Consider $\overline{M}_{g+1}(C,1)$ where $C$ is a (fixed) genus g curve. Geometrically, the points in this moduli space correspond to maps $E\cup_p C \to C$ whose domain is the union of an elliptic curve $E$ attached to $C$ at a node $p$. The map is an isomorphism on $C$ and collapses the component $E $ to the point $p\in C$. Thus the moduli space is isomorphic to $\overline{M}_{1,1} \times C$

which is of the expected dimension 2. Consequently, the virtual class is just equal to the usual class and the map

$ \overline{M}\_{1,1} \times C \to \overline{M}_{g+1}$

is an inclusion and its image is clearly an explicit stratum in the boundary.

P.S. I can't figure out why I can't get this last equation to typeset correctly.

Consider $\overline{M}_{g+1}(C,1)$ where $C$ is a (fixed) genus g curve. Geometrically, the points in this moduli space correspond to maps $E\cup_p C \to C$ whose domain is the union of an elliptic curve $E$ attached to $C$ at a node $p$. The map is an isomorphism on $C$ and collapses the component $E $ to the point $p\in C$. Thus the moduli space is isomorphic to $\overline{M}_{1,1} \times C$

which is of the expected dimension 2. Consequently, the virtual class is just equal to the usual class and the map

$ \overline{M}\_{1,1} \times C \to \overline{M}_{g+1}$

is an inclusion and its image is clearly an explicit stratum in the boundary.

Consider $\overline{M}_{g+1}(C,1)$ where $C$ is a (fixed) genus g curve. Geometrically, the points in this moduli space correspond to maps $E\cup_p C \to C$ whose domain is the union of an elliptic curve $E$ attached to $C$ at a node $p$. The map is an isomorphism on $C$ and collapses the component $E $ to the point $p\in C$. Thus the moduli space is isomorphic to $\overline{M}_{1,1} \times C$

which is of the expected dimension 2. Consequently, the virtual class is just equal to the usual class and the map

$ \overline{M}_{1,1} \times C \to \overline{M}_{g+1}$

is an inclusion and its image is clearly an explicit stratum in the boundary.

P.S. I can't figure out why I can't get this last equation to typeset correctly.

Consider $\overline{M}_{g+1}(C,1)$ where $C$ is a (fixed) genus g curve. Geometrically, the points in this moduli space correspond to maps $E\cup_p C \to C$ whose domain is the union of an elliptic curve $E$ attached to $C$ at a node $p$. The map is an isomorphism on $C$ and collapses the component $E $ to the point $p\in C$. Thus the moduli space is isomorphic to $\overline{M}_{1,1} \times C$

which is of the expected dimension 2. Consequently, the virtual class is just equal to the usual class and the map

$ \overline{M}\_{1,1} \times C \to \overline{M}_{g+1}$

is an inclusion and its image is clearly an explicit stratum in the boundary.

P.S. I can't figure out why I can't get this last equation to typeset correctly.