On the Moduli space $\overline{M}_{g}$ of genus $g$ stable curves the Hodge class $\lambda$ induces a morphism $f$ on a projective variety contracting the boundary, that is the exceptional locus of $f$ coincides with the boundary of the moduli space.

Is there a line bundle $L$ on the moduli space of pointed curves (as instance on $\overline{M}_{2,1}$ and on the moduli space of $2$-pointed elliptic curves) with the same property (i.e. a line bundle which induces a birational morphism whose exceptional locus coincides with the boundary) ?

On the Moduli space $\overline{M}_{g}$ of genus $g$ stable curves the Hodge class $\lambda$ induces a birational morphism $f$ on a projective variety contracting the boundary, that is the exceptional locus of $f$ coincides with the boundary of the moduli space.

Is there a line bundle $L$ on the moduli space of pointed curves (as instance on $\overline{M}_{2,1}$ and on the moduli space of $2$-pointed elliptic curves) with the same property (i.e. a line bundle which induces a birational morphism whose exceptional locus coincides with the boundary) ?