# Canonical bundle of the moduli space of curves

By the pointed canonical bundle formula the canonical bundle of $\overline{M}_{g,n}$ is given by $$K_{\overline{M}_{g,n}} = 13\lambda+\psi-2\delta-\sum_{I}\delta_{1,I}$$ where $\lambda$ is the Hodge class, $\psi = \sum_{i=1}^{n}\psi_{i}$ is the sum of the psi-classes and $\delta$ is total boundary divisor.

Are there constraints involving $g,n$ which imply that $K_{\overline{M}_{g,n}}$ is ample ?

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The cotangent bundle of $\overline M_{g,n}$ is never ample. You can see this for instance by restricting it to the hyperelliptic locus.

There has been a lot of work in the last 5-10 years on running the (log) minimal model program on the moduli space of curves. You can read about this to get a feeling for how far from true it is that $K_{\overline M_{g,n}}$ is ample. There's a survey of Fedorchuk-Smyth in the Handbook of Moduli, for instance.

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