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 $\bar M_g\setminus M_g$?
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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. |
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