# cartierIndex -- the Cartier index of a Weil divisor

## Synopsis

• Usage:
N = cartierIndex(L,F)
• Inputs:
• L, a list
• F, an instance of the type Fan, a pure and full dimensional fan
• Outputs:

## Description

L must be a list of weights, exactly one for each ray of the fan. Then the Cartier index is the smallest strictly positive natural number $N$ such that $N$ times the Weil divisor is Cartier. If the Weil divisor defined by these weights is not QQ-Cartier, then $N$ would be infinity. In this case cartierIndex returns an error. Otherwise it returns $N$.

 i1 : F = fan posHull matrix {{1,5},{5,1}} o1 = {ambient dimension => 2 } number of generating cones => 1 number of rays => 2 top dimension of the cones => 2 o1 : Fan i2 : L = {2,2} o2 = {2, 2} o2 : List i3 : cartierIndex(L,F) o3 = 3

If we change the Weil divisor we get a different Cartier index:

 i4 : L = {3,3} o4 = {3, 3} o4 : List i5 : cartierIndex(L,F) o5 = 2