I would like to understand if a certain variety is $\mathbb{Q}$-factorial (i.e., if every Weil divisor $D$ has a multiple $mD$ that is Cartier). This property can be deduced by a local picture around the singularities; is there some computational way to find it out just looking at the local ring of the singularity?
The cases I'm interested in all come from G.I.T. quotients of a representation of the multiplicative group $\mathbb{G_m}$ by $\mathbb{G}_m$ itself. For instance, one example is $$L_2\oplus L_{2}\oplus L_{-1}\oplus L_{-3}\ //\ \mathbb{G_m}$$ where of course the action of $t\in \mathbb{G}_m$ on the affine line $L_i$ is the multiplication by $t^i$. It is not very hard to produce the coordinate ring in these cases. In the situation above, it is just the weight zero component of the ring $k[x_1,x_2,x_3,x_4]$ where $x_1$ and $x_2$ have weight $2$, $x_3$ has weight $-1$ and $x_4$ has weight $-3$.
Thank you all very much!
Francesco
