Question

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) ?

Answer 1

A detailed description of the nef cones of the examples you mention is given in the thesis of William Rulla (The birational geometry of $\overline{M}_3$ and $\overline{M}_{2,1}$, The University of Texas at Austin, 2001.) From this, it follows that there are no such line bundles for the two examples that you mention. I would guess that the same holds in general as well.

Answer 2

The Hodge class $\lambda_{2,1}$ on $\overline{M}_{2,1}$

is the pull-back of the Hodge class $\lambda$ on $\overline{M}_{2}$ via the forgetful morphism

$$\pi:\overline{M}_{2,1}\rightarrow\overline{M}_2,$$

that is $\lambda_{2,1} = \pi^{*}\lambda$.

So $\lambda_{1,2}$ is nef but not big on $\overline{M}_{2,1}$.