The line bundle of the divisor at infinity of the moduli stack of stable curves of genus $g \ge 2$

Question

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 $m$?

Answer 1

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.