Every curve of genus $g\leq 2$ has non trivial automorphisms. So the fiber of the forgetful morphism $\pi:\bar M\_{g,1}\rightarrow\bar M\_{g}$ over $[C]$ is not isomorphic to $C$ but to $C/Aut(C)$ (I am considering the coarse moduli space instead of the stack).


The general fiber of $\pi:\bar M\_{1,2}\rightarrow\bar M\_{1,1}$ is isomorphic to $\mathbb{P}^{1}$ and the same is true for $\pi:\bar M\_{2,1}\rightarrow\bar M\_{2}$. Now $\bar M\_{1,2}$ is a rational surface and $\bar M\_{2,1}$ is rational $4$-fold.


The question is the following:
**Does anyone know an explicit description of $\bar M\_{1,2}$ and $\bar M\_{2,1} ?$**
My feeling is that $\bar M\_{1,2}$ should be a blow-up of a ruled surface.



Moduli of pointed Curves
The moduli space $\overline{M}\_{1,2}$ can not be the blow up of $\mathbb{P}^{2}$ in $2$ points because its rational Picard group has rank $2$, indeed it is generated by the divisor parametrizing genus $0$ irreducible nodal curves with $2$ marked points and the divisor parametrizing reducible curves whose components are a smooth genus $1$ curve and a smooth genus $0$ curve with $2$ marked points.


The moduli space of genus $1$ stable curves with $2$ marked points is a rational surface with four singular points. Two singular points lie in $M\_{1,2}$, and are:


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 $[0:1:1]$.


The remaining two singular points lie on the boundary divisor $\Delta\_{0,2}$, and are:


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.


From this one can prove that:


**The moduli space of genus $1$ stable curves with $2$ marked points is isomorphic to a weighted blow up of the weighted projective plane $\mathbb{P}(1,2,3)$ in its smooth point $[1:0:0]$. In particular $\overline{M}\_{1,2}$ is a toric variety.**


