# NormalToricVariety _ ZZ -- make an irreducible torus-invariant divisor

## Synopsis

• Usage:
X_i
• Operator: _
• Inputs:
• i, an integer, indexing a ray in the fan associated to X
• Outputs:
• , namely the irreducible torus-invariant divisor associated to the i-th ray in the fan of X

## Description

The irreducible torus-invariant divisors on a normal toric variety correspond to the rays in the associated fan. In this package, the rays are ordered and indexed by the nonnegative integers. Given a normal toric variety and nonnegative integer, this method returns the corresponding irreducible torus-invariant divisor. The most convenient way to make a general torus-invariant Weil divisor is to simply write the appropriate linear combination of these torus-invariant Weil divisors.

There are three irreducible torus-invariant divisors on the projective plane.

 `i1 : PP2 = toricProjectiveSpace 2;` ```i2 : PP2_0 o2 = PP2 0 o2 : ToricDivisor on normalToricVariety((({{-1, -1}, {1, 0}, {0, 1}}(,({{0, 1}, {0, 2}, {1, 2}} )))))``` ```i3 : PP2_1 o3 = PP2 1 o3 : ToricDivisor on normalToricVariety((({{-1, -1}, {1, 0}, {0, 1}}(,({{0, 1}, {0, 2}, {1, 2}} )))))``` ```i4 : PP2_2 o4 = PP2 2 o4 : ToricDivisor on normalToricVariety((({{-1, -1}, {1, 0}, {0, 1}}(,({{0, 1}, {0, 2}, {1, 2}} )))))``` `i5 : assert (- PP2_0 - PP2_1 - PP2_2 === toricDivisor PP2)`

A torus-invariant Weil divisor is irreducible if and only if its support has a single element.

 `i6 : X = normalToricVariety (id_(ZZ^3) | -id_(ZZ^3));` ```i7 : X_0 o7 = X 0 o7 : ToricDivisor on normalToricVariety((({{1, 1, 1}, {-1, 1, 1}, {1, -1, 1}, {-1, -1, 1}, {1, 1, -1}, {-1, 1, -1}, {1, -1, -1}, {-1, -1, -1}}(,({{0, 1, 2, 3}, {0, 1, 4, 5}, {0, 2, 4, 6}, {1, 3, 5, 7}, {2, 3, 6, 7}, {4, 5, 6, 7}} )))))``` ```i8 : support X_0 o8 = {0} o8 : List``` `i9 : assert( # support X_0 === 1)` ```i10 : K = toricDivisor X o10 = - X - X - X - X - X - X - X - X 0 1 2 3 4 5 6 7 o10 : ToricDivisor on normalToricVariety((({{1, 1, 1}, {-1, 1, 1}, {1, -1, 1}, {-1, -1, 1}, {1, 1, -1}, {-1, 1, -1}, {1, -1, -1}, {-1, -1, -1}}(,({{0, 1, 2, 3}, {0, 1, 4, 5}, {0, 2, 4, 6}, {1, 3, 5, 7}, {2, 3, 6, 7}, {4, 5, 6, 7}} )))))``` ```i11 : support K o11 = {0, 1, 2, 3, 4, 5, 6, 7} o11 : List```