On a complete normal toric variety, the polyhedron associated to a Cartier divisor is a lattice polytope. Given a torus-invariant Cartier divisor on a normal toric variety, this method returns an integer matrix whose columns correspond to the lattices points contained in the associated polytope. For a non-effective Cartier divisor, this method returns
null.
On the projective plane, the associate polytope is either empty, a point, or a triangle.
PP2 = projectiveSpace 2; |
vertices (-PP2_0) |
null === vertices (- PP2_0) |
latticePoints (0*PP2_0) |
isAmple PP2_0 |
V1 = latticePoints (PP2_0) |
X1 = normalToricVariety V1; |
set rays X1 === set rays PP2 |
max X1 === max PP2 |
isAmple (2*PP2_0) |
V2 = latticePoints (2*PP2_0) |
X2 = normalToricVariety(V2, MinimalGenerators => true); |
rays X2 === rays X1 |
max X2 === max X1 |
In this singular example, we see that all the lattice points in the polytope arising from a divisor
2D do not come from the lattice points in the polytope arising from
D.
Y = normalToricVariety matrix {{0,1,0,0,1},{0,0,1,0,1},{0,0,0,1,1},{0,0,0,0,3}}; |
D = 3*Y_0; |
latticePoints D |
latticePoints (2*D) |