hashtable indexing the torus orbits (a.k.a. cones in the fan)
- picardGroup -- see picardGroup(NormalToricVariety) -- make the Picard group
- smallAmpleToricDivisor -- see smallAmpleToricDivisor(ZZ,ZZ) -- get a very ample toric divisor from the database
- smoothFanoToricVariety -- see smoothFanoToricVariety(ZZ,ZZ) -- get a smooth Fano toric variety from database
- toricBlowup -- see toricBlowup(List,NormalToricVariety,List) -- makes the toricBlowup of a normal toric variety along a torus orbit closure
- toricDivisor -- see toricDivisor(List,NormalToricVariety) -- make a torus-invariant Weil divisor
- toricProjectiveSpace -- see toricProjectiveSpace(ZZ) -- make a projective space as a normal toric variety
- weightedProjectiveSpace -- see weightedProjectiveSpace(List) -- make a weighted projective space
- weilDivisorGroup -- see weilDivisorGroup(NormalToricVariety) -- make the group of torus-invariant Weil divisors
Methods
- abstractSheaf(NormalToricVariety,AbstractVariety,CoherentSheaf) -- make the corresponding abstract sheaf
- abstractSheaf(NormalToricVariety,CoherentSheaf) -- see abstractSheaf(NormalToricVariety,AbstractVariety,CoherentSheaf) -- make the corresponding abstract sheaf
- abstractSheaf(NormalToricVariety,AbstractVariety,ToricDivisor) -- make the corresponding abstract sheaf
- abstractSheaf(NormalToricVariety,ToricDivisor) -- see abstractSheaf(NormalToricVariety,AbstractVariety,ToricDivisor) -- make the corresponding abstract sheaf
- abstractVariety(NormalToricVariety) -- see abstractVariety(NormalToricVariety,AbstractVariety) -- make the corresponding abstract variety
- abstractVariety(NormalToricVariety,AbstractVariety) -- make the corresponding abstract variety
- affineSpace(ZZ) -- make an affine space as a normal toric variety
- cartesianProduct(NormalToricVariety) -- see cartesianProduct(Sequence) -- make the Cartesian product of normal toric varieties
- cartesianProduct(Sequence) -- make the Cartesian product of normal toric varieties
- cartierDivisorGroup(NormalToricVariety) -- compute the group of torus-invariant Cartier divisors
- cartierDivisorGroup(ToricMap) -- make the induced map between groups of Cartier divisors.
- ch(CoherentSheaf) -- see ch(ZZ,CoherentSheaf) -- compute the Chern character of a coherent sheaf
- ch(ZZ,CoherentSheaf) -- compute the Chern character of a coherent sheaf
- chern(CoherentSheaf) -- see chern(ZZ,CoherentSheaf) -- compute the Chern class of a coherent sheaf
- chern(ZZ,CoherentSheaf) -- compute the Chern class of a coherent sheaf
- chi(CoherentSheaf) -- compute the Euler characteristic of a coherent sheaf
- intersectionRing(NormalToricVariety) -- see Chow ring -- make the rational Chow ring
- intersectionRing(NormalToricVariety,AbstractVariety) -- see Chow ring -- make the rational Chow ring
- classGroup(NormalToricVariety) -- make the class group
- classGroup(ToricMap) -- make the induced map between class groups
- components(NormalToricVariety) -- list the factors in a product
- cotangentSheaf(List,NormalToricVariety) -- see cotangentSheaf(NormalToricVariety) -- make the sheaf of Zariski 1-forms
- cotangentSheaf(NormalToricVariety) -- make the sheaf of Zariski 1-forms
- cotangentSheaf(ZZ,NormalToricVariety) -- see cotangentSheaf(NormalToricVariety) -- make the sheaf of Zariski 1-forms
- ctop(CoherentSheaf) -- compute the top Chern class of a coherent sheaf
- degree(ToricDivisor) -- make the degree of the associated rank-one reflexive sheaf
- diagonalToricMap(NormalToricVariety) -- see diagonalToricMap -- make a diagonal map into a Cartesian product
- diagonalToricMap(NormalToricVariety,ZZ) -- see diagonalToricMap -- make a diagonal map into a Cartesian product
- diagonalToricMap(NormalToricVariety,ZZ,Array) -- see diagonalToricMap -- make a diagonal map into a Cartesian product
- dim(NormalToricVariety) -- get the dimension of a normal toric variety
- - ToricDivisor -- see divisor arithmetic -- perform arithmetic on toric divisors
- ToricDivisor + ToricDivisor -- see divisor arithmetic -- perform arithmetic on toric divisors
- ToricDivisor - ToricDivisor -- see divisor arithmetic -- perform arithmetic on toric divisors
- ZZ * ToricDivisor -- see divisor arithmetic -- perform arithmetic on toric divisors
