# 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 $\bar M_g\setminus M_g$?

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. Commented Nov 6, 2012 at 22:00

The irreducible components of the normalization $\mathcal{B}^{\nu}$ of the boundary $\mathcal{B}\subset\overline{\mathcal{M}}_{g,n}$ are finite images of the moduli spaces:

• $\overline{\mathcal{M}}_{g_{1},S_{1}\cup\{n_{1}+1\}}\times\overline{\mathcal{M}}_{g_{2},S_{2}\cup\{n_{2}+1\}}$, where $g_{1}+g_{2} = g$ and $S_{1},S_{2}$ is a partition of $\{1,...,n\}$,
• $\overline{\mathcal{M}}_{g-1,n+2}$,

Therefore any component of the boundary can be interpred as a moduli space of curves and you question can be rewritten in this way: For which $(g,n)$ is $\overline{\mathcal{M}}_{g,n}$ uniruled?

It is well known that $\overline{\mathcal{M}}_{g,n}$ is of general type for $g\geq 24, n\geq 0$. Then it is not uniruled. In lower genus, at the best of my knowledge:

• $\overline{\mathcal{M}}_{0,n}$ is rational for any $n\geq 3$.
• $\overline{\mathcal{M}}_{1,n}$ is rational for any $1\leq n\leq 10$, P. Belorousski, Chow rings of moduli spaces of pointed elliptic curves, Ph.D. thesis, Chicago, 1998.
• $\overline{\mathcal{M}}_{g,n}$ for $g = 2$ and $1\leq n\leq 12$, $g = 3$ and $1\leq n\leq 14$, $g = 4$ and $1\leq n\leq 15$, $g = 5$ and $1\leq n\leq 12$, http://arxiv.org/abs/math/0504249.
• $\overline{\mathcal{M}}_{g,n}$ is rational for $g=6$ and $1\leq n\leq 18$, and it is unirational for $g=8$ and $1\leq n\leq 11$, $g=10$ and $1\leq n\leq 3$, $g=12$ and $n=1$, http://arxiv.org/abs/math/0701475.
• In Section $7$ of A. Logan, The Kodaira dimension of moduli spaces of curves with marked points, Am. J. Math. 125 (2003), 105–138, the author determines for $g = 2, . . . , 9, 11$ an integer $f(g)$ such that $\overline{\mathcal{M}}_{g,n}$ is unirational for $n \leq f(g)$.
• $\overline{\mathcal{M}}_{g}$ is unirational for $g\leq 14$, $\overline{\mathcal{M}}_{15}$ is rationally connected, $\overline{\mathcal{M}}_{16}$ is uniruled, http://arxiv.org/abs/0805.2424.
• $\overline{\mathcal{M}}_{22}$ is of general type. The Kodaira dimension of $\overline{\mathcal{M}}_{g}$ for $17\leq g\leq 21$ and $g = 23$ is not know, http://arxiv.org/abs/0805.2424.
• $\overline{\mathcal{M}}_{g,n}$ is uniruled for $g = 12$ and $n \leq 5$, $g = 13$ and $n \leq 3$, $g = 15$ and $n \leq 2$, http://arxiv.org/abs/1206.1424.