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 -- 0.132243 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)) -- 0.000991989 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 -- 0.095068 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)) -- 0.000877565 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 -- 7.69666 seconds elapsed o11 = 7909 |
i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3)) -- 0.0507367 seconds elapsed |
By exploiting latticePoints, this method function avoids using the basis function.