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NormalToricVarieties :: ToricDivisor

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

See also

Functions and methods returning a toric divisor :

Methods that use a toric divisor :

For the programmer

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