On the Arakelov Geometry of Moduli Spaces of Curves

In this paper we consider some problems in the Arakelov geometry of Mg, the moduli space of smooth projective curves of genus g over C. Specifically, we are interested in naturally metrized line bundles over Mg and their extensions to Mg, the Deligne-Mumford compactification of Mg. These line bundles typically occur when computing the archimedean height of a curve. Here they arise in computations of the archimedean height of the algebraic cycle C −C in the jacobian of a genus g curve C (cf. [6]). As is well known, PicMg is of rank 1 when g ≥ 3. It is generated by the determinant bundle L := det π∗Ω 1 C/Mg , where C denotes the universal curve over Mg. It has a natural metric which is induced from the standard one on π∗Ω 1 C/Mg : ‖ω‖ = i 2 ∫