top
|
index
|
Macaulay2 web site
NormalToricVarieties : Table of Contents
NormalToricVarieties
-- normal toric varieties
affineSpace
-- make an affine space
Basic invariants and properties of normal toric varieties
blowup
-- makes the blowup of a normal toric variety along a torus orbit closure
cDiv
-- make the group of torus-invariant Cartier divisors
cl
-- make the class group
cotangentSheaf(NormalToricVariety)
-- make the sheaf of Zariski 1-forms
dim(NormalToricVariety)
-- get the dimension of a normal toric variety
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
-- get the map from Cartier divisors to the Picard group
fromCDivToWDiv
-- get the map from Cartier divisors to Weil divisors
fromPicToCl
-- get the map from Picard group to class group
fromWDivToCl
-- get the map from Weil divisors to the class group
HH^ZZ(NormalToricVariety,CoherentSheaf)
-- compute the cohomology of a coherent sheaf
hirzebruchSurface
-- make a Hirzebruch surface
ideal(NormalToricVariety)
-- make the irrelevant ideal
isAmple
-- whether a torus-invariant Weil divisor is ample
isCartier
-- whether a torus-invariant Weil divisor is Cartier
isComplete(NormalToricVariety)
-- whether a toric variety is complete
isDegenerate
-- whether a toric variety is degenerate
isEffective
-- whether a torus-invariant Weil divisor is effective
isFano
-- whether a normal toric variety is Fano
isNef
-- whether a torus-invariant Weil divisor is nef
isProjective
-- whether a toric variety is projective
isQQCartier
-- whether a torus-invariant Weil divisor is QQ-Cartier
isSimplicial(NormalToricVariety)
-- whether a toric variety is simplicial
isSmooth(NormalToricVariety)
-- whether a toric variety is smooth
isVeryAmple(ToricDivisor)
-- whether a torus-invariant Weil divisor is very ample
isWellDefined(NormalToricVariety)
-- whether a toric variety is well-defined
kleinschmidt
-- make a smooth toric variety with Picard rank two
latticePoints(ToricDivisor)
-- computes the lattice points in the associated polytope
makeSimplicial
-- make a simplicial toric variety
makeSmooth
-- make a birational smooth toric variety
Making normal toric varieties
max(NormalToricVariety)
-- get the maximal cones in the associated fan
NormalToricVariety
-- the class of all normal toric varieties
normalToricVariety
-- make a normal toric variety
NormalToricVariety ** NormalToricVariety
-- Cartesian product
NormalToricVariety _ ZZ
-- make a torus-invariant prime divisor
normalToricVariety(Fan)
-- make a normal toric variety from a 'Polyhedra' fan
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
-- make a hashtable indexing the proper torus orbits
orbits(NormalToricVariety,ZZ)
-- get a list of the torus orbits of a given dimension
pic
-- make the Picard group
polytope(ToricDivisor)
-- makes the associated 'Polyhedra' polyhedron
projectiveSpace
-- make a 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
smoothFanoToricVariety
-- get a smooth Fano toric variety from database
support(ToricDivisor)
-- make the list of prime divisors with nonzero coefficients
ToricDivisor
-- the class of all torus-invariant Weil divisors
toricDivisor
-- make a torus-invariant Weil divisor
ToricDivisor + ToricDivisor
-- arithmetic of toric divisors
toricDivisor(NormalToricVariety)
-- make the canonical divisor
Total coordinate rings and coherent sheaves
variety(ToricDivisor)
-- get the underlying normal toric variety
vector(ToricDivisor)
-- make the vector of coefficients
vertices(ToricDivisor)
-- computes the vertices of the associated polytope
wDiv
-- make the group of torus-invariant Weil divisors
weightedProjectiveSpace
-- make a weighted projective space
Working with divisors and their associated groups