ToricDivisor -- the class of all torus-invariant Weil divisors

Description

A torus-invariant Weil divisor on a normal toric variety is an integral linear combination of the irreducible torus-invariant divisors. The irreducible torus-invariant divisors correspond to the rays. In this package, the rays are ordered and indexed by the nonnegative integers.

The first examples illustrates some torus-invariant Weil divisors on projective $2$-space.

i1 : PP2 = toricProjectiveSpace 2;

i2 : D1 = toricDivisor ({2,-7,3}, PP2)

o2 = 2*PP2  - 7*PP2  + 3*PP2
          0        1        2

o2 : ToricDivisor on PP2

i3 : D2 = 2*PP2_0 + 4*PP2_2

o3 = 2*PP2  + 4*PP2
          0        2

o3 : ToricDivisor on PP2

i4 : D1+D2

o4 = 4*PP2  - 7*PP2  + 7*PP2
          0        1        2

o4 : ToricDivisor on PP2

i5 : D1-D2

o5 = - 7*PP2  - PP2
            1      2

o5 : ToricDivisor on PP2

i6 : K = toricDivisor PP2

o6 = - PP2  - PP2  - PP2
          0      1      2

o6 : ToricDivisor on PP2

One can easily extract individual coefficients or the list of coefficients.

i7 : D1#0

o7 = 2

i8 : D1#1

o8 = -7

i9 : D1#2

o9 = 3

i10 : entries D1

o10 = {2, -7, 3}

o10 : List

i11 : entries K

o11 = {-1, -1, -1}

o11 : List

Functions and methods returning a toric divisor :

- 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
NormalToricVariety _ ZZ -- make an irreducible torus-invariant divisor
pullback(ToricMap,ToricDivisor) -- make the pullback of a Cartier divisor under a toric map
smallAmpleToricDivisor(ZZ,ZZ) -- get a very ample toric divisor from the database
toricDivisor -- see toricDivisor(List,NormalToricVariety) -- make a torus-invariant Weil divisor
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

Methods that use a toric divisor :

abstractSheaf(NormalToricVariety,AbstractVariety,ToricDivisor) -- make the corresponding abstract sheaf
abstractSheaf(NormalToricVariety,ToricDivisor) -- see abstractSheaf(NormalToricVariety,AbstractVariety,ToricDivisor) -- make the corresponding abstract sheaf
degree(ToricDivisor) -- make the degree of the associated rank-one reflexive sheaf
entries(ToricDivisor) -- get the list of coefficients
expression(ToricDivisor) -- get the expression used to format for printing
isAmple(ToricDivisor) -- whether a torus-invariant Weil divisor is ample
isCartier(ToricDivisor) -- whether a torus-invariant Weil divisor is Cartier
isEffective(ToricDivisor) -- whether a torus-invariant Weil divisor is effective
isNef(ToricDivisor) -- whether a torus-invariant Weil divisor is nef
isQQCartier(ToricDivisor) -- whether a torus-invariant Weil divisor is QQ-Cartier
isVeryAmple(ToricDivisor) -- whether a torus-invariant Weil divisor is very ample
isWellDefined(ToricDivisor) -- whether a toric divisor is well-defined
latticePoints(ToricDivisor) -- compute the lattice points in the associated polytope
monomials(ToricDivisor) -- list the monomials that span the linear series
OO ToricDivisor -- make the associated rank-one reflexive sheaf
polytope(ToricDivisor) -- makes the associated 'Polyhedra' polyhedron
support(ToricDivisor) -- make the list of irreducible divisors with nonzero coefficients
ToricDivisor == ToricDivisor -- equality of toric divisors
ToricDivisor == ZZ -- see ToricDivisor == ToricDivisor -- equality of toric divisors
ZZ == ToricDivisor -- see ToricDivisor == ToricDivisor -- equality of toric divisors
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

For the programmer

The object ToricDivisor is a type, with ancestor classes HashTable < Thing.

ToricDivisor -- the class of all torus-invariant Weil divisors

Description

See also

Functions and methods returning a toric divisor :

Methods that use a toric divisor :

For the programmer