isCartier(ToricDivisor) -- whether a torus-invariant Weil divisor is Cartier

Synopsis

Function: isCartier
Usage:

isCartier D
Inputs:
- D, a toric divisor,
Outputs:
- a Boolean value, that is true if the divisor is Cartier

Description

A torus-invariant Weil divisor $D$ on a normal toric variety $X$ is Cartier if it is locally principal, meaning that $X$ has an open cover $\{U_i\}$ such that $D|_{U_i}$ is principal in $U_i$ for every $i$.

On a smooth variety, every Weil divisor is Cartier.

i1 : PP3 = toricProjectiveSpace 3;

i2 : assert all (3, i -> isCartier PP3_i)

On a simplicial toric variety, every torus-invariant Weil divisor is $\QQ$-Cartier, which means that every torus-invariant Weil divisor has a positive integer multiple that is Cartier.

i3 : W = weightedProjectiveSpace {2,5,7};

i4 : assert isSimplicial W

i5 : assert not isCartier W_0

i6 : assert isQQCartier W_0

i7 : assert isCartier (35*W_0)

In general, the Cartier divisors are only a subgroup of the Weil divisors.

i8 : X = normalToricVariety (id_(ZZ^3) | -id_(ZZ^3));

i9 : assert not isCartier X_0

i10 : assert not isQQCartier X_0

i11 : K = toricDivisor X;

o11 : ToricDivisor on X

i12 : assert isCartier K

Ways to use this method: