By identifying the coefficients of an effective irreducible torus-invariant divisors with exponents of the generators of the total coordinate ring, each toric divisor on a NormalToricVariety corresponds to a monomial. This method function returns all of the monomials corresponding to linear equivalent toric divisors.
This method function assumes that the underlying toric variety is projective.
Projective space is especially simple.
i1 : PP2 = toricProjectiveSpace 2; |
i2 : D1 = 5*PP2_0
o2 = 5*PP2
0
o2 : ToricDivisor on PP2
|
i3 : M1 = elapsedTime monomials D1
-- 0.347121 seconds elapsed
5 4 4 2 3 3 2 3 3 2 2 2 2 2 3 2 4
o3 = {x , x x , x x , x x , x x x , x x , x x , x x x , x x x , x x , x x ,
2 1 2 0 2 1 2 0 1 2 0 2 1 2 0 1 2 0 1 2 0 2 1 2
------------------------------------------------------------------------
3 2 2 3 4 5 4 2 3 3 2 4 5
x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x }
0 1 2 0 1 2 0 1 2 0 2 1 0 1 0 1 0 1 0 1 0
o3 : List
|
i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring variety D1)) -- 0.000966894 seconds elapsed |
Toric varieties of Picard-rank 2 are slightly more interesting.
i5 : FF2 = hirzebruchSurface 2; |
i6 : D2 = 2*FF2_0 + 3 * FF2_1
o6 = 2*FF2 + 3*FF2
0 1
o6 : ToricDivisor on FF2
|
i7 : M2 = elapsedTime monomials D2
-- 0.124263 seconds elapsed
2 3 2 3 2 3
o7 = {x x , x x , x x x , x x }
1 3 1 2 0 1 2 0 1
o7 : List
|
i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring variety D2)) -- 0.000911619 seconds elapsed |
i9 : X = kleinschmidt (5, {1,2,3});
|
i10 : D3 = 3*X_0 + 5*X_1
o10 = 3*X + 5*X
0 1
o10 : ToricDivisor on X
|
i11 : m3 = elapsedTime # monomials D3 -- 10.3401 seconds elapsed o11 = 7909 |
i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3)) -- 0.044155 seconds elapsed |
By exploiting latticePoints, this method function avoids using the basis function.