The moduli space of stable $2$-pointed genus $1$ curves has cyclic quotient singularities, so any Weil divisor on $\overline{M}\_{1,2}$ is $\mathbb{Q}$-Cartier.


**Is one of the two boundary divisors $\Delta\_{irr}$ and $\Delta\_{0,2}$ of $\overline{M}\_{1,2}$ Cartier ?** 



Cartier divisors on the moduli space of two pointed elliptic curves
The singularities of $\overline{M}\_{1,2}$ are located as follows:


a singularity of type $\frac{1}{4}(2,3)$ representing an elliptic curve of Weierstrass representation $C\_{4}$ with marked points $[0:1:0]$ and $[0:0:1]$;


a singularity of type $\frac{1}{3}(2,4)$ representing an elliptic curve of Weierstrass representation $C\_{6}$ with marked points $[0:1:0]$ and $[0:1:1]$.


a singularity of type $\frac{1}{6}(2,4)$ representing a reducible curve whose irreducible components are an elliptic curve of type $C\_{6}$ and a smooth rational curve connected by a node;


a singularity of type $\frac{1}{4}(2,6)$ representing a reducible curve whose irreducible components are an elliptic curve of type $C\_{4}$ and a smooth rational curve connected by a node.


**The divisor $\Delta\_{irr}$ is contained in the smooth locus of $\overline{M}\_{1,2}$ and it is Cartier, while the other boundary divisor contains two singular points, it is $\mathbb{Q}$-Cartier but not Cartier.**


