## Rational curved lying in the boundary of Deligne-Mumford compactification $\bar M_g$

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.
I see now my mistake (that led me to the question). I was assuming (without any reason) that Satake compactification of $M_g$ is a smooth space. And since Deligne-Mumford is birational to Stake compactification I was thinking that boundary divisors in $\bar M_g$ should be ruled since they have larger dimension than in Satke. This sounded strange to me. But now I see that Satake is singular and so not surprisingly its blow up does not need to contain additional rational curves. Thank you for the answer again. – Dmitri Nov 6 at 22:00