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NormalToricVarieties :: divisor arithmetic

divisor arithmetic -- perform arithmetic on toric divisors

Synopsis

The set of torus-invariant Weil divisors forms an abelian group under addition. The basic operations arising from this structure, including addition, subtraction, negation, and scalar multiplication by integers, are available.

We illustrate a few of the possibilities on one variety.

i1 : X = normalToricVariety(id_(ZZ^3) | -id_(ZZ^3));
i2 : # rays X

o2 = 8
i3 : D = toricDivisor({2,-7,3,0,7,5,8,-8}, X)

o3 = 2*X  - 7*X  + 3*X  + 7*X  + 5*X  + 8*X  - 8*X
        0      1      2      4      5      6      7

o3 : ToricDivisor on X
i4 : K = toricDivisor X

o4 = - X  - X  - X  - X  - X  - X  - X  - X
        0    1    2    3    4    5    6    7

o4 : ToricDivisor on X
i5 : D + K

o5 = X  - 8*X  + 2*X  - X  + 6*X  + 4*X  + 7*X  - 9*X
      0      1      2    3      4      5      6      7

o5 : ToricDivisor on X
i6 : assert(D + K == K + D)
i7 : D - K

o7 = 3*X  - 6*X  + 4*X  + X  + 8*X  + 6*X  + 9*X  - 7*X
        0      1      2    3      4      5      6      7

o7 : ToricDivisor on X
i8 : assert(D - K == -(K-D))
i9 : - K

o9 = X  + X  + X  + X  + X  + X  + X  + X
      0    1    2    3    4    5    6    7

o9 : ToricDivisor on X
i10 : assert(-K == (-1)*K)
i11 : 7*D

o11 = 14*X  - 49*X  + 21*X  + 49*X  + 35*X  + 56*X  - 56*X
          0       1       2       4       5       6       7

o11 : ToricDivisor on X
i12 : assert(7*D == (3+4)*D)
i13 : assert(7*D == 3*D + 4*D)
i14 : -3*D + 7*K

o14 = - 13*X  + 14*X  - 16*X  - 7*X  - 28*X  - 22*X  - 31*X  + 17*X
            0       1       2      3       4       5       6       7

o14 : ToricDivisor on X
i15 : assert(-3*D+7*K == (-2*D+8*K) + (-D-K))

See also