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NormalToricVarieties :: latticePoints(ToricDivisor)

latticePoints(ToricDivisor) -- computes the lattice points in the associated polytope

Synopsis

Description

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)

See also