Let \(\mathcal M_g\) denote the moduli space of smooth projective curves of genus \(g\geq3\) over \(\mathbb C\). It is well known that the Picard group \(\text{Pic}\mathcal M_g\) has rank 1, generated by the class of the determinant of the Hodge bundle \(\mathcal L:=\det\pi_{*}\Omega^1_{\mathcal C/\mathcal M_g}\), where \(\mathcal C\) denotes the universal curve over \(\mathcal M_g\).
This paper studies the biextension line bundle \(\mathcal B\), associated to the algebraic 1-cycle \(C-C^{-}\) in the Jacobian of a curve \(C\) of genus \(g\). This bundle is isomorphic to \(\mathcal L^{\otimes(8g+4)}\). The bundle \(\mathcal B\) also carries a natural metric, but the isomorphism \(\mathcal B\cong\mathcal L^{\otimes(8g+4)}\) is not an isometry. Let \(\beta_g:\mathcal M_g\to\mathbb R\) denote the logarithm of the ratio of the metrics; it is well defined up to an additive constant.
This paper gives asymptotic estimates of \(\beta_g\) near the boundary of \(\mathcal M_g\) in certain cases. Specifically, if \(X\to\mathbb D\) is a proper family of stable curves of genus \(g\geq3\) over the unit disk, such that \(X\) is smooth, \(X_t\) is smooth for all \(t\neq0\), and \(X_0\) has only one node, then asymptotics for \(\beta_g(X_t)\) are obtained as \(t\to0\).
These asymptotics differ from corresponding estimates for Faltings’ \(\delta\)-function obtained by J. Jorgensen, in: Proc. AMS-IMS-SIAM Jt. Summer Res. Conf. Schottky Probl., Amherst/AM (USA) 1990, Contemp. Math. 136, 255–281 (1992; Zbl 0767.30033)] and [R. Wentworth, Commun. Math. Phys. 137, 427–459 (1991; Zbl 0820.14017)]. Therefore, as a corollary, it follows that the exact 1-forms \(d\delta_g\) and \(d\beta_g\) on \(\mathcal M_g\) are linearly independent over \(\mathbb R\).
More generally, the authors show that \(\mathcal B\) extends naturally to a line bundle \(\overline{\mathcal B}\) on \(\overline{\mathcal M}_g\). The metric extends to a smooth metric over \(\overline{\mathcal M}_g\setminus\Delta_0\), and extends continuously to the restriction of \(\overline{\mathcal B}\) to any holomorphic arc meeting \(\Delta_0\) transversely. Also, the first Chern class of \(\overline{\mathcal B}\) in \(\text{Pic}(\overline{\mathcal M}_g)\) is computed explicitly.