- entries(ToricDivisor) -- get the list of coefficients
- expression(NormalToricVariety) -- get the expression used to format for printing
- expression(ToricDivisor) -- get the expression used to format for printing
- fan(NormalToricVariety) -- make the 'Polyhedra' fan associated to the normal toric variety
- fromCDivToPic(NormalToricVariety) -- get the map from Cartier divisors to the Picard group
- fromCDivToWDiv(NormalToricVariety) -- get the map from Cartier divisors to Weil divisors
- fromPicToCl(NormalToricVariety) -- get the map from Picard group to class group
- fromWDivToCl(NormalToricVariety) -- get the map from the group of Weil divisors to the class group
- HH^ZZ(NormalToricVariety,CoherentSheaf) -- compute the cohomology of a coherent sheaf
- HH^ZZ(NormalToricVariety,SheafOfRings) -- see HH^ZZ(NormalToricVariety,CoherentSheaf) -- compute the cohomology of a coherent sheaf
- hilbertPolynomial(NormalToricVariety) -- compute the multivariate Hilbert polynomial
- hilbertPolynomial(NormalToricVariety,Ring) -- see hilbertPolynomial(NormalToricVariety) -- compute the multivariate Hilbert polynomial
- hilbertPolynomial(NormalToricVariety,SheafOfRings) -- see hilbertPolynomial(NormalToricVariety) -- compute the multivariate Hilbert polynomial
- hilbertPolynomial(NormalToricVariety,CoherentSheaf) -- compute the multivariate Hilbert polynomial
- hilbertPolynomial(NormalToricVariety,Ideal) -- see hilbertPolynomial(NormalToricVariety,CoherentSheaf) -- compute the multivariate Hilbert polynomial
- hilbertPolynomial(NormalToricVariety,Module) -- see hilbertPolynomial(NormalToricVariety,CoherentSheaf) -- compute the multivariate Hilbert polynomial
- hirzebruchSurface(ZZ) -- make any Hirzebruch surface
- ideal(NormalToricVariety) -- make the irrelevant ideal
- monomialIdeal(NormalToricVariety) -- see ideal(NormalToricVariety) -- make the irrelevant ideal
- ideal(ToricMap) -- make the ideal defining the closure of the image
- inducedMap(ToricMap) -- make the induced map between total coordinate rings (a.k.a. Cox rings)
- isAmple(ToricDivisor) -- whether a torus-invariant Weil divisor is ample
- isCartier(ToricDivisor) -- whether a torus-invariant Weil divisor is Cartier
- isComplete(NormalToricVariety) -- whether a toric variety is complete
- isDegenerate(NormalToricVariety) -- whether a toric variety is degenerate
- isDominant(ToricMap) -- whether a toric map is dominant
- isEffective(ToricDivisor) -- whether a torus-invariant Weil divisor is effective
- isFano(NormalToricVariety) -- whether a normal toric variety is Fano
- isFibration(ToricMap) -- whether a toric map is a fibration
- isNef(ToricDivisor) -- whether a torus-invariant Weil divisor is nef
- isProjective(NormalToricVariety) -- whether a toric variety is projective
- isProper(ToricMap) -- whether a toric map is proper
- isQQCartier(ToricDivisor) -- whether a torus-invariant Weil divisor is QQ-Cartier
- isSimplicial(NormalToricVariety) -- whether a normal toric variety is simplicial
- isSmooth(NormalToricVariety) -- whether a normal toric variety is smooth
- isSurjective(ToricMap) -- whether a toric map is surjective
- isVeryAmple(ToricDivisor) -- whether a torus-invariant Weil divisor is very ample
- isWellDefined(NormalToricVariety) -- whether a toric variety is well-defined
- isWellDefined(ToricDivisor) -- whether a toric divisor is well-defined
- isWellDefined(ToricMap) -- whether a toric map is well defined
- kleinschmidt(ZZ,List) -- make any smooth normal toric variety having Picard rank two
- latticePoints(ToricDivisor) -- compute the lattice points in the associated polytope
- makeSimplicial(NormalToricVariety) -- make a birational simplicial toric variety
- makeSmooth(NormalToricVariety) -- make a birational smooth toric variety
- map(NormalToricVariety,NormalToricVariety,Matrix) -- make a torus-equivariant map between normal toric varieties
- map(NormalToricVariety,NormalToricVariety,ZZ) -- make a torus-equivariant map between normal toric varieties
- matrix(ToricMap) -- get the underlying map of lattices
- max(NormalToricVariety) -- get the maximal cones in the associated fan
- monomials(ToricDivisor) -- list the monomials that span the linear series
- nefGenerators(NormalToricVariety) -- compute generators of the nef cone
