toricDivisor(Polyhedron) -- make the toric divisor associated to a polyhedron

Synopsis

Function: toricDivisor
Usage:

toricDivisor P
Inputs:
- P, a convex polyhedron, whose vertices are lattice points
Optional inputs:
- CoefficientRing => a ring, default value QQ, that determines the coefficient ring of the total coordinate ring of the underlying normal toric variety
- Variable => a symbol, default value x, that specifies the base symbol for the indexed variables in the total coordinate ring of the underlying normal toric variety
- WeilToClass => a matrix, default value null, that specifies the map from the group of torus-invariant Weil divisors to the class group of the underlying normal toric variety
Outputs:
- a toric divisor, corresponding to the given convex lattice polytope on the normal toric variety defined by the inner normal fan

Description

A convex lattice polytope corresponds to a pair: the normal toric variety determined by its normal fan and toric divisor. The coefficient of the $i$-th irreducible torus-invariant divisor is determined by the supporting hyperplane to the polytope whose normal vector is the minimal lattice point on the $i$-th ray.

Our example demonstrates how different triangles correspond to toric divisors on the projective plane.

i1 : P1 = convexHull matrix{{0,1,0},{0,0,1}};

i2 : D1 = toricDivisor P1

o2 = D
      2

o2 : ToricDivisor on normalToricVariety ({{1, 0}, {0, 1}, {-1, -1}}, {{0, 1}, {0, 2}, {1, 2}})

i3 : X = variety D1;

i4 : D1

o4 = X
      2

o4 : ToricDivisor on X

i5 : P2 = convexHull matrix{{-1,0,-1},{0,0,1}};

i6 : D2 = toricDivisor P2

o6 = D
      0

o6 : ToricDivisor on normalToricVariety ({{1, 0}, {0, 1}, {-1, -1}}, {{0, 1}, {0, 2}, {1, 2}})

i7 : P3 = convexHull matrix{{0,1,0},{-1,-1,0}};

i8 : D3 = toricDivisor P3

o8 = D
      1

o8 : ToricDivisor on normalToricVariety ({{1, 0}, {0, 1}, {-1, -1}}, {{0, 1}, {0, 2}, {1, 2}})

i9 : P4 = convexHull matrix{{-1,2,-1},{-1,-1,2}};

i10 : D4 = toricDivisor(P4, CoefficientRing => ZZ/2)

o10 = D  + D  + D
       0    1    2

o10 : ToricDivisor on normalToricVariety ({{1, 0}, {0, 1}, {-1, -1}}, {{0, 1}, {0, 2}, {1, 2}})

i11 : ring variety D4

      ZZ
o11 = --[x ..x ]
       2  0   2

o11 : PolynomialRing

This method function creates both the toric divisor and the underlying normal toric variety.

Ways to use this method: