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NormalToricVarieties :: monomials(ToricDivisor)

monomials(ToricDivisor) -- list the monomials that span the linear series

Synopsis

Description

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
 -- 0.0712048 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))
 -- 0.000728887 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
 -- 0.0756581 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))
 -- 0.000800225 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
 -- 9.09946 seconds elapsed

o11 = 7909
i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
 -- 0.0388467 seconds elapsed

By exploiting latticePoints, this method function avoids using the basis function.

See also