toricDivisor(NormalToricVariety) -- make the canonical divisor

Synopsis

Function: toricDivisor
Usage:

toricDivisor X
Inputs:
- X, a normal toric variety,
Optional inputs:
- CoefficientRing => ..., default value QQ, make the toric divisor associated to a polyhedron
- Variable => ..., default value x, make the toric divisor associated to a polyhedron
- WeilToClass => ..., default value null, make the toric divisor associated to a polyhedron
Outputs:
- a toric divisor, specifically minus the sum of all the irreducible torus-invariant divisors

Description

On a smooth normal toric variety, the canonical divisor equals minus the sum of all the torus-invariant irreducible divisors. For a singular toric variety, this divisor may not be Cartier or even $\QQ$-Cartier. Nevertheless, the associated coherent sheaf, whose local sections are rational functions with at least simple zeros along the irreducible divisors, is the dualizing sheaf.

The first example illustrates the canonical divisor on projective space.

i1 : PP3 = toricProjectiveSpace 3;

i2 : assert(isSmooth PP3 and isProjective PP3)

i3 : K = toricDivisor PP3

o3 = - PP3  - PP3  - PP3  - PP3
          0      1      2      3

o3 : ToricDivisor on PP3

i4 : assert(all(entries K, i -> i === -1) and isWellDefined K)

i5 : omega = OO K

          1
o5 = OO    (-4)
       PP3

o5 : coherent sheaf on PP3, free of rank 1

i6 : assert(HH^3(PP3, OO_PP3(-7) ** omega) === HH^0(PP3, OO_PP3(7)))

The second example illustrates that duality also holds on complete singular nonprojective toric varieties.

i7 : X = normalToricVariety({{1,0,0},{0,1,0},{0,0,1},{0,-1,-1},{-1,0,-1},{-2,-1,0}},{{0,1,2},{0,1,3},{1,3,4},{1,2,4},{2,4,5},{0,2,5},{0,3,5},{3,4,5}});

i8 : assert(isComplete X and not isProjective X and not isSmooth X)

i9 : KX = toricDivisor X

o9 = - X  - X  - X  - X  - X  - X
        0    1    2    3    4    5

o9 : ToricDivisor on X

i10 : assert(all(entries KX, i -> i === -1) and isWellDefined KX)

i11 : isCartier KX

o11 = false

i12 : omegaX = OO KX

         1
o12 = OO   (-3, -3, -4)
        X

o12 : coherent sheaf on X, free of rank 1

i13 : assert( HH^0(X, OO_X(1,2,5)) === HH^3(X, OO_X(-1,-2,-5) ** omegaX) )

Ways to use this method: