top
|
index
|
Macaulay2 web site
NormalToricVarieties : Table of Contents
NormalToricVarieties
-- a package for working with normal toric varieties
abstractSheaf(NormalToricVariety,AbstractVariety,CoherentSheaf)
-- make the corresponding abstract sheaf
abstractSheaf(NormalToricVariety,AbstractVariety,ToricDivisor)
-- make the corresponding abstract sheaf
abstractVariety(NormalToricVariety,AbstractVariety)
-- make the corresponding abstract variety
affineSpace(ZZ)
-- make an affine space
Basic invariants and properties of normal toric varieties
cartierDivisorGroup(NormalToricVariety)
-- compute the group of torus-invariant Cartier divisors
ch(ZZ,CoherentSheaf)
-- compute the Chern character 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
classGroup(NormalToricVariety)
-- make the class group
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
dim(NormalToricVariety)
-- get the dimension of a normal toric variety
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
hilbertPolynomial(NormalToricVariety)
-- compute the multivariate Hilbert polynomial
hilbertPolynomial(NormalToricVariety,CoherentSheaf)
-- compute the multivariate Hilbert polynomial
hirzebruchSurface(ZZ)
-- make a Hirzebruch surface
ideal(NormalToricVariety)
-- make the irrelevant ideal
intersectionRing(NormalToricVariety,AbstractVariety)
-- make the rational Chow ring
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
isEffective(ToricDivisor)
-- whether a torus-invariant Weil divisor is effective
isFano(NormalToricVariety)
-- whether a normal toric variety is Fano
isNef(ToricDivisor)
-- whether a torus-invariant Weil divisor is nef
isProjective(NormalToricVariety)
-- whether a toric variety is projective
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
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
kleinschmidt(ZZ,List)
-- make a smooth normal toric variety with 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
Making normal toric varieties
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
-- the class of all normal toric varieties
NormalToricVariety ** NormalToricVariety
-- make the Cartesian product of normal toric varieties
NormalToricVariety ^** ZZ
-- make the Cartesian power of a normal toric variety
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
polytope(ToricDivisor)
-- makes the associated 'Polyhedra' polyhedron
projective space
-- various methods for constructing projective space
rays(NormalToricVariety)
-- get the rays of the associated fan
Resolution of singularities
ring(NormalToricVariety)
-- make the total coordinate ring (a.k.a. Cox ring)
sheaf(NormalToricVariety,Module)
-- make a coherent sheaf
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
support(ToricDivisor)
-- make the list of irreducible divisors with nonzero coefficients
todd(CoherentSheaf)
-- compute the Todd class of a coherent sheaf
toricBlowup(List,NormalToricVariety,List)
-- makes the toricBlowup of a normal toric variety along a torus orbit closure
ToricDivisor
-- the class of all torus-invariant Weil divisors
ToricDivisor + ToricDivisor
-- perform arithmetic on toric divisors
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
toricProjectiveSpace(ZZ)
-- make a projective space
Total coordinate rings and coherent sheaves
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
Working with divisors and their associated groups