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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 ?

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  • $\begingroup$ Is it possible to give some more less explicit description of $\bar M_{1,2}$? $\endgroup$ – Dmitri Panov Jun 13 '11 at 22:42
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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.

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