Let $\overline{\mathscr{M}}_g$ be the $\mathbb{Z}$-algebraic stack of stable curves of genus $g \ge 2$, as constructed in the paper of Deligne and Mumford. The degeneracy locus of the universal stable curve is a closed substack $D \subset \overline{\mathscr{M}}_g$ that is a relative effective Cartier divisor over $\mathrm{Spec}(\mathbb{Z})$, so $D$ gives rise to a line bundle $\mathcal{L}$ on $\overline{\mathscr{M}}_g$. Is there a relation between $\mathcal{L}$ and the line bundle $\lambda$ defined as the determinant bundle of the pushforward of the relative dualizing sheaf of the universal stable curve? Specifically, is $\mathcal{L} \simeq \lambda^{\otimes m}$ for some $m$, and, if so, for which $m$?

  • $\begingroup$ I'm pretty sure that the answer is no, and you can probably check this by finding a compact curve in $\mathcal M_g$ and observing that $\lambda$ is nontrivial on it but $\mathcal L$ is trivial. $\endgroup$ – Will Sawin Aug 19 '15 at 3:24

The correct formula (written additively in terms of divisor classes) is $\kappa = 12 \lambda - \delta$ where $\delta$ is the divisor at infinity and $\kappa$ is the class $\pi_\ast c_1(\omega_{C/\overline M_g})^2$, the pushforward of the square of the first Chern class of the relative dualizing sheaf of the universal family. This identity goes back to Mumford. It suffices to prove this after tensoring by $\mathbf Q$ since the Picard group of $\overline M_g$ is torsion free, so you can do it by an application of Grothendieck-Riemann-Roch theorem.

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  • $\begingroup$ Thank you for the answer, now I have a clearer idea of what to expect. I have some questions about $\kappa$ though: how to make sense of it over an arbitrary (nonfield) base, as would be required to answer my question? Also, how does GRR apply given that we are working over $\mathbb{Z}$? (I thought GRR is for smooth (or lci) varieties over fields.) $\endgroup$ – O-Ren Ishii Aug 19 '15 at 13:04

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