- NormalToricVariety ** NormalToricVariety -- make the Cartesian product of two normal toric varieties
- NormalToricVariety ^ Array -- make a canonical projection map
- NormalToricVariety ^** ZZ -- make the Cartesian power of a normal toric variety
- NormalToricVariety _ Array -- make a canonical inclusion into a product
- NormalToricVariety _ ZZ -- make an irreducible torus-invariant divisor
- normalToricVariety(Fan) -- make a normal toric variety from a 'Polyhedra' fan
- normalToricVariety(List,List) -- make a normal toric variety
- normalToricVariety(Matrix) -- make a normal toric variety from a polytope
- normalToricVariety(Polyhedron) -- make a normal toric variety from a 'Polyhedra' polyhedron
- normalToricVariety(Ring) -- get the associated normal toric variety
- OO ToricDivisor -- make the associated rank-one reflexive sheaf
- orbits(NormalToricVariety) -- make a hashtable indexing the torus orbits (a.k.a. cones in the fan)
- orbits(NormalToricVariety,ZZ) -- get a list of the torus orbits (a.k.a. cones in the fan) of a given dimension
- picardGroup(NormalToricVariety) -- make the Picard group
- picardGroup(ToricMap) -- make the induced map between Picard groups
- polytope(ToricDivisor) -- makes the associated 'Polyhedra' polyhedron
- pullback(ToricMap,CoherentSheaf) -- make the pullback of a coherent sheaf under a toric map
- pullback(ToricMap,Module) -- see pullback(ToricMap,CoherentSheaf) -- make the pullback of a coherent sheaf under a toric map
- pullback(ToricMap,ToricDivisor) -- make the pullback of a Cartier divisor under a toric map
- rays(NormalToricVariety) -- get the rays of the associated fan
- ring(NormalToricVariety) -- make the total coordinate ring (a.k.a. Cox ring)
- sheaf(NormalToricVariety,Module) -- make a coherent sheaf
- OO _ NormalToricVariety -- see sheaf(NormalToricVariety,Ring) -- make a coherent sheaf of rings
- sheaf(NormalToricVariety) -- see sheaf(NormalToricVariety,Ring) -- make a coherent sheaf of rings
- sheaf(NormalToricVariety,Ring) -- make a coherent sheaf of rings
- smallAmpleToricDivisor(ZZ,ZZ) -- get a very ample toric divisor from the database
- smoothFanoToricVariety(ZZ,ZZ) -- get a smooth Fano toric variety from database
- source(ToricMap) -- get the source of the map
- support(ToricDivisor) -- make the list of irreducible divisors with nonzero coefficients
- target(ToricMap) -- get the target of the map
- todd(CoherentSheaf) -- compute the Todd class of a coherent sheaf
- todd(NormalToricVariety) -- see todd(CoherentSheaf) -- compute the Todd class of a coherent sheaf
- toricBlowup(List,NormalToricVariety) -- see toricBlowup(List,NormalToricVariety,List) -- makes the toricBlowup of a normal toric variety along a torus orbit closure
- toricBlowup(List,NormalToricVariety,List) -- makes the toricBlowup of a normal toric variety along a torus orbit closure
- ToricDivisor == ToricDivisor -- equality of toric divisors
- ToricDivisor == ZZ -- see ToricDivisor == ToricDivisor -- equality of toric divisors
- ZZ == ToricDivisor -- see ToricDivisor == ToricDivisor -- equality of toric divisors
- toricDivisor(List,NormalToricVariety) -- make a torus-invariant Weil divisor
- toricDivisor(NormalToricVariety) -- make the canonical divisor
- toricDivisor(Polyhedron) -- make the toric divisor associated to a polyhedron
- ToricMap * ToricMap -- make the composition of two toric maps
- ToricMap == ToricMap -- whether to toric maps are equal
- toricProjectiveSpace(ZZ) -- make a projective space as a normal toric variety
- normalToricVariety(ToricDivisor) -- see variety(ToricDivisor) -- get the underlying normal toric variety
- variety(ToricDivisor) -- get the underlying normal toric variety
- vector(ToricDivisor) -- make the vector of coefficients
- vertices(ToricDivisor) -- compute the vertices of the associated polytope
- weightedProjectiveSpace(List) -- make a weighted projective space
- weilDivisorGroup(NormalToricVariety) -- make the group of torus-invariant Weil divisors
- weilDivisorGroup(ToricMap) -- make the induced map between groups of Weil divisors
Symbols
- WeilToClass -- see normalToricVariety(List,List) -- make a normal toric variety

For the programmer

The object NormalToricVarieties is a package.