Let $\bar M\_g$ be the Deligne-Mumford compactifiction of the moduli space of complex genus $g$ curves $M\_g$. Is this correct that through every point of the boundary $\bar M\_g\setminus M\_g$ passes a rational curve that lies in the boundary $\bar M\_g\setminus M\_g$?



Rational curved lying in the boundary of Deligne-Mumford compactification $\bar M_g$
It seems unlikely. Say $g \geq 24$ (or so). There's a divisor $\Delta\_g$ on the boundary $M\_{2g+1}$ corresponding to curves with a single node consisting of two genus $g$ curves glued at a point. This component is birational to $M\_{g,1} \times M\_{g,1}$, which is of general type by the assumption on $g$. Thus through a general point of $\Delta\_g$ there is no rational curve. For that matter there is no rational curve on $\bar{M}\_g$ through this point at all.


