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) ?**



Birational Contractions on Moduli of pointed Curves
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